text:
User talk:Tonius

Image:Game.com pocket pro.PNG
Hi! Thanks for uploading Image:Game.com pocket pro.PNG. I notice it currently doesn't have an image copyright tag. Could you add one to let us know its copyright status? (You can use if you release it under the GFDL, or  if you claim fair use, etc.) If you don't know what any of this means, just let me know where you got the images and I'll tag them for you. Thanks so much,   – Quadell (talk) (help)   16:52, Dec 8, 2004 (UTC)

Unverified images
Hi! Thanks for uploading the following images:

 * mage:Indrema.PNG
 * mage:Infogrames.JPG
 * mage:Infogrames.PNG

I notice it currently doesn't have an image copyright tag. Could you add one to let us know its copyright status? (You can use if you release it under the GNU Free Documentation License,  if you claim fair use, etc.) If you don't know what any of this means, just let me know at my talk page where you got the images and I'll tag them for you. Thanks so much. [[User:Poccil|Peter O. (Talk, automation script)]] 03:32, Dec 10, 2004 (UTC)

P.S. You can help tag other images at User:Yann/Untagged_Images. Thanks again.

Unverified images
Hi! Thanks for uploading the following images:

 * Image:Indrema.PNG
 * Image:Infogrames.JPG
 * Image:Infogrames.PNG

I notice it currently doesn't have an image copyright tag. Could you add one to let us know its copyright status? (You can use if you release it under the GNU Free Documentation License,  if you claim fair use, etc.) If you don't know what any of this means, just let me know at my talk page where you got the images and I'll tag them for you. Thanks so much. [[User:Poccil|Peter O. (Talk, automation script)]] 03:58, Dec 10, 2004 (UTC)

P.S. You can help tag other images at User:Yann/Untagged_Images. Thanks again.

Article Licensing
Hi, I've started a drive to get users to multi-license all of their contributions that they've made to either (1) all U.S. state, county, and city articles or (2) all articles, using the Creative Commons Attribution-Share Alike (CC-by-sa) v1.0 and v2.0 Licenses or into the public domain if they prefer. The CC-by-sa license is a true free documentation license that is similar to Wikipedia's license, the GFDL, but it allows other projects, such as WikiTravel, to use our articles. Since you are among the top 2000 Wikipedians by edits, I was wondering if you would be willing to multi-license all of your contributions or at minimum those on the geographic articles. Over 90% of people asked have agreed. For More Information:
 * Multi-Licensing FAQ - Lots of questions answered
 * Multi-Licensing Guide
 * Free the Rambot Articles Project

To allow us to track those users who muli-license their contributions, many users copy and paste the " " template into their user page, but there are other options at Template messages/User namespace. The following examples could also copied and pasted into your user page:

 * Option 1
 * I agree to multi-license all my contributions, with the exception of my user pages, as described below:

OR
 * Option 2
 * I agree to multi-license all my contributions to any U.S. state, county, or city article as described below:

Or if you wanted to place your work into the public domain, you could replace " " with "  ". If you only prefer using the GFDL, I would like to know that too. Please let me know what you think at my talk page. It's important to know either way so no one keeps asking. -- Ram-Man (comment| talk)

Image Tagging Image:Gb printer.PNG
Thanks for uploading Image:Gb printer.PNG. I notice the 'image' page currently doesn't specify who created the content, so the copyright status is therefore unclear. If you have not created this media yourself then you need to argue that we have the right to use the media on Wikipedia (see copyright tagging below). If you have not created the media yourself then you should also specify where you found it, i.e., in most cases link to the website where you got it, and the terms of use for content from that page.

If the media also doesn't have a copyright tag then you must also add one. If you created/took the picture, audio, or video then you can use GFDL to release it under the GFDL. If you believe the media qualifies as fair use, please read fair use, and then use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other media, please check that you have specified their source and copyright tagged them, too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any unsourced and untagged images will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. Thank you. Soothing R  09:08, 12 February 2006 (UTC)

Image Tagging for Image:Neo_geo_pocket_color.PNG
Thanks for uploading Image:Neo_geo_pocket_color.PNG. The image page currently doesn't specify who created the image, so the copyright status is therefore unclear. If you have not created the image yourself then you need to indicate why we have the right to use the image on Wikipedia (see copyright tagging below). If you have not created the image yourself then you should also specify where you found it, i.e., in most cases link to the website where you got it, and the terms of use for content from that page.

If the image also doesn't have a copyright tag then you must also add one. If you created/took the picture then you can use GFDL to release it under the GFDL. If you believe the image qualifies under Wikipedia's fair use guidelines, please read fair use, and then use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use. If you want the image to be deleted, tag it as db-unksource.

If you have uploaded other images, please check that you have specified their source and copyright tagged them, too. You can find a list of image pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any unsourced and untagged images will be deleted one week after they have been uploaded, as described on criteria for speedy deletion.

This is an automated notice by OrphanBot. If you have any concerns, contact the bot's owner: Carnildo. 06:17, 25 February 2006 (UTC)

Image tagging for Image:Wonderswan_color.jpg
Thanks for uploading Image:Wonderswan_color.jpg. The image has been identified as not specifying the source and creator of the image, which is required by Wikipedia's policy on images. If you don't indicate the source and creator of the image on the image's description page, it may be deleted some time in the next seven days. If you have uploaded other images, please verify that you have provided source information for them as well.

For more information on using images, see the following pages:
 * Image use policy
 * Image copyright tags

This is an automated notice by OrphanBot. For assistance on the image use policy, see Media copyright questions. 11:25, 20 July 2006 (UTC)

Deleting your Talk page
Hello! You can't speedy delete your Talk page. Please read [http://en.wikipedia.org/wiki/WP:USERPAGE#How_do_I_delete_my_user_and_user_talk_pages.3F] for the correct method of deletion. You shouldn't remove warning templates from your Talk page; please don't do this in future.  (aeropagitica)   (talk)  22:20, 30 July 2006 (UTC)

User:Tonius
Your userpage has been deleted, as requested.  (aeropagitica)   (talk)  22:22, 30 July 2006 (UTC)

Image:Q .PNG
Thanks for uploading Image:Q .PNG. I notice the 'image' page specifies that the image is being used under fair use, but its use in Wikipedia articles fails our first fair use criterion in that it illustrates a subject for which a freely licensed image could reasonably be found or created. If you believe this image is not replaceable, please:

 * 1) Go to the image description page and edit it to add, without deleting the original Replaceable fair use template.
 * 2) On the image discussion page, write the reason why this image is not replaceable at all.

Alternatively, you can also choose to replace the fair use image by finding a freely licensed image of its subject or by taking a picture of it yourself.

If you have uploaded other fair use media, consider checking that you have specified how these images fully satisfy our fair use criteria. You can find a list of 'image' pages you have edited by clicking on [ this link]. Note that any fair use images which are replaceable by free-licensed alternatives will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you. Nilfanion (talk) 18:45, 17 November 2006 (UTC)

Fair use rationale for Image:Bleem! 2.PNG
Thanks for uploading Image:Bleem! 2.PNG. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in Wikipedia articles constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale.

If you have uploaded other fair use media, consider checking that you have specified the fair use rationale on those pages too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 23:52, 17 May 2007 (UTC)

Unspecified source for Image:Bleem! 2.PNG
Thanks for uploading Image:Bleem! 2.PNG. I noticed that the file's description page currently doesn't specify who created the content, so the copyright status is unclear. If you did not create this file yourself, then you will need to specify the owner of the copyright. If you obtained it from a website, then a link to the website from which it was taken, together with a restatement of that website's terms of use of its content, is usually sufficient information. However, if the copyright holder is different from the website's publisher, then their copyright should also be acknowledged.

As well as adding the source, please add a proper copyright licensing tag if the file doesn't have one already. If you created/took the picture, audio, or video then the GFDL-self-no-disclaimers tag can be used to release it under the GFDL. If you believe the media meets the criteria at Fair use, use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other files, consider checking that you have specified their source and tagged them, too. You can find a list of files you have uploaded by following [ this link]. Unsourced and untagged images may be deleted one week after they have been tagged, as described on criteria for speedy deletion. If the image is copyrighted under a non-free license (per Fair use) then the image will be deleted 48 hours after 23:52, 17 May 2007 (UTC). If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 23:52, 17 May 2007 (UTC)

Fair use rationale for Image:Indrema.PNG
Thanks for uploading Image:Indrema.PNG. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in Wikipedia articles constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale.

If you have uploaded other fair use media, consider checking that you have specified the fair use rationale on those pages too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 23:57, 17 May 2007 (UTC)

Unspecified source for Image:Indrema.PNG
Thanks for uploading Image:Indrema.PNG. I noticed that the file's description page currently doesn't specify who created the content, so the copyright status is unclear. If you did not create this file yourself, then you will need to specify the owner of the copyright. If you obtained it from a website, then a link to the website from which it was taken, together with a restatement of that website's terms of use of its content, is usually sufficient information. However, if the copyright holder is different from the website's publisher, then their copyright should also be acknowledged.

As well as adding the source, please add a proper copyright licensing tag if the file doesn't have one already. If you created/took the picture, audio, or video then the GFDL-self-no-disclaimers tag can be used to release it under the GFDL. If you believe the media meets the criteria at Fair use, use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other files, consider checking that you have specified their source and tagged them, too. You can find a list of files you have uploaded by following [ this link]. Unsourced and untagged images may be deleted one week after they have been tagged, as described on criteria for speedy deletion. If the image is copyrighted under a non-free license (per Fair use) then the image will be deleted 48 hours after 23:57, 17 May 2007 (UTC). If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 23:57, 17 May 2007 (UTC)

Fair use rationale for Image:Pong.PNG
Thanks for uploading Image:Pong.PNG. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in Wikipedia articles constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale.

If you have uploaded other fair use media, consider checking that you have specified the fair use rationale on those pages too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 00:03, 18 May 2007 (UTC)

Unspecified source for Image:Pong.PNG
Thanks for uploading Image:Pong.PNG. I noticed that the file's description page currently doesn't specify who created the content, so the copyright status is unclear. If you did not create this file yourself, then you will need to specify the owner of the copyright. If you obtained it from a website, then a link to the website from which it was taken, together with a restatement of that website's terms of use of its content, is usually sufficient information. However, if the copyright holder is different from the website's publisher, then their copyright should also be acknowledged.

As well as adding the source, please add a proper copyright licensing tag if the file doesn't have one already. If you created/took the picture, audio, or video then the GFDL-self-no-disclaimers tag can be used to release it under the GFDL. If you believe the media meets the criteria at Fair use, use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other files, consider checking that you have specified their source and tagged them, too. You can find a list of files you have uploaded by following [ this link]. Unsourced and untagged images may be deleted one week after they have been tagged, as described on criteria for speedy deletion. If the image is copyrighted under a non-free license (per Fair use) then the image will be deleted 48 hours after 00:03, 18 May 2007 (UTC). If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 00:03, 18 May 2007 (UTC)

Fair use rationale for Image:Tiger 2.PNG
Thanks for uploading Image:Tiger 2.PNG. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in Wikipedia articles constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale.

If you have uploaded other fair use media, consider checking that you have specified the fair use rationale on those pages too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 00:06, 18 May 2007 (UTC)

Unspecified source for Image:Tiger 2.PNG
Thanks for uploading Image:Tiger 2.PNG. I noticed that the file's description page currently doesn't specify who created the content, so the copyright status is unclear. If you did not create this file yourself, then you will need to specify the owner of the copyright. If you obtained it from a website, then a link to the website from which it was taken, together with a restatement of that website's terms of use of its content, is usually sufficient information. However, if the copyright holder is different from the website's publisher, then their copyright should also be acknowledged.

As well as adding the source, please add a proper copyright licensing tag if the file doesn't have one already. If you created/took the picture, audio, or video then the GFDL-self-no-disclaimers tag can be used to release it under the GFDL. If you believe the media meets the criteria at Fair use, use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other files, consider checking that you have specified their source and tagged them, too. You can find a list of files you have uploaded by following [ this link]. Unsourced and untagged images may be deleted one week after they have been tagged, as described on criteria for speedy deletion. If the image is copyrighted under a non-free license (per Fair use) then the image will be deleted 48 hours after 00:06, 18 May 2007 (UTC). If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 00:06, 18 May 2007 (UTC)

Fair use rationale for Image:Virtual game station.PNG
Thanks for uploading Image:Virtual game station.PNG. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in Wikipedia articles constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale.

If you have uploaded other fair use media, consider checking that you have specified the fair use rationale on those pages too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 00:06, 18 May 2007 (UTC)

Unspecified source for Image:Virtual game station.PNG
Thanks for uploading Image:Virtual game station.PNG. I noticed that the file's description page currently doesn't specify who created the content, so the copyright status is unclear. If you did not create this file yourself, then you will need to specify the owner of the copyright. If you obtained it from a website, then a link to the website from which it was taken, together with a restatement of that website's terms of use of its content, is usually sufficient information. However, if the copyright holder is different from the website's publisher, then their copyright should also be acknowledged.

As well as adding the source, please add a proper copyright licensing tag if the file doesn't have one already. If you created/took the picture, audio, or video then the GFDL-self-no-disclaimers tag can be used to release it under the GFDL. If you believe the media meets the criteria at Fair use, use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other files, consider checking that you have specified their source and tagged them, too. You can find a list of files you have uploaded by following [ this link]. Unsourced and untagged images may be deleted one week after they have been tagged, as described on criteria for speedy deletion. If the image is copyrighted under a non-free license (per Fair use) then the image will be deleted 48 hours after 00:06, 18 May 2007 (UTC). If you have any questions please ask them at the Media copyright questions page. Thank you.Betacommand (talk • contribs • Bot) 00:06, 18 May 2007 (UTC)

Orphaned non-free image (Image:Pong.PNG)
Thanks for uploading Image:Pong.PNG. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

If you have uploaded other unlicensed media, please check whether they're used in any articles or not. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any non-free images not used in any articles will be deleted after seven days, as described on criteria for speedy deletion. Thank you. User:Gay Cdn (talk) (Contr) 20:39, 18 May 2007 (UTC)

Orphaned non-free image (Image:Bleem! 2.PNG)
Thanks for uploading Image:Bleem! 2.PNG. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

If you have uploaded other unlicensed media, please check whether they're used in any articles or not. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any non-free images not used in any articles will be deleted after seven days, as described on criteria for speedy deletion. Thank you. BetacommandBot 01:46, 20 May 2007 (UTC)

Orphaned non-free image (Image:Indrema.PNG)
Thanks for uploading Image:Indrema.PNG. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

If you have uploaded other unlicensed media, please check whether they're used in any articles or not. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any non-free images not used in any articles will be deleted after seven days, as described on criteria for speedy deletion. Thank you. BetacommandBot 01:48, 20 May 2007 (UTC)

Orphaned non-free image (Image:Tiger 2.PNG)
Thanks for uploading Image:Tiger 2.PNG. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

If you have uploaded other unlicensed media, please check whether they're used in any articles or not. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any non-free images not used in any articles will be deleted after seven days, as described on criteria for speedy deletion. Thank you. BetacommandBot 01:52, 20 May 2007 (UTC)

Thanks for uploading Image:Bleem! 2.PNG. However, there is a concern that the rationale you have provided for using this image under "fair use" may be invalid. Please read carefully the instructions at Non-free content and then go to the image description page and clarify why you think the image qualifies.

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Disputed fair use rationale for Image:Indrema.PNG
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If it is determined that the image does not qualify under fair use, it will be deleted within a couple of days according to our criteria for speedy deletion. If you have any questions please ask them at the media copyright questions page. Thank you.BetacommandBot (talk) 23:38, 13 February 2008 (UTC)

ArbCom elections are now open!
MediaWiki message delivery (talk) 08:53, 23 November 2015 (UTC)

Game Boy Camera & Printer listed at Redirects for discussion
An editor has asked for a discussion to address the redirect Game Boy Camera &. Since you had some involvement with the Game Boy Camera & Printer redirect, you might want to participate in the redirect discussion if you have not already done so. Steel1943 (talk) 20:19, 15 February 2018 (UTC)

Nomination of Game Boy Advance SP for deletion
A discussion is taking place as to whether the article Game Boy Advance SP is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The article will be discussed at Articles for deletion/Game Boy Advance SP until a consensus is reached, and anyone, including you, is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the article during the discussion, including to improve the article to address concerns raised in the discussion. However, do not remove the article-for-deletion notice from the top of the article until the discussion has finished. Grandmaster Huon (talk) 03:26, 29 July 2023 (UTC)

title: User talk:Tonius


text:
Talk:Limit (music)

Prime vs whole
Hmm, I'm not sure about this - isn't it the highest prime number (or at least the highest odd number), rather than the highest whole number? So in 3-limit, for example, a 9:8 is fine (or at the very least, a 4:3 is fine). --Camembert

You're absolutely right, but now I have to figure out why.--Hyacinth Found out!
 * I think highest prime divisor doesn't make sense. With high enough powers you can do almost anything.  For example, 36:29 and 210:36 are both pretty close to a tritone.  To get completely ridiculous, 2243:3153 is even closer. Phr 23:56, 15 February 2006 (UTC)

 * Yes, 729/512 and 1024/729 are both 3-limit (aka Pythagorean) tritones. What's your point? —Keenan Pepper 04:33, 16 February 2006 (UTC)
 * The point is that there is no such thing as an interval that can't be very closely approximated with high enough powers of 2 and 3. So describing an interval as 3-limit is meaningless unless the exponent is bounded somehow, preferably to low numbers.  Am I missing something? Phr 23:42, 17 February 2006 (UTC)

 * Exactly. A rational number has a definite prime limit, but the ear cannot discern intervals if they are too complex. But the important thing is that any 3-limit interval can be precisely tuned by using a succesion of perfect fifths and octaves (for example 729/512 occurs in the Pythagorean diatonic scale), whereas an interval of any higher limit cannot. —Keenan Pepper 00:12, 18 February 2006 (UTC)
 * I'm taking the view that just intonation is only musically relevant because the ear can distinguish between, say, 1.5:1 (perfect fifth) and 1.4983:1 (equal tempered fifth). But if the ratios are sufficiently close, say within .0001 of each other, the ear cannot distinguish them and therefore they are for musical purposes exactly the same.  So since every interval is has a 3-limit ratio within .0001 of it, for musical purposes there is no such thing as a non-Pythagorean interval and the mathematical distinction between 3-limit and non-3-limit intervals is meaningful for theoretical purposes only.  Is that a fair description?  (If that's the intended meaning, maybe the article should say so).  Phr 03:06, 18 February 2006 (UTC)

 * From a mathematical point of view, it is fair to say that any tone can be accurately expressed as an infinite series of pythagorean tunings. However, it is not possible in tuning practice to go through such a series. To tune a pythagorean (3-limit) major third, for instance, you need to tune first 3:2, then 9:8 based on that 3:2, now 27:16, now 81:64. Thus it takes 5 tunings to get to that major third, each of which takes time and care. You cannot simply jump to the ratio 81:64, because it is not -audibly- tuneable. Can you understand how impractical it would be to have to tune everything from series' of fifths? - Rainwarrior 23:10, 7 May 2006 (UTC)

Rewrite by Namrevlis
I see some improvements, but I don't see any mention of odd limits or the difference between odd and prime limits, which to me is a crucial omission. —Keenan Pepper 13:40, 15 May 2006 (UTC)

Please, feel free to add information about odd limits. I rewrote the article in a broad attempt to replace much of the arithmetic with descriptive musical information.Namrevlis 02:37, 16 May 2006 (UTC)

Set notation

 * The following was originally written as a reply to the above. Headline inserted because the previous is one year old; the article has been substantially rewritten since. — Sebastian 18:28, 9 May 2007 (UTC)

Hope you'll be patient to my English... It is possible rewrite so: In just intonation, any given interval can be expressed as the ratio between two frequencies, type 4/3 for the perfect fourth or 10/9 for the minor tone. If these ratios to factorize, the limits for such intervals are defined as follows: The odd limit selects a set of the notes of one name in all octaves and is the biggest multiplication of the all odd prime members of the factorization with same exponent sign and exponent modulus taken. If to suppose frequency of the note C as 1, then C:1 = {C2:2-2, C1:2-1, C:20, c:21, c1:22, …, c5:26} in fact is a set of the notes С in all octaves, and each note of this set has the odd limit 1. The notes F:4/3 = C:1×3-1×22 = {F2:3-1×20, F1:3-1×21, F:3-1×22, f:3-1×:23, f1:3-1×24, …, f5:3-1×28} have the odd limit 3 = 1×3undefined; the notes D-:10/9 = C:1×51×3-2×21 = {D2-:51×3-2×2-1, D1-:51×3-2×20, D-:51×3-2×21, d-:51×3-2×22, d1-:51×3-2×23, …, d5-:51×3-2×27} have the odd limit 9 = 1×3undefined, because 1×3undefined > 1×5undefined. The prime limit is a largest prime in the factorization. The prime limit of the perfect fourth is 3 (the same as the odd limit), because in the factorization 3-1×22 the largest prime is 3, but the minor tone has a prime limit of 5, because in the factorization 51×3-2×21 the largest prime is 5. Commator 18:18, 7 May 2007 (UTC)

 * In principle, it is possible to write a set notation. However, I don't see how it would improve the article. If you want to be mathematically rigorous, it seems that modular arithmetic would be a more promising and elegant approach. The sets require a lot of terms for relatively little added value. Moreover, to be rigorous you would have to use infinite sets. — Sebastian 18:28, 9 May 2007 (UTC)

Psychoacoustic?
How is odd-limit tuning supported by modern psychoacoustics? I checked the reference, and the graph seems to support odd-limit tuning. But how did they arrive at the graph? Is it a theoretical model, or some kind of experiment? tedneeman (talk) 04:28, 10 October 2009 (UTC)

Failed verification: Partch's Genesis
While I don't think it's necessary for the exact word "complexity" to be found, a similar word should do, I have also failed to verify the reference by finding a description of "odd limit" or any description of "limit". Hyacinth (talk) 07:16, 29 May 2010 (UTC)
 * Yes, I agree with all of this. Partch uses the word "limit" frequently throughout Genesis without ever actually defining the term. It leaves the impression that he is using a concept he got from somewhere else, and assumes widespread familiarity on the part of his readers. However, it is still possible that Partch originated the term in one of his earlier writings, and is simply assuming that, by 1949 (the year of the original edition of Genesis), anyone reading his new book would be familiar with his previous work. It is possible to infer a great deal about this from Genesis (for example, all of the index entries for "limit" have an odd number appended: "limit of 3", "limit of 5", etc., and on page 88, Partch takes care to explain in a footnote that he means odd numbers rather than prime ones, because "the multiple 9 also assists in the delineation of tonality"). However, it is still true that he never plainly sets forth this principal in Genesis, so we must look elsewhere for it—whether in Partch's own writings, or in those of other authors. As to "complexity", the problem may be compounded by the associated references to Bart Hopkin and Paul Erlich, one or both of whom may equate Partch's scale of consonance/dissonance with simple/complex. Erlich should be easy to check, since his essay appears to be available online; Hopkin may take a little longer.—Jerome Kohl (talk) 19:06, 29 May 2010 (UTC)

While not necessarily intended in as precise a technical sense, there seems to be some precedent from William Smythe Babcock Mathews 1881: "The ninth is the genetic limit of harmony". —David Eppstein (talk) 19:39, 29 May 2010 (UTC)
 * Thanks for the pointer, which at least indicates caution needs to be exercised with respect to the origins of the term "limit" in this sense. In the meantime, I have, I think, straightened out the reference to "complex" and its equating to higher-limit systems. Unfortunately, in the process I also uncovered another failed claim verification.—Jerome Kohl (talk) 20:00, 29 May 2010 (UTC)

"Just major tenth" is linked to a wrong file
I cannot supply a this file myslef, so I leave it to the original uploader. —Preceding unsigned comment added by 90.177.201.163 (talk) 17:39, 20 February 2011 (UTC)

Etymology of odd limit and prime limit, clarification
I searched through Genesis of a Music trying to find the official definition of "odd limit" and I can't find the term used anywhere. Was this invented by someone other than Partch? The references would seem to need work. Can we find a reference for the first usage of these terms? Is anything found in Genesis of a Music then a reference to odd limit and not prime limit?

If it's modulo octaves, then the interval 10:3 would be in the 5 odd-limit and 5 prime-limit? If not for the modulo octaves, this would be in the 3 odd-limit?

11:7 is 11 odd-limit and 11 prime-limit? 15:7 is 15 odd-limit and 7 prime-limit? 12:1 is in the 3 odd-limit and 3 prime-limit? 96.224.67.227 (talk) 18:30, 4 September 2014 (UTC)

 * Odd limit is the greatest odd factor, so octaves do not make a difference in either case. The odd limit of 10:3 is 5, because it is 2·5:3. Basically, if either side of the ratio is even, divide it by 2 until you find an odd number, and then compare it to the other.

 * I could have sworn that Partch mentioned both but only really used one. It's been a while since I've read GoaM. — Gwalla | Talk

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Tetrads
Linking the appearance of tetrads (7th chords) as the main building block of harmony to jazz rather than to impressionist music seems rather surprising. For instance Debussy’s very popular first Arabesque was composed in 1890. Satie’s Gymopédies date 1888. JX Bardant (talk) 12:27, 9 December 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Limit (music)


text:
1971 in Afghanistan

The following details notable events from the year 1971 in Afghanistan. The Afghan Islamic Republic, is a landlocked country located in the centre of Asia, forming part of Central Asia, South Asia, and the Greater Middle East.

Political life in the capital is dominated by squabbles between the administration and the People's Council. The 1969 elections had returned assembly members who were for the most part quite unfamiliar with parliamentary methods and procedure, at least as laid down in the constitution. The final brush comes over a widely supported demand that questions concerning the administration be dealt with by the minister concerned at the time they are asked. This may not seem serious, but it follows a series of deliberate refusals to pass bills that the government regarded as essential. These refusals were based less on any difference over principles than on the rivalry between groups headed by individual parliamentarians, who, in the absence of political parties, formed cabals linked by a determination to promote local interests. To secure the passage of essential legislation, the king is obliged to intervene, with a threat to dissolve parliament.

Incumbents

 * Monarch - Mohammed Zahir Shah
 * Prime Minister - Mohammad Nur Ahmad Etemadi (until 9 June), Abdul Zahir (starting 9 June)

May 17, 1971
Prime Minister Nur Ahmad Etemadi, after being continually frustrated in his efforts to modernize the administration and bring the country forward, resigns. He is persuaded to continue as head of the government until his successor, Abdul Zahir, who was ambassador in Rome, is confirmed as prime minister on June 9. Zahir makes earnest efforts to reach a good understanding with parliament; and when he presents his list of ministers to the king in July, he is able to put forward the general desire that the administration concentrate attention on the difficulties of low-income groups. The king gives an assurance that the wishes of the legislators will be respected.

August 1971
The government takes the unprecedented step of launching a worldwide appeal for food after the most serious drought in the country's history. The economic life of the country is severely affected; it is feared that almost three-quarters of the nation's sheep, the main meat staple, might have perished. Large numbers of people migrated in search of food and water. The response, especially from Pakistan and Iran, is generous. The government undertakes a massive campaign of relief operations to deal with the emergency, but is hampered by the traditionally independent attitude of remote outlying areas.

September 1971
Afghanistan's ties with the Muslim world are strengthened by participation in the Islamic conference of foreign ministers that meets in Kabul.

title: 1971 in Afghanistan


text:
Talk:List of Hungarians

[Untitled]
I alphabetized the list, but it is clearly not the best way to organize it. There are conflicts between Hungarian and English alphabetical order as well as name changes (e.g. Harry Houdini et al).

Categorizing could also be problematic as some people would cross over, although I think that this would be a less serious problem.

Any thoughts? -- Jay Allen

How can alphabetization on last name be problematic? I am a bit dismayed that someone took information from our Website and didn't even mention it. Anyway, here it is [The Hungary Page, Nobel Prize Winners and Famous Hungarians] -- [[hipcat@hungary.org]

Maybe you should change to a notation like this:
 * Surename, Given Name

It will be closer to the Hungarian name order too...

-- Szilvási, Ádám (talk) 22:21, 1 June 2010 (UTC)

Here is the current list categorized for discussion...

Math and Sciences

 * Donát Bánki, engineer.
 * Ottó Bláthy, engineer.
 * Janos Bolyai, mathematician, dicoverer of non-Euclidean geometry.
 * Miksa Déri, engineer.
 * Paul Erdős , mathematician.
 * Dennis Gabor (Gábor Dénes), Nobel-prize winner physicist, inventor of holography.
 * István Gyarmati, physicist
 * Kálmán Kandó, engineer.
 * Tódor Kármán, engineer.
 * John von Neumann (Neumann János), mathematician.
 * Ernö Rubik, inventor of Rubik's cube
 * Charles Simonyi, computer scientist; father of Hungarian notation
 * Albert Szent-Györgyi, chemist, Nobel Prize winner in 1937.
 * Leo Szilard, physicist.
 * Edward Teller, physicist.
 * Eugene Wigner, physicist, mathematician.
 * Károly Zipernowsky, engineer.

Arts, Entertainment and Literature

 * Béla Bartók, composer.
 * William Fox, founder of Fox studios
 * Eva Gabor, actress
 * Magda Gabor, actress
 * Zsa Zsa Gabor, actress
 * Harry Houdini, magician
 * Imre Kertész, writer, Nobel Prize winner in 2002.
 * Zoltán Kodály, composer.
 * Lászlo Kovács, cinematographer and multiple Oscar award winner
 * György Ligeti, composer.
 * Ferenc Liszt, composer (known to most of the world as Franz Liszt).
 * Peter Lorre, actor
 * Bela Lugosi, actor
 * '''Sebestyen Marta, singer
 * George Pál, producer, inventor of stop-motion animation and multiple Oscar award winner
 * Joseph Pulitzer, publisher and namesake of Pulitzer Prize
 * Elie Wiesel, author and Nobel Peace Prize Winner
 * Vilmos Zsigmond, cinematographer and multiple Oscar award winner
 * Adolph Zukor, founder of Paramount Pictures; produced first full-length motion picture

Sports

 * Bela Karoly, premier gymnastics coach
 * Ferenc Puskas, football player, center forward.

History and Politics

 * Count Miklós Zrínyi, (1508-1566), Hungarian militant and hero defended against Ottoman Turks.
 * Count Miklos Zrinyi, (1620-1664), Hungarian warrior, statesman and poet.

Business and Finance

 * Andy Grove, former President/CEO and Chairman of Intel
 * George Soros, entrepreneur and one of the world's wealthiest men

-- Jay Allen

Since it will probably take months to get a comment on the above, I just went ahead and did it...

 * It looks good to me. I'd say use the most common name in English for alphabetical purposes, and if somebody falls into more than one category, I don't think there's anything wrong with listing them in more than one. --Camembert

Ha! Guess I was wrong. You're fast, Camembert! Anyway, I listed everyone in the most appropriate (i.e. what they are most famous for) category and used their most famous name (which is subjective, I know...). I also put birth names (when available) as a notation.

The funny thing about a list like this is where do you stop? There are a lot of Hungarian physicists and mathmaticians who are famous to mathmaticians and physicists (and Hungarians), but to virtually no one else... -- Jay Allen

 * Yes, that is a bit of a poser. A few months ago there was s a very long and boring argument about it at Talk:List of famous Canadians (I don't recommend reading it), at the end of which I don't think anybody agreed on anything. I'd say just list whoever you like - if they're famous enough to have a Wikipedia article, they're probably famous enough to be listed here. But that's just me, and others might disagree (by the way, if you type three tildes in a row, thus:, you should get an automatic link to your user page). --Camembert

Who qualifies here? I put in Catherine Schell, and now I think I'm going to take her out again. She was born in Hungary to a German family, left Hungary at the age of 4, and became known as an actress in Britain and the U.S.  -- Dominus 11:05, 23 Sep 2003 (UTC)

 * I'd say people who already have a Wikipedia article about them, and
 * consider themselves as Hungarians, or
 * born as Hungarians and have the ability to understand Hungarian, or
 * born as Hungarians and have lived there for at least 15 years.
 * Why? Just because.
 * Oh, by the way I don't know who Catherine Schell is, but she doesn't have an article anyway. --grin 17:05, 24 Sep 2003 (UTC)

Shouldn't Polya (the mathematician) be here also?

List needs serious work
There are two problems with this list:

1) it is inaccurate (e.g Alanis Morrisette is listed as American rather than Canadian) 2) it is clearly not just a list of Hungarians, but includes numerous people born and raised in another country with their main connection to Hungary being a single Hungarian parent. For example, to claim that Alanis Morrisette or Jamie Lee Curtis are "Hungarian" is rather ridiculous. You may as well claim that John Kerry is Hungarian (he had a Hungarian grandparent).

I propose, at the very least, separating the list into Hungarians (this can be easily defined for those still living e.g possessing or having at some point in their lives possessed Hungarian citizenship) and "people with (some degree of) Hungarian ancestry". Scott Moore 19:48, 26 Jan 2005 (UTC)

Backing up Hungarian descent
I looked through and checked the list, and I found that there is no mentioning to the ethnic Hungarian descent of the following people, or it is not enough to categorize them into some specific category. The articles of the below people should be backed up or made more obvious from this point of view, or they should be removed from the list:

 * Tony Curtis
 * William Fox (producer)
 * Ernie Kovacs
 * George Pataki
 * Theodor Herzl

Adam78 23:42, 29 Apr 2005 (UTC)

As for Tony Curtis, he was a child of a Hungarian immigrant (according to various sources, e.g. IMDB - Biography from Leonard Martin's Movie Encyclopedia) born in Bronx, New York.

German Wikipedia mentions that Theodor Herzl was born in Budapest to Jewish parents, who oriented their son towards German culture (incl. language), so it might be reasonable to remove him from there, and put him on the list of Austrians (& Austrian Jews) instead.

Listed? lásd...

 * List of Prime Ministers of Hungary
 * György Sándor(ja)
 * Hans Swarovsky
 * Victor L. Berger
 * Leslie Mahaffy
 * Homolka people
 * Géza Maróczy
 * Victor Ambrus

Category:Hungarian people

And see also my de:Liste bedeutender Ungarn!
 * --Sheynhertzגעשׁ״ך 17:50, 28 September 2005 (UTC)

also
 * Wilhelm Georg Berger, etc saxons
 * Richard Hönigswald
 * Iván T. Berend, scholar
 * János Harmatta

?
 * Gregory Nagy
 * Blaise Nagy

シャーンディ・ガーボルさんから
I was surprised to see myself in this list - I hardly see myself as someone famous or worth mentioning, I wonder how my name got in here in the first place.

But being new among Wikipedia editors, I wonder if it's appropriate to remove myself - I am hardly an unbiased observer.

Any comments?
 * by (Sándi Gábor)

See Wikipédia:Mi nem való a Wikipédiába or (What Wikipedia is not in English) to see what belongs to Wikipedia and what doesn't. (To be frank, I don't know anything about you, so I can't decide whether your article belongs to Wikipedia.) Adam78 21:06, 7 October 2005 (UTC)

Köszi, elolvastam. Azután kivettem magamat - azért Széchenyi Istvánnal mégsem szerepelhetek egy listân!

(Thank you. I read what you advised me to read, then took myself off the list. It's really not appropriate for me to figure on the same list as István Széchenyi)

 * Actually the article is titled "List of Hungarians", not "List of famous Hungarians". I might as well add my name to it. :-) Alensha 10:07, 8 October 2005 (UTC)

Writers

 * Andras Petocz
 * Gyula Illyes
 * Istvan Fekete
 * Janos Kodolanyi
 * Kalman Toth
 * Deszo Kosztolanyi
 * Lorinc Szabo
 * Klaman Mikszath
 * Kalman Thaly

FIlm Artists

 * Imre Gyongossy
 * Gabor Brody
 * Gyorgy Illes
 * Geza Radvanyi
 * Marton Keleti
 * Zoltan Fabri

Scientists and Engineers

 * Oszkar Asboth
 * Gabor Bernath
 * Imre Brody
 * Janos Csonka
 * Jeno Fejes
 * Albert Fono
 * József Galamb
 * Istvan Vedres
 * Denes Mihaly
 * Jozsef Mihalyi
 * Odon Riszdorfer
 * Pal Dienes
 * Janos Kabay
 * Mihaly Csikzsentmihalyi
 * Jozsef Dallos
 * Lajos Loczy
 * Eszter Pecsi
 * Gabor A. Somorjai American chemist. Winner of the 2008 Priestly Medal, the highest award of the American Chemical Society.  —Preceding unsigned comment added by 136.152.154.201 (talk) 23:12, 20 April 2011 (UTC)
 * Mihaly Somogyi
 * Antal Bejczy
 * Imre Gyula Izsak
 * Madeleine Forro Barnothy
 * Imre Frivaldzsky
 * Sandor Gaal
 * Gyula Fenyi
 * Istvan Dekany
 * Oscar Jaszi
 * Sandor Szalai
 * Balazs Fekete
 * Charles Vörösmarty

Composers
 * György Kosa

Artists
 * Jozsef Koszta
 * Simon Hollosy
 * Oszkar Glatz
 * Bela Kondor
 * Alajoz Hauszmann, architect
 * Gyula Derkovits
 * Aurel Bernath
 * Istvan Szonyi
 * Bela Czobel
 * Istvan Nagy
 * Ede Bohacsek
 * Istvan Zador
 * Istvan Szonyi
 * Jozsef Egry
 * Janos Nagy Balogh
 * Adolf Fenyes
 * Lili Orszag
 * Menyhert Toth
 * Sandor Bortnyik
 * Laszlo Mednyanszky
 * Jozsef Rippl-Ronai
 * Tamas Lossonczy
 * Lajos Gulascy
 * Karoly Ferenczy
 * Istvan Reti
 * Janos Thorma
 * Dezso Korniss
 * Vilmos Aba-Novak
 * Miklos Farkashazy
 * Mihaly Zichy
 * VIlmos Huszar
 * Bertalan Szekely
 * Istvan Csok
 * Gyula Benczur
 * Jeno Barcsay
 * Janos Kmetty
 * Gyula Czimra
 * Jeno Gyarfas
 * Lajos Gulacsy

This page needs consistency regarding whether to write names in the Hungarian or English style
This page needs consistency regarding whether to write names in the Hungarian (Eotvos Lorand) or English (Lorand Eotvos) style. In my opinion either is acceptable, but the style must be applied consistently. The first section favours the Hungarian style, and then subsequent sections mix and match styles. —Preceding unsigned comment added by Dc197 (talk • contribs) 14:47, 11 January 2008 (UTC)

Merge
I propose that List of Hungarians who were born outside present-day Hungary be merged into List of Hungarians. Firstly, according to the List of Hungarians who were born outside present-day Hungary article, "The borders of Hungary have changed substantially in the past century." The borders of Poland/Russia (Russian Empire, Soviet Russia, Soviet Union, Russian Federation) etc. also changed "substantially in the past century", but there is no a separate article, named "List of Polish/Russian people who were born outside present-day [borders of] Poland/Russia." Secondly, the listed people mainly were born in the Hungarian part of the Habsburg Monarchy or Austria-Hungary, so it is a bit anachronistic that Dezső Kosztolányi was born in Serbia, beacuase he was born in Austria-Hungary. I think we should merge them.--Rovibroni (talk) 13:43, 2 May 2014 (UTC)
 * Support. It does make sense for me. Fakirbakir (talk) 22:07, 2 May 2014 (UTC)
 * Support. Agreed that this article seems to be unique for Wikipedia and doesn't warrant a stand-alone list (particularly in light of confusing borders, empires et al). --Iryna Harpy (talk) 23:25, 2 May 2014 (UTC)

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Cube root

In mathematics, a cube root of a number $x$ is a number $y$ such that $y^{3} = x$. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of $8$, denoted $$\sqrt[3]8$$, is $2$, because $2^{3} = 8$, while the other cube roots of $8$ are $$-1+i\sqrt 3$$ and $$-1-i\sqrt 3$$. The three cube roots of $−27i$ are:
 * $$3i, \quad \frac{3\sqrt{3}}{2}-\frac{3}{2}i, \quad \text{and} \quad -\frac{3\sqrt{3}}{2}-\frac{3}{2}i.  $$

In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign $$\sqrt[3]{~^~}.$$ The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always $$\left(\sqrt[3]x\right)^3 =x,$$ the cube of a nonzero number has more than one complex cube root and its principal cube root may not be the number that was cubed. For example, $$(-1+i\sqrt 3)^3=8$$, but $$\sqrt[3]8=2.$$

Formal definition
The cube roots of a number x are the numbers y which satisfy the equation

 * $$y^3 = x.\ $$

Real numbers
For any real number x, there is one real number y such that y3 = x. The cube function is increasing, so does not give the same result for two different inputs, and it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number.

If x and y are allowed to be complex, then there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are:

 * $$ 1, \quad -\frac{1}{2}+\frac{\sqrt{3}}{2}i, \quad -\frac{1}{2}-\frac{\sqrt{3}}{2}i. $$

The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.

Complex numbers

For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula

 * $$x^{\frac13} = \exp \left( \frac13 \ln{x} \right).$$

If we write x as

 * $$x = r \exp(i \theta)\,$$

where r is a non-negative real number and θ lies in the range

 * $$-\pi < \theta \le \pi$$,

then the principal complex cube root is

 * $$\sqrt[3]{x} = \sqrt[3]{r}\exp \left(\frac {i\theta}{3} \right).$$

This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance $\sqrt{2|3}$ will not be −2, but rather 1 + i√3.

This difficulty can also be solved by considering the cube root as a multivalued function: if we write the original complex number x in three equivalent forms, namely

 * $$x = \begin{cases} r \exp (i \theta ), \\[3px] r \exp (i \theta + 2i\pi ), \\[3px] r \exp ( i \theta - 2i\pi ). \end{cases} $$

The principal complex cube roots of these three forms are then respectively

 * $$\sqrt[3]{x} = \begin{cases}

\sqrt[3]{r}\exp \left( \frac{i\theta}{3}\right), \\ \sqrt[3]{r}\exp \left(\frac{i\theta}{3} + \frac{2i \pi}{3} \right), \\ \sqrt[3]{r}\exp \left(\frac{i\theta}{3} - \frac{2i \pi}{3} \right). \end{cases} $$

Unless x = 0, these three complex numbers are distinct, even though the three representations of x were equivalent. For example, $\sqrt{−8|3}$ may then be calculated to be −2, 1 + i√3, or 1 − i√3.

This is related with the concept of monodromy: if one follows by continuity the function cube root along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by $$e^{2i\pi/3}.$$

Impossibility of compass-and-straightedge construction
Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837 Pierre Wantzel proved that neither of these can be done with a compass-and-straightedge construction.

Numerical methods
Newton's method is an iterative method that can be used to calculate the cube root. For real floating-point numbers this method reduces to the following iterative algorithm to produce successively better approximations of the cube root of a:

 * $$x_{n+1} = \frac{1}{3} \left(\frac{a}{x_n^2} + 2x_n\right).$$

The method is simply averaging three factors chosen such that
 * $$ x_n \times x_n \times \frac{a}{x_n^2}=a $$

at each iteration.

Halley's method improves upon this with an algorithm that converges more quickly with each iteration, albeit with more work per iteration:

 * $$x_{n+1} = x_n \left(\frac{x_n^3 + 2a}{2x_n^3 + a}\right).$$

This converges cubically, so two iterations do as much work as three iterations of Newton's method. Each iteration of Newton's method costs two multiplications, one addition and one division, assuming that $3}$ is precomputed, so three iterations plus the precomputation require seven multiplications, three additions, and three divisions.

Each iteration of Halley's method requires three multiplications, three additions, and one division, so two iterations cost six multiplications, six additions, and two divisions. Thus, Halley's method has the potential to be faster if one division is more expensive than three additions.

With either method a poor initial approximation of $x = 0$ can give very poor algorithm performance, and coming up with a good initial approximation is somewhat of a black art. Some implementations manipulate the exponent bits of the floating-point number; i.e. they arrive at an initial approximation by dividing the exponent by 3.

Also useful is this generalized continued fraction, based on the nth root method:

If x is a good first approximation to the cube root of a and y = a − x3, then:

 * $$\sqrt[3]{a} = \sqrt[3]{x^3+y} = x+\cfrac{y} {3x^2+\cfrac{2y} {2x+\cfrac{4y} {9x^2+\cfrac{5y} {2x+\cfrac{7y} {15x^2+\cfrac{8y} {2x+\ddots}}}}}}$$
 * $$= x+\cfrac{2x \cdot y} {3(2x^3+y)-y-\cfrac{2\cdot 4y^2} {9(2x^3+y)-\cfrac{5\cdot 7y^2} {15(2x^3+y)-\cfrac{8\cdot 10y^2} {21(2x^3+y)-\ddots}}}}.$$

The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence.

Appearance in solutions of third and fourth degree equations
Cubic equations, which are polynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a rational number). If two of the solutions are complex numbers, then all three solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number.

Quartic equations can also be solved in terms of cube roots and square roots.

History
The calculation of cube roots can be traced back to Babylonian mathematicians from as early as 1800 BCE. In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube; this required the construction, now known to be impossible, of the length $\sqrt{−8|3}$.

A method for extracting cube roots appears in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the second century BCE and commented on by Liu Hui in the third century CE. The Greek mathematician Hero of Alexandria devised a method for calculating cube roots in the first century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes. In 499 CE Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave a method for finding the cube root of numbers having many digits in the Aryabhatiya (section 2.5).

title: Cube root


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Princess Caroline of Monaco

Princess Caroline Louise Marguerite of Monaco (born 23 January 1957) is Princess of Hanover by marriage to Prince Ernst August. As the eldest child of Rainier III, Prince of Monaco, and Grace Kelly, she is the elder sister of Albert II, Prince of Monaco, and Princess Stéphanie.

She was Hereditary Princess of Monaco and heir presumptive to the Monegasque throne from her birth in 1957 until her brother Albert was born the following year, and again from Albert's accession in 2005 until the birth of his twins, her niece Gabriella and nephew Jacques, in 2014.

Family and early life
Caroline was born on 23 January 1957 in the Prince's Palace, Monaco. She is the eldest child of Rainier III, Prince of Monaco, and his wife, former American actress Grace Kelly. Christened Caroline Louise Marguerite, she belongs to the House of Grimaldi. She was the heiress presumptive from her birth to 14 March 1958, when her brother Prince Albert was born. On 1 February 1965, her younger sister Princess Stéphanie was born. Caroline is a legitimate patrilineal descendant of the Dukes of Polignac, and as such belongs to the historical French nobility. Through her mother, she is of Irish and German descent.

In an interview for People in April 1982, shortly before her death, Grace described Caroline and Stéphanie as "warm, bright, amusing, intelligent and capable girls. They're very much in tune with their era. Besides being good students, they are good athletes – excellent skiers and swimmers. Both can cook and sew and play the piano and ride a horse. But, above all, my children are good sports, conscious of their position and considerate of others. They are sympathetic to the problems and concerns in the world today."

As a child, Caroline spent time at the home of her maternal grandparents, John B. Kelly, Sr. and Margaret Kelly (née, Majer), in Philadelphia. In addition to visiting her mother's family in the United States, she spent the summer of 1971 at Camp Oneka in the Poconos at the age of 14. While there, unbeknownst to her parents, Caroline was protected by the United States Secret Service.

Princess Grace died on 14 September 1982, the day after suffering a stroke while driving herself and Princess Stéphanie home to Monaco from a visit to France, resulting in an accident in which both were injured.

Education
The princess received her French baccalauréat in 1974 with honours. She was also educated at St Mary's School Ascot. After a semester at Sciences Po, Caroline continued her studies at the Sorbonne University, where she received a diploma in philosophy and minors in psychology and biology. She is fluent in French, English, Spanish, German and Italian.

Activities

In 1979, Princess Caroline was appointed by her parents as the president of the Monégasque Committee for the International Year of the Child. Two years later, in 1981, Caroline founded Jeune J'écoute association. The association set up a 'youth hotline' where young people can talk about their problems on the telephone with qualified people trained in dealing problems young people faced. Other philanthropic organizations Caroline has been involved with include the World Association of Children's Friends (AMADE Mondiale), the Princess Grace Foundation, the Prince Pierre Foundation, and UNICEF.

Caroline is the Patron of Peter Le Marchant Trust, an organization that operates canal boat trips for ill and disabled people. Her other patronages include the International School of Paris, Les Ballets de Monte Carlo, which she also founded, the Monte-Carlo Philharmonic Orchestra, the Association des Guides et Scouts de Monaco, the Monte Carlo Garden Club and The Spring Arts Festival. In 1992, she was appointed the president of the International Contemporary Art Prize.

Following her mother's death in 1982, Caroline served as de facto first lady of Monaco until her brother married Charlene Wittstock in 2011. She regularly attends important social events in Monaco related to the Monégasque Princely Family, such as the National Day celebrations, the annual Rose Ball, the Red Cross Ball and the Formula One competition Monaco Grand Prix. Due to her commitment to philanthropy and arts, Caroline was named a UNESCO Goodwill Ambassador on 2 December 2003. The UNICEF honoured her with Children's Champion Award on 20 May 2006. The next year, she travelled to the Republic of South Africa to meet its former president Nelson Mandela. In December 2011, the World Association of Children's Friends honoured her for "tireless endeavours in continuing the organisation's legacy". Her personal friend and the Chanel head designer Karl Lagerfeld presented her the award. Caroline had also previously been given the Grand Cross of the Order of St. Charles, and had been appointed as the Commander of the Order of Cultural Merit.

Personal and media life
Caroline's personal interests include horseback riding, swimming and skiing. Since her youth, she has been considered an international fashion icon and one of the best dressed women in the world. In November 2011, an exhibition honouring Princess Caroline was opened at the National Museum of Monaco.

Caroline was romantically linked to many famous men, including Guillermo Vilas; Henri Giscard d'Estaing, the son of former President of France Valéry Giscard d'Estaing; French singer Philippe Lavil; and Bobby Shriver, nephew of former U.S. President John F. Kennedy. Following her divorce from Philippe Junot, she was briefly engaged to Robertino Rossellini, the son of Roberto Rossellini and Ingrid Bergman. Between her second and third marriages, Caroline had a relationship with French actor Vincent Lindon.

First marriage
Princess Caroline's first husband was Philippe Junot (born 19 April 1940), a Parisian banker. They were married civilly in Monaco on 28 June 1978, and religiously on 29 June 1978. Their lavish wedding ceremony was attended by some 650 guests, including Hollywood stars Ava Gardner, Cary Grant and Frank Sinatra.

The couple divorced, childless, on 9 October 1980. In 1992, the Catholic Church granted the princess an annulment.

Second marriage
Her second husband was Stefano Casiraghi (8 September 1960 – 3 October 1990), the sportsman heir to an Italian industrial fortune. They were married civilly in Monaco on 29 December 1983, and had three children:

 * Andrea Albert Pierre Casiraghi (born on 8 June 1984 at Princess Grace Hospital Centre in Monaco). Married Tatiana Santo Domingo on 31 August 2013, at the Prince's Palace in Monaco-Ville. The couple have three children:
 * Alexandre "Sasha" Andrea Stefano Casiraghi (born on 21 March 2013 at Portland Hospital in London; entered the line of succession to the Monegasque throne when his parents married).
 * India Julia Casiraghi (born on 12 April 2015 in London).
 * Maximilian Rainier Casiraghi (born on 19 April 2018 in London).
 * Charlotte Marie Pomeline Casiraghi (born on 3 August 1986 at Princess Grace Hospital Centre). Has a son with her former partner, the French actor and comedian Gad Elmaleh, and a second son with her husband, the French producer Dimitri Rassam:
 * Raphaël Elmaleh (born on 17 December 2013 at Princess Grace Hospital Centre).
 * Balthazar Rassam (born on 23 October 2018 at Princess Grace Hospital Centre).
 * Pierre Rainier Stefano Casiraghi (born on 5 September 1987 at Princess Grace Hospital Centre). He married Beatrice Borromeo in a civil ceremony on 25 July 2015, in the gardens of the Prince's Palace of Monaco. They have two children:
 * Stefano Ercole Carlo Casiraghi (born on 28 February 2017 at Princess Grace Hospital Centre).
 * Francesco Carlo Albert Casiraghi (born on 21 May 2018 at Princess Grace Hospital Centre).

The two younger children are named for their maternal great-grandparents, Princess Charlotte and Prince Pierre, while Andrea was named for a childhood friend of his father's. Stefano Casiraghi was killed in a speed-boating accident in 1990, aged 30 years.

Even though their parents had not married in the Church, as required under canon law, their marriage was convalidated by Pope John Paul II in February 1993, eight months after their mother's marriage to Junot had been annulled in June 1992.

Lagerfeld photographed Caroline and Casiraghi and their three children at the Villa La Vigie, Roquebrune-Cap-Martin for their fifth wedding anniversary in 1989.

Third marriage
Caroline's third and current husband is Prince Ernst August of Hanover, Duke of Brunswick, head of the House of Hanover, which lost its throne in 1866. From 1913 to 1918, his family ruled the sovereign Duchy of Brunswick.

The couple married in Monaco on 23 January 1999. Ernst August had previously divorced his first wife Chantal Hochuli, with whom he had sons Prince Ernst August and Prince Christian, and who had been Caroline's friend.

The couple have one daughter together:
 * Princess Alexandra Charlotte Ulrike Maryam Virginia of Hanover (born 20 July 1999 in Vöcklabruck, Austria)

Her husband's title as Duke of Brunswick is honorific since the ruling family of that state was removed by the Weimar Republic in 1918, along with all royal and noble German ruling families, which were still allowed to retain their titles. Neither she nor her husband has royal rank in Germany, but Monaco recognizes the Hanoverians' former German royal titles, attributing to the couple the style of Royal Highness. On 11 January 1999, shortly before Caroline and Ernst's wedding, his third cousin once removed (Queen Victoria was their common ancestor), Queen Elizabeth II of the United Kingdom, issued this Order in Council, "My Lords, I do hereby declare My Consent to a Contract of Matrimony between His Royal Highness Prince Ernst August Albert of Hanover, Duke of Brunswick-Luneburg and Her Serene Highness Princess Caroline Louise Marguerite of Monaco...". As a legitimate male-line descendant of George II, Ernst August was subject to the Royal Marriages Act 1772 (repealed in 2015). Prior to the repeal of the Act, the revised form of which limits those who must gain permission to the first six people in the line of British succession, marrying without the Queen's Royal Assent would have meant their marriage would be void in Britain, where Ernst August's family owned substantial property and he holds (dual) citizenship.

Likewise, the Monégasque court officially notified France of Caroline's contemplated marriage to Prince Ernst August and received assurance that there was no objection, in compliance with Article 2 of the 1918 Franco-Monégasque Treaty. Despite obtaining the official approval of the governments of France, Monaco and the United Kingdom, upon Caroline's marriage to Ernst August he forfeited his own place in Britain's order of succession. He is also subject to the Act of Settlement 1701, which imposes that consequence upon British dynasts who marry Roman Catholics. The Succession to the Crown Act of 2013 however removed that consequence of marrying a Roman Catholic, and would place him back in the order of succession.

In 2009, it was reported that Caroline had separated from Ernst August and returned to live in Monaco.

Privacy cases
Caroline has had a bad relationship with media and paparazzi since her youth, when she complained she "could not live the life of a normal student". On 24 June 2004, the Princess obtained a judgement from the European Court of Human Rights condemning Germany for non-respect of her right to private life under Article 8 of the European Convention on Human Rights.

Caroline invoked the judgment in combination with articles 1(1) and 2(1) of the Basic Law (human dignity and personal freedom, respectively) as well as § 22 of the German Art and Photography Copyright Act or KunstUrhG (no publication of personal images without permission) in a new domestic case, attempting to get the courts to prohibit publication of certain images of her in a private setting. The Supreme Court accepted her claim with regard to two images, but did not prohibit publication of a third, stating that the image accompanied an article about a subject of public interest, which allows publication without permission per § 23 of the KunstUrhG. Caroline appealed to the Federal Constitutional Court, which affirmed the Supreme Court's judgement. Unsatisfied with this result, Caroline filed a new complaint with the European Court of Human Rights. This time, the court found that the domestic courts had properly weighed the competing interests of Caroline's privacy and the press' right to freedom of expression, and thus found that there had been no violation of Article 8.

Writings
In April 1981, the Princess penned an essay, entitled "Home" and published in the International Herald Tribune's supplement. The byline was "Caroline de Grimaldi." In the essay, she wrote: "I long for the Mediterranean ... I feel in my bones that I belong in Monaco." The article was titled, "A Compulsive Need for Blue."

Succession issues
Princess Caroline was heiress presumptive to the crown of Monaco until the birth of her brother's legitimate children.

There is precedent for a Monégasque prince to adopt his own illegitimate child and thereby place that child at the head of the line of succession to the Monegasque throne, as was done for Caroline's grandmother, Princess Charlotte, Duchess of Valentinois. However, because of changes to the constitution of Monaco in 2002, this was no longer an option.

Albert's lack of legitimate children until the 2010s prompted Prince Rainier III to change the constitution so as to ensure there would be a successor to the throne, which strengthened the places of Caroline and her descendants in the line of succession. On 2 April 2002, Monaco passed Princely Law 1.249, which provides that if the Sovereign Prince assumes the throne and then dies without a legitimate direct heir, the throne will pass to his dynastic siblings and their descendants according to the rule of male-preference cognatic primogeniture. The law was then ratified by France, as required by a 1918 Franco-Monégasque Treaty, on 4 October 2005. Before this change, the crown of Monaco could pass only to a descendant of the last reigning prince, excluding such collateral relations as siblings, nephews, and nieces.

Titles and styles

 * 23 January 1957 – 23 January 1999: Her Serene Highness Princess Caroline of Monaco
 * 23 January 1999 – present: Her Royal Highness The Princess of Hanover

National honours

 * 🇲🇨 Monaco: Knight Grand Cross of the Order of Saint-Charles
 * 🇲🇨 Monaco: 10 November 2005 Knight Commander of the Order of Cultural Merit
 * House of Hanover: Knight of the Order of Saint George

Foreign honours

 * 🇧🇷 Brazil: Grand Officer of the Order of Rio Branco (1 December 2022)
 * 🇫🇷 France: Commander of the Order of Agricultural Merit (3 July 2014)
 * 🇫🇷 France: Commander of the Order of Arts and Letters (20 May 2014)
 * 🇸🇪 Sweden: Recipient of the 50th Birthday Medal of King Carl XVI Gustaf (30 April 1996)

International

 * 2 December 2003 UNESCO Goodwill Ambassador

title: Princess Caroline of Monaco


text:
User:Seshat~enwiki

So far I have only contributed a reference to the article on the Maximum Entropy Principle. More serious work will be attempted in the future ;)

I have posted comments on your talk page. Cyan 04:02, 10 Aug 2003 (UTC)

title: User:Seshat~enwiki


text:
Anna Maria Luisa de' Medici

Anna Maria Luisa de' Medici (11 August 1667 – 18 February 1743) was an Italian noblewoman who was the last lineal descendant of the main branch of the House of Medici. A patron of the arts, she bequeathed the Medicis' large art collection, including the contents of the Uffizi, Palazzo Pitti and the Medici villas, which she inherited upon her brother Gian Gastone's death in 1737, and her Palatine treasures to the Tuscan state, on the condition that no part of it could be removed from "the Capital of the grand ducal State....[and from] the succession of His Serene Grand Duke."

Anna Maria Luisa was the only daughter of Cosimo III de' Medici, Grand Duke of Tuscany, and Marguerite Louise d'Orléans, a niece of Louis XIII of France. On her marriage to Johann Wilhelm, Elector Palatine, she became Electress of the Palatinate, and, by patronising musicians, she earned for the contemporary Palatine court the reputation of an important music centre. As Johann Wilhelm had syphilis the union produced no offspring, which, combined with her siblings' barrenness, meant that the Medici were on the verge of extinction.

In 1713 Cosimo III altered the Tuscan laws of succession to allow the accession of his daughter, and spent his final years canvassing the European powers to agree to recognise this statute. However, in 1735, as part of a territorial arrangement, the European powers appointed Duke Francis Stephen of Lorraine as heir, and he duly ascended the Tuscan throne in her stead. After the death of Johann Wilhelm, Anna Maria Luisa returned to Florence, where she enjoyed the rank of first lady until the accession of her brother Gian Gastone, who banished her to the Villa La Quiete. When Gian Gastone died in 1737, Francis Stephen's envoy offered Anna Maria Luisa the position of nominal regent of Tuscany, but she declined. Her death, in 1743, brought the grand ducal House of Medici to an end. Her remains were interred in the Medicean necropolis, Basilica of San Lorenzo, Florence, which she helped complete.

Early life
Despite her mother's efforts to induce a miscarriage by means of riding, Anna Maria Luisa de' Medici, the only daughter and second child of Cosimo III de' Medici, Grand Duke of Tuscany, and his consort, Marguerite Louise d'Orléans, was born in Florence on 11 August 1667. She was named after her maternal aunt Anne Marie Louise d'Orléans, Duchess of Montpensier.

Her parents' relationship was quarrelsome; Marguerite Louise took every chance to humiliate Cosimo. On one documented occasion, she branded him "a poor groom" in the presence of the Papal nuncio. The enmity between them continued until 26 December 1674; after all attempts at conciliation failed, a stressed Cosimo consented to his wife's departure for the Convent of Montmartre, France. The contract created that day revoked her privileges as a petite fille de France, and declared that upon her death all her assets were to be inherited by her children. Cosimo granted her a pension of 80,000 livres in compensation. She abandoned Tuscany in June 1675; Anna Maria Luisa, who was less than eight years old at that time, never saw her again. Although Cosimo doted on his daughter, she was raised by her paternal grandmother, Vittoria della Rovere.

Electress of the Palatinate
In 1669, Anna Maria Luisa was considered as a potential bride to Louis, le Grand Dauphin, the heir-apparent of Louis XIV of France. Cosimo III did not like the idea of a French marriage, and never devoted himself fully to the cause (she was later rejected). Instead, Cosimo offered her to his first choice, Peter II of Portugal. Peter's ministers declined, fearing that Anna Maria Luisa might have inherited her mother's temperament, and may seek to dominate Peter II while being herself intractable to reason. In fact, contemporaries thought her traits to be a combination of those of her father and paternal grandmother, Vittoria della Rovere.

Following refusals from Portugal, France, Spain and Savoy, James II of England put forward his brother-in-law, Francesco II d'Este, Duke of Modena, but the Princess deemed a duke too lowly in terms of protocol for the daughter of a grand duke. Finally, it was Leopold I, Holy Roman Emperor who suggested Johann Wilhelm, Elector Palatine. The Elector Palatine obtained the style Royal Highness from the Holy Roman Emperor for Cosimo III in February 1691. (Cosimo had hitherto been outranked by the Duke of Savoy — much to his anger—who derived royal status from his successful pretendership to the abolished Cypriot throne). Consequently, Johann Wilhelm was ultimately chosen. He and Anna Maria Luisa were married by proxy on 29 April 1691. At the accompanying festivities, a contemporary describes the Electress's physical attributes: "In her person, she is tall, her complexion was fair, her eyes large and expressive, both those and her hair were black; her mouth was small, with a fullness of the lips; her teeth were as white as ivory...."

She departed for Düsseldorf, her husband’s capital, on 6 May 1691, accompanied by her younger brother, Gian Gastone. Johann Wilhelm surprised her at Innsbruck, where they officially married. The Palatinate Anna Maria Luisa arrived in was ravaged by the ongoing Nine Years' War, in which Louis XIV assaulted the Palatinate on behalf of his brother, Philippe of France, Duke of Orléans, occupying the city of Philippsburg in the process.

The Electress became pregnant in 1692; however, she miscarried. It is thought incorrectly by some historians that soon after arrival she contracted syphilis from the Elector, which they think explains why Anna Maria Luisa and Johann Wilhelm failed to produce any children. Anna Maria Luisa and Johann Wilhelm, notwithstanding, shared a harmonious marriage. The Electress spent her time enjoying balls, musical performances and other festivities. He commissioned a theatre for her where the comedies of French playwright Molière were performed. Because Anna Maria Luisa patronised many musicians, the contemporary Palatine court enjoyed regard as an international centre of music. She invited Fortunato Chelleri to court and appointed him maestro di cappella ("music teacher"). Agostino Steffani, a polymath, was sponsored by the Electress from his arrival in Düsseldorf, in 1703, until her return to Tuscany; the Conservatorio Luigi Cherubini library in Florence houses two editions of his chamber duets.

Anna Maria Luisa arranged a marriage for her younger brother at the instigation of their father: On 2 July 1697 Gian Gastone de' Medici married Anna Maria Franziska of Saxe-Lauenburg, heiress of the eponymous duchy, in Düsseldorf. Gian Gastone's wife repulsed him, and for that reason, they separated in 1708.

The same year as Gian Gastone's marriage, the Peace of Ryswick ended the Nine Years' War: French troops withdrew from the Electoral Palatinate and Johann Wilhelm received the County of Megen. Following the revocation of the Edict of Nantes, a document which had hitherto given rights to Calvinists, in 1685, 2,000 French Huguenots emigrated to the Electoral Palatinate. Johann Wilhelm, under criticism for his treatment of the Palatine Protestants from the Elector of Brandenburg introduced a Religionsdeklaration in 1705, which sanctioned religious freedom.

Tuscan succession
Cosimo III wished to alter the male-only Tuscan line of succession so as to allow the accession of his daughter, Anna Maria Luisa, in the event of a male-line succession failure. But his plan was met with fierce opposition from the European powers. Charles VI, Holy Roman Emperor, Tuscany's nominal feudal over-lord, subscribed, but only if he should succeed her. Cosimo and she were at odds with the proposal. Without a concord in sight, the "Tuscan question" became dormant.

Some years later, as the question of the succession became more urgent, Cardinal Francesco Maria de' Medici, Cosimo III's brother, was released from his vows and coerced into marrying Vincenzo Gonzaga, Duke of Guastalla's elder daughter, Eleanor, in 1709. The Electress urged him to care for his health and "give us the consolation of a little prince." However, two years later, he died without issue, taking with him any hope of an heir.

Following the death of his heir apparent, Ferdinando, in 1713, Cosimo deposited a bill in the Senate, Tuscany's titular legislature, promulgating that if Cosimo and his new heir apparent, Gian Gastone, were to predecease the Electress, she would ascend the throne. Charles VI was furious; he replied that the grand duchy was an imperial fief and therefore he alone possessed the prerogative to alter the laws of succession. To complicate things further, Elisabeth Farnese, heiress of the Duchy of Parma, the second wife of Philip V of Spain, as a great-granddaughter of Margherita de' Medici, exercised a claim to Tuscany. In May 1716, Charles VI, who constantly changed his stance on the issue, told Florence that the Electress's succession was unquestioned, but added that Austria and Tuscany must soon reach an agreement regarding which royal house was to follow the Medici.

In June 1717, Cosimo declared his wish that the House of Este should succeed the Electress. Charles VI had previously offered the Grand Duke territorial compensation—in the form of the State of Presidi—if he chose quickly, but reneged. In 1718, Charles VI repudiated Cosimo's decision, declaring a union of Tuscany and Modena (the Este lands) unacceptable. Hereafter, a stalemate existed between them.

Return to Florence
The Elector Palatine died in June 1716. His widow, Anna Maria Luisa, returned to Florence in October 1717. Dowager Grand Princess Violante Beatrice, her brother Ferdinando's widow, and Anna Maria Luisa did not enjoy an amiable relationship. Upon hearing of Anna Maria Luisa's intention to return, Violante Beatrice prepared to depart for Munich, her brother's capital, but Gian Gastone wished her to stay, so she did. To keep the two ladies from quarrelling over precedence, Cosimo III defined Violante Beatrice's status just before the Electress's arrival by appointing her Governess of Siena.

On 4 April 1718 Great Britain, France and the Dutch Republic (and later Austria) selected Infante Charles of Spain, the elder child of Elisabeth Farnese and Philip V of Spain, as the Tuscan heir (with no mention of Anna Maria Luisa). By 1722, the Electress was not even acknowledged as heiress, and Cosimo was reduced to a spectator at the conferences for Tuscany's future. In the midst of this, Marguerite Louise, Anna Maria Luisa's mother, died. Instead of willing her valuables to her children, as prescribed by the 1674 agreement, they went to the Princess of Epinoy, a distant relative.

On 25 October 1723, six days before his death, Cosimo III distributed a final proclamation commanding that Tuscany shall stay independent; Anna Maria Luisa shall succeed uninhibited after Gian Gastone; the Grand Duke reserves the right to choose his successor. Unfortunately for Cosimo, Europe completely ignored it. Gian Gastone, now the Grand Duke, and Anna Maria Luisa were not on good terms. He despised the Electress for engineering his unhappy marriage with Anna Maria Franziska of Saxe-Lauenburg, while she detested his liberal policies: he repealed all of his father's anti-Semitic statutes and revelled in upsetting her. Consequently, the Electress was compelled to abandon her apartment in the left wing of the Palazzo Pitti, for the Villa La Quiete. She refurbished La Quiete's house and gardens with the assistance of Sebastiano Rapi, the gardener of the Boboli Gardens, and the architects Giovanni Battista Foggini and Paolo Giovanozzi. In the period 1722–1725, the Electress embellished the villa further by commissioning twelve statues of various religious figures.

In spite of their mutual dislike, the Electress and Violante Beatrice attempted to improve Gian Gastone's poor public image together. Rumours abounded that the Grand Duke had died; it was a rarity for the public to see him. To dispel the said rumours, the Electress compelled him to make an appearance—his last one—in 1729, on the feast day of the patron saint of Florence, John the Baptist. The Ruspanti, Gian Gastone's morally corrupt entourage, hated the Electress; and she, them. Violante Beatrice tried to withdraw the Grand Duke from their sphere of influence by organising banquets. His conduct at these literally sent those in attendance scrambling for their carriages: he vomited repeatedly into his napkin, belched and told rude jokes. These distractions ceased upon Violante Beatrice's death in 1731.

In 1736, during the War of the Polish Succession, Infante Charles was banished from Tuscany as part of a territorial swap, and Francis III of Lorraine was made heir in his stead. In January 1737, the Spanish troops, who had occupied Tuscany since 1731, withdrew; 6,000 Austrian soldiers took their place.

Gian Gastone died from "an accumulation of diseases" on 9 July 1737, surrounded by prelates and his sister. Anna Maria Luisa was offered a nominal regency by the Prince de Craon, the Grand Duke's envoy, until Francis III could arrive in Florence, but declined. At Gian Gastone's demise, all the House of Medici's allodial possessions, including £2,000,000 liquid cash, a vast art collection, robes of state and lands in the former Duchy of Urbino, were conferred on Anna Maria Luisa. In regards to this, her most notable act was the Patto di Famiglia ("Family Pact"), signed on 31 October 1737. In collaboration with the Holy Roman Emperor and Francis of Lorraine, she willed all the personal property of the Medici's to the Tuscan state, provided that nothing was ever removed from Florence.

Death and legacy
The "Lorrainers," as the occupying forces were dubbed, were popularly loathed. The Viceroy, the Prince de Craon, whom the Electress disliked for his "vulgar" court, allowed the Electress to live undisturbed in her own wing of the Palazzo Pitti, living in virtual seclusion, only on occasion receiving a select-number of guests under a black dais in her silver-clad audience room. She occupied herself financing and overseeing the construction of the Cappella dei Principi, started in 1604 by Ferdinando I de' Medici, Grand Duke of Tuscany—to the tune of 1,000 crowns per week, and she donated much of her fortune to charity: £4,000 per annum. This is equivalent to £ in present-day terms. On 18 February 1743, Anna Maria Luisa de' Medici, Dowager Electress Palatine, died of an "oppression on the breast". Sir Horace Mann, 1st Baronet, a British resident in Florence, recalled in a letter that "The common people are convinced she went off in a hurricane of wind; a most violent one began this morning and lasted for about two hours, and now the sun shines as bright as ever..." The royal line of the House of Medici became extinct with her death. Her will, having been completed just months before, according to Sir Horace Mann, left £500,000 worth of jewellery to the Grand Duke Francis and her lands in the former Duchy of Urbino to the Marquis Rinuccini, her main executor and a minister under her father, Cosimo III. She was interred in the crypt that she helped to complete in San Lorenzo; although not entirely finished at the time of her death, her testament stipulated that part of the revenue of her estate should "be used to continue, finish and perfect...the said famous chapel San Lorenzo".

Anna Maria Luisa's single most enduring act was the Family Pact. It ensured that all the Medicean art and treasures collected over nearly three centuries of political ascendancy remained in Florence. Cynthia Miller Lawrence, an American art historian, argues that Anna Maria Luisa thus provisioned for Tuscany's future economy through tourism. Sixteen years after her death, the Uffizi Gallery, built by Cosimo the Great, the founder of the grand duchy, was made open to public viewing.

In 2012 after concern caused by the 1966 Flood of the Arno River, her bones were exhumed. A scientific examination found no traces of syphilis, which she had long been thought to have died from.

title: Anna Maria Luisa de' Medici