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Armenian language

Armenian (endonym: հայերեն,, ) is an Indo-European language and the sole member of the independent branch of the Armenian language family. It is the native language of the Armenian people and the official language of Armenia. Historically spoken in the Armenian highlands, today Armenian is also widely spoken throughout the Armenian diaspora. Armenian is written in its own writing system, the Armenian alphabet, introduced in 405 AD by Saint Mesrop Mashtots. The estimated number of Armenian speakers worldwide is between five and seven million.

Classification and origins
Armenian is an independent branch of the Indo-European languages. It is of interest to linguists for its distinctive phonological changes within that family. Armenian exhibits more satemization than centumization, although it is not classified as belonging to either of these subgroups. Some linguists tentatively conclude that Armenian, Greek (and Phrygian), European Albanian and Indo-Iranian were dialectally close to each other;  within this hypothetical dialect group, Proto-Armenian was situated between Proto-Greek (centum subgroup) and Proto-Indo-Iranian (satem subgroup). Ronald I. Kim has noted unique morphological developments connecting Armenian to Balto-Slavic languages.

The Armenian language has a long literary history, with a 5th-century Bible translation as its oldest surviving text. Another text translated into Armenian early on, and also in the 5th-century, was the Armenian Alexander Romance. The vocabulary of the language has historically been influenced by Western Middle Iranian languages, particularly Parthian; its derivational morphology and syntax were also affected by language contact with Parthian, but to a lesser extent. Contact with Greek, Persian, and Syriac also resulted in a number of loanwords. There are two standardized modern literary forms, Eastern Armenian (spoken mainly in Armenia) and Western Armenian (spoken originally mainly in modern-day Turkey and, since the Armenian genocide, mostly in the diaspora). The differences between them are considerable but they are mutually intelligible after significant exposure. Some subdialects such as Homshetsi are not mutually intelligible with other varieties.

Although Armenians were known to history much earlier (for example, they were mentioned in the 6th-century BC Behistun Inscription and in Xenophon's 4th century BC history, The Anabasis), the oldest surviving Armenian-language writing is etched in stone on Armenian temples and is called Mehenagir. The Armenian alphabet was created by Mesrop Mashtots in 405, at which time it had 36 letters. He is also credited by some with the creation of the Georgian alphabet and the Caucasian Albanian alphabet.

While Armenian constitutes the sole member of the Armenian branch of the Indo-European family, Aram Kossian has suggested that the hypothetical Mushki language may have been a (now extinct) Armenic language.

Early contacts
W. M. Austin (1942) concluded that there was early contact between Armenian and Anatolian languages, based on what he considered common archaisms, such as the lack of a feminine gender and the absence of inherited long vowels. Unlike shared innovations (or synapomorphies), the common retention of archaisms (or symplesiomorphy) is not considered conclusive evidence of a period of common isolated development. There are words used in Armenian that are generally believed to have been borrowed from Anatolian languages, particularly from Luwian, although some researchers have identified possible Hittite loanwords as well. One notable loanword from Anatolian is Armenian xalam, "skull", cognate to Hittite ḫalanta, "head".

In 1985, the Soviet linguist Igor M. Diakonoff noted the presence in Classical Armenian of what he calls a "Caucasian substratum" identified by earlier scholars, consisting of loans from the Kartvelian and Northeast Caucasian languages. Noting that Hurro-Urartian-speaking peoples inhabited the Armenian homeland in the second millennium BC, Diakonoff identifies in Armenian a Hurro-Urartian substratum of social, cultural, and animal and plant terms such as ałaxin "slave girl" ( ← Hurr. al(l)a(e)ḫḫenne), cov "sea" ( ← Urart. ṣûǝ "(inland) sea"), ułt "camel" ( ← Hurr. uḷtu), and xnjor "apple (tree)" ( ← Hurr. ḫinzuri). Some of the terms he gives admittedly have an Akkadian or Sumerian provenance, but he suggests they were borrowed through Hurrian or Urartian. Given that these borrowings do not undergo sound changes characteristic of the development of Armenian from Proto-Indo-European, he dates their borrowing to a time before the written record but after the Proto-Armenian language stage.

Contemporary linguists, such as Hrach Martirosyan, have rejected many of the Hurro-Urartian and Northeast Caucasian origins for these words and instead suggest native Armenian etymologies, leaving the possibility that these words may have been loaned into Hurro-Urartian and Caucasian languages from Armenian, and not vice versa. A notable example is arciv, meaning "eagle", believed to have been the origin of Urartian Arṣibi and Northeast Caucasian arzu. This word is derived from Proto-Indo-European *h₂r̥ǵipyós, with cognates in Sanskrit (ऋजिप्य, ṛjipyá), Avestan (ərəzifiia), and Greek (αἰγίπιος, aigípios). Hrach Martirosyan and Armen Petrosyan propose additional borrowed words of Armenian origin loaned into Urartian and vice versa, including grammatical words and parts of speech, such as Urartian eue ("and"), attested in the earliest Urartian texts and likely a loan from Armenian (compare to Armenian եւ, ultimately from Proto-Indo-European *h₁epi). Other loans from Armenian into Urartian includes personal names, toponyms, and names of deities.

Loan words from Iranian languages, along with the other ancient accounts such as that of Xenophon above, initially led some linguists to erroneously classify Armenian as an Iranian language. Scholars such as Paul de Lagarde and F. Müller believed that the similarities between the two languages meant that Armenian belonged to the Iranian language family. The distinctness of Armenian was recognized when philologist Heinrich Hübschmann (1875) used the comparative method to distinguish two layers of Iranian words from the older Armenian vocabulary. He showed that Armenian often had two morphemes for one concept, that the non-Iranian components yielded a consistent Proto-Indo-European pattern distinct from Iranian, and that the inflectional morphology was different from that of Iranian languages.

Graeco-Armenian hypothesis
The hypothesis that Greek is Armenian's closest living relative originates with Holger Pedersen (1924), who noted that the number of Greek-Armenian lexical cognates is greater than that of agreements between Armenian and any other Indo-European language. Antoine Meillet (1925, 1927) further investigated morphological and phonological agreement and postulated that the parent languages of Greek and Armenian were dialects in immediate geographical proximity during the Proto-Indo-European period. Meillet's hypothesis became popular in the wake of his book Esquisse d'une histoire de la langue latine (1936). Georg Renatus Solta (1960) does not go as far as postulating a Proto-Graeco-Armenian stage, but he concludes that considering both the lexicon and morphology, Greek is clearly the dialect to be most closely related to Armenian. Eric P. Hamp (1976, 91) supports the Graeco-Armenian thesis and even anticipates a time "when we should speak of Helleno-Armenian" (meaning the postulate of a Graeco-Armenian proto-language). Armenian shares the augment and a negator derived from the set phrase in the Proto-Indo-European language ("never anything" or "always nothing"), the representation of word-initial laryngeals by prothetic vowels, and other phonological and morphological peculiarities with Greek. Nevertheless, as Fortson (2004) comments, "by the time we reach our earliest Armenian records in the 5th century AD, the evidence of any such early kinship has been reduced to a few tantalizing pieces".

Greco-Armeno-Aryan hypothesis
Graeco-(Armeno)-Aryan is a hypothetical clade within the Indo-European family, ancestral to the Greek language, the Armenian language, and the Indo-Iranian languages. Graeco-Aryan unity would have become divided into Proto-Greek and Proto-Indo-Iranian by the mid-3rd millennium BC. Conceivably, Proto-Armenian would have been located between Proto-Greek and Proto-Indo-Iranian, consistent with the fact that Armenian shares certain features only with Indo-Iranian (the satem change) but others only with Greek (s > h).

Graeco-Aryan has comparatively wide support among Indo-Europeanists who believe the Indo-European homeland to be located in the Armenian Highlands, the "Armenian hypothesis". Early and strong evidence was given by Euler's 1979 examination on shared features in Greek and Sanskrit nominal flection.

Used in tandem with the Graeco-Armenian hypothesis, the Armenian language would also be included under the label Aryano-Greco-Armenic, splitting into Proto-Greek/Phrygian and "Armeno-Aryan" (ancestor of Armenian and Indo-Iranian).

Evolution
Classical Armenian (Arm: grabar), attested from the 5th century to the 19th century as the literary standard (up to the 11th century also as a spoken language with different varieties), was partially superseded by Middle Armenian, attested from the 12th century to the 18th century. Specialized literature prefers "Old Armenian" for grabar as a whole, and designates as "Classical" the language used in the 5th century literature, "Post-Classical" from the late 5th to 8th centuries, and "Late Grabar" that of the period covering the 8th to 11th centuries. Later, it was used mainly in religious and specialized literature, with the exception of a revival during the early modern period, when attempts were made to establish it as the language of a literary renaissance, with neoclassical inclinations, through the creation and dissemination of literature in varied genres, especially by the Mekhitarists. The first Armenian periodical, Azdarar, was published in grabar in 1794.

The classical form borrowed numerous words from Middle Iranian languages, primarily Parthian, and contains smaller inventories of loanwords from Greek, Syriac, Aramaic, Arabic, Mongol, Persian, and indigenous languages such as Urartian. An effort to modernize the language in Bagratid Armenia and the Armenian Kingdom of Cilicia (11–14th centuries) resulted in the addition of two more characters to the alphabet ("օ" and "ֆ"), bringing the total number to 38.

The Book of Lamentations by Gregory of Narek (951–1003) is an example of the development of a literature and writing style of Old Armenian by the 10th century. In addition to elevating the literary style and vocabulary of the Armenian language by adding well above a thousand new words, through his other hymns and poems Gregory paved the way for his successors to include secular themes and vernacular language in their writings. The thematic shift from mainly religious texts to writings with secular outlooks further enhanced and enriched the vocabulary. "A Word of Wisdom", a poem by Hovhannes Sargavak devoted to a starling, legitimizes poetry devoted to nature, love, or female beauty. Gradually, the interests of the population at large were reflected in other literary works as well. Konsdantin Yerzinkatsi and several others took the unusual step of criticizing the ecclesiastic establishment and addressing the social issues of the Armenian homeland. These changes represented the nature of the literary style and syntax, but they did not constitute immense changes to the fundamentals of the grammar or the morphology of the language. Often, when writers codify a spoken dialect, other language users are then encouraged to imitate that structure through the literary device known as parallelism.

In the 19th century, the traditional Armenian homeland was once again divided. This time Eastern Armenia was conquered from Qajar Iran by the Russian Empire, while Western Armenia, containing two thirds of historical Armenia, remained under Ottoman control. The antagonistic relationship between the Russian and Ottoman empires led to creation of two separate and different environments under which Armenians lived. Halfway through the 19th century, two important concentrations of Armenian communities were further consolidated. Because of persecutions or the search for better economic opportunities, many Armenians living under Ottoman rule gradually moved to Istanbul, whereas Tbilisi became the center of Armenians living under Russian rule. These two cosmopolitan cities very soon became the primary poles of Armenian intellectual and cultural life.

The introduction of new literary forms and styles, as well as many new ideas sweeping Europe, reached Armenians living in both regions. This created an ever-growing need to elevate the vernacular, Ashkharhabar, to the dignity of a modern literary language, in contrast to the now-anachronistic Grabar. Numerous dialects existed in the traditional Armenian regions, which, different as they were, had certain morphological and phonetic features in common. On the basis of these features two major standards emerged:

 * Western standard: The influx of immigrants from different parts of the traditional Armenian homeland to Istanbul crystallized the common elements of the regional dialects, paving the way for a style of writing that required a shorter and more flexible learning curve than Grabar.
 * Eastern standard: The Yerevan dialect provided the primary elements of Eastern Armenian, centered in Tbilisi, Georgia. Similar to the Western Armenian variant, the Modern Eastern was in many ways more practical and accessible to the masses than Grabar.

Both centers vigorously pursued the promotion of Ashkharhabar. The proliferation of newspapers in both versions (Eastern & Western) and the development of a network of schools where modern Armenian was taught, dramatically increased the rate of literacy (in spite of the obstacles by the colonial administrators), even in remote rural areas. The emergence of literary works entirely written in the modern versions increasingly legitimized the language's existence. By the turn of the 20th century both varieties of the one modern Armenian language prevailed over Grabar and opened the path to a new and simplified grammatical structure of the language in the two different cultural spheres. Apart from several morphological, phonetic, and grammatical differences, the largely common vocabulary and generally analogous rules of grammatical fundamentals allows users of one variant to understand the other as long as they are fluent in one of the literary standards.

After World War I, the existence of the two modern versions of the same language was sanctioned even more clearly. The Armenian Soviet Socialist Republic (1920–1990) used Eastern Armenian as its official language, whereas the diaspora created after the Armenian genocide preserved the Western Armenian dialect.

The two modern literary dialects, Western (originally associated with writers in the Ottoman Empire) and Eastern (originally associated with writers in the Russian Empire), removed almost all of their Turkish lexical influences in the 20th century, primarily following the Armenian genocide.

Geographic distribution
In addition to Armenia and Turkey, where it is indigenous, Armenian is spoken among the diaspora. According to Ethnologue, globally there are million Western Armenian speakers and  million Eastern Armenian speakers, totalling  million Armenian speakers.

In Georgia, Armenian speakers are concentrated in Ninotsminda and Akhalkalaki districts where they represent over 90% of the population.

Status and usage
The short-lived First Republic of Armenia declared Armenian its official language. Eastern Armenian was then dominating in institutions and among the population. When Armenia was incorporated into the USSR, the Armenian Soviet Socialist Republic made Eastern Armenian the language of the courts, government institutions and schools. Armenia was also russified. The current Republic of Armenia upholds the official status of the Armenian language. Eastern Armenian is the official variant used, making it the prestige variety while other variants have been excluded from national institutions. Indeed, Western Armenian is perceived by some as a mere dialect. Armenian was also official in the Republic of Artsakh. It is recognized as an official language of the Eurasian Economic Union although Russian is the working language.

Armenian (without reference to a specific variety) is officially recognized as a minority language in Cyprus, Hungary, Iraq, Poland,  Romania, and Ukraine. It is recognized as a minority language and protected in Turkey by the 1923 Treaty of Lausanne.

Western Armenian is the language of the diaspora, it is the medium of instruction in the majority of Armenian-language schools outside Armenia. In particular, in the Samtskhe-Javakheti region of Georgia, although Armenian has no legal status, there were 144 state-funded schools in the area as of 2010 where Armenian is the main language of instruction. The Lebanese curriculum allows Armenian schools to teach the Armenian language as a basic language. In California, home to a large Armenian American community, various state government agencies provide Armenian translations of their documents: the California Department of Social Services, California Department of Motor Vehicles, California superior courts. In the city of Glendale, there are street signs in Armenian.

In Iran, article 15 of the constitution allows the use of "regional and tribal languages" in the mass media as well as within the schools. However, these languages do not receive formal status and are not officially regulated by the authorities. Iranian Armenians are de facto the only non-Persian ethnic group in Iran enjoying this right. They have their own private schools, where Armenian is the medium of instruction.

Phonology

Proto-Indo-European voiceless stop consonants are aspirated in the Proto-Armenian language, one of the circumstances that is often linked to the glottalic theory, a version of which postulated that some voiceless occlusives of Proto-Indo-European were aspirated.

Stress
In Armenian, the stress falls on the last syllable unless the last syllable contains the definite article or, and the possessive articles ս and դ, in which case it falls on the penultimate one. For instance, ախորժակ, մաղադանոս, գինի but Վահագն and դաշտը. Exceptions to this rule are some words with the final letter է (ե in the reformed orthography) (մի՛թէ, մի՛գուցե, ո՛րեւէ) and sometimes the ordinal numerals (վե՛ցերորդ, տա՛սներորդ, etc.), as well as նաեւ, նամանաւանդ, հիմա, այժմ, and a small number of other words.

Vowels
Eastern Armenian has six monophthongs. Each vowel phoneme in the table is represented by three symbols. The first is the sounds transcription in the International Phonetic Alphabet (IPA). After that appears the corresponding letter of the Armenian alphabet. The last symbol is its Latin transliteration.

 * Western and other dialects may also have /, /.

Consonants
The following table lists the Eastern Armenian consonantal system. The occlusives and affricates have an aspirated series, commonly transcribed with a reversed apostrophe after the letter. Each phoneme in the table is represented by IPA, Armenian script and romanization.

The major phonetic difference between dialects is in the reflexes of Classical Armenian voice-onset time. The seven dialect types have the following correspondences, illustrated with the t–d series:
 * {| class="wikitable" style=text-align:center

!Armenian Letter !Indo-European !Karin, Sebastia !Istanbul !Kharberd, Middle Armenian !Malatya, SWA !Classical Armenian, Agulis, SEA, Yerevan !Van, Artsakh
 * +Correspondence in initial position
 * Թ
 * Տ
 * Դ
 * rowspan="6" |
 * colspan=2|
 * rowspan="2" |
 * colspan=2|
 * }

Morphology
Armenian corresponds with other Indo-European languages in its structure, but it shares distinctive sounds and features of its grammar with neighboring languages of the Caucasus region. The Armenian orthography is rich in combinations of consonants, but in pronunciation, this is broken up with schwas. Both classical Armenian and the modern spoken and literary dialects have a system of noun declension, with six or seven noun cases but no gender. In modern Armenian, the use of auxiliary verbs to show tense (comparable to will in "he will go") has generally supplanted the inflected verbs of Classical Armenian. Negative verbs are conjugated differently from positive ones (as in English "he goes" and "he does not go") in many tenses, otherwise adding only the negative չ to the positive conjugation. Grammatically, early forms of Armenian had much in common with classical Greek and Latin, but the modern language, like modern Greek, has undergone many transformations, adding some analytic features.

Noun
Armenian has no grammatical gender, not even in the pronoun, but there is a feminine suffix (-ուհի "-uhi"). For example, ուսուցիչ (usucʻičʻ, "teacher") becomes ուսուցչուհի (usucʻčʻuhi, female teacher). This suffix does not have a grammatical effect on the sentence. The nominal inflection reserves several types of inherited stem classes. Historically, nouns were declined for one of seven cases: nominative (ուղղական uġġakan), accusative (հայցական haycʻakan), locative (ներգոյական nergoyakan), genitive (սեռական seṙakan), dative (տրական trakan), ablative (բացառական bacʻaṙakan), or instrumental (գործիական gorciakan), but in the modern language, the nominative and accusative cases, as well as the dative and genitive cases, have merged.

 * Examples of noun declension in Eastern Armenian

Which case the direct object takes is split based on animacy (a phenomenon more generally known as differential object marking). Inanimate nouns take the nominative, while animate nouns take the dative. Additionally, animate nouns can never take the locative case.

 * Examples of noun declension in Western Armenian

Verb
Verbs in Armenian have an expansive system of conjugation with two main verb types in Eastern Armenian and three in Western Armenian changing form based on tense, mood and aspect.

Dialects
Armenian is a pluricentric language, having two modern standardized forms: Eastern Armenian and Western Armenian. The most distinctive feature of Western Armenian is that it has undergone several phonetic mergers; these may be due to proximity to Arabic- and Turkish-speaking communities.

Classical Armenian (Grabar), which remained the standard until the 18th century, was quite homogeneous across the different regions that works in it were written; it may have been a cross-regional standard. The Middle Armenian variety used in the court of Cilician Armenia (1080–1375) provides a window into the development of Western Armenian, which came to be based on what became the dialect of Istanbul, while the standard for Eastern Armenian was based on the dialect around Mount Ararat and Yerevan. Although the Armenian language is often divided into "east" and "west", the two standards are actually relatively close to each other in light of wealth of the diversity present among regional non-standard Armenian dialects. The different dialects have experienced different degrees of language contact effects, often with Turkic and Caucasian languages; for some, the result has been significant phonological and syntactic changes. Fortson notes that the modern standard as well has now attained a subordinate clausal structure that greatly resembles a Turkic language.

Eastern Armenian speakers pronounce (թ) as [tʰ], (դ) as [d], and (տ) as a tenuis occlusive [t˭]. Western Armenian has simplified the occlusive system into a simple division between voiced occlusives and aspirated ones; the first series corresponds to the tenuis series of Eastern Armenian, and the second corresponds to the Eastern voiced and aspirated series. Thus, the Western dialect pronounces both (թ) and (դ) as [tʰ], and the (տ) letter as [d].

There is no precise linguistic border between one dialect and another because there is nearly always a dialect transition zone of some size between pairs of geographically identified dialects.

Armenian can be divided into two major dialectal blocks and those blocks into individual dialects, though many of the Western Armenian dialects have become extinct due to the effects of the Armenian genocide. In addition, neither dialect is completely homogeneous: any dialect can be subdivided into several subdialects. Although Western and Eastern Armenian are often described as different dialects of the same language, many subdialects are not readily mutually intelligible. Nevertheless, a fluent speaker of one of two greatly varying dialects who is also literate in one of the standards, when exposed to the other dialect for a period of time will be able to understand the other with relative ease.

Distinct Western Armenian varieties currently in use include Homshetsi, spoken by the Hemshin peoples; the dialects of Armenians of Kessab (Քեսապի բարբառ), Latakia and Jisr al-Shughur (Syria), Anjar, Lebanon, and Vakıflı, Samandağ (Turkey), part of the "Sueidia" dialect (Սուէտիայի բարբառ).

Forms of the Karin dialect of Western Armenian are spoken by several hundred thousand people in Northern Armenia, mostly in Gyumri, Artik, Akhuryan, and around 130 villages in Shirak Province, and by Armenians in Samtskhe–Javakheti province of Georgia (Akhalkalaki, Akhaltsikhe).

Nakhichevan-on-Don Armenians speak another Western Armenian variety based on the dialect of Armenians in Crimea, where they came from in order to establish the town and surrounding villages in 1779 (Նոր Նախիջևանի բարբառ).

Western Armenian dialects are currently spoken also in Gavar (formerly Nor Bayazet and Kamo, on the western shore of Lake Sevan), Aparan, and Talin in Armenia (Mush dialect), and by the large Armenian population residing in Abkhazia, where they are considered to be the first or second ethnic minority, or even equal in number to the local Abkhaz population

Orthography
The Armenian alphabet (Հայոց գրեր or Հայոց այբուբեն) is a graphically unique alphabetical writing system that is used to write the Armenian language. It was introduced around AD 405 by Mesrop Mashtots, an Armenian linguist and ecclesiastical leader, and originally contained 36 letters. Two more letters, օ (ō) and ֆ (f), were added in the Middle Ages.

During the 1920s orthography reform in Soviet Armenia, a new letter և (capital ԵՎ) was added, which was a ligature before ե+ւ, whereas the letter Ւ ւ was discarded and reintroduced as part of a new letter ՈՒ ու (which was a digraph before). This alphabet and associated orthography is used by most Armenian speakers of Armenia and the countries of the former Soviet Union. Neither the alphabet nor the orthography has been adopted by Diaspora Armenians, including Eastern Armenian speakers of Iran and all Western Armenian speakers, who keep using the traditional alphabet and spelling.

Indo-European cognates
Armenian is an Indo-European language, so many of its Proto-Indo-European-descended words are cognates of words in other Indo-European languages such as English, Latin, Greek, and Sanskrit.

Due to extensive loaning, only around 1,500 words (G. Jahukyan) are known to have been inherited from Indo-European by the Classical Armenian stage; the rest were lost, a fact that presents a major challenge to endeavors to better understand Proto-Armenian and its place within the family, especially as many of the sound changes along the way from Indo-European to Armenian remain quite difficult to analyze.

This table lists some of the more recognizable cognates that Armenian shares with English words descended from Old English.

title: Armenian language


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Additive synthesis

Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.

The timbre of musical instruments can be considered in the light of Fourier theory to consist of multiple harmonic or inharmonic partials or overtones. Each partial is a sine wave of different frequency and amplitude that swells and decays over time due to modulation from an ADSR envelope or low frequency oscillator.

Additive synthesis most directly generates sound by adding the output of multiple sine wave generators. Alternative implementations may use pre-computed wavetables or the inverse fast Fourier transform.

Explanation
The sounds that are heard in everyday life are not characterized by a single frequency. Instead, they consist of a sum of pure sine frequencies, each one at a different amplitude. When humans hear these frequencies simultaneously, we can recognize the sound. This is true for both "non-musical" sounds (e.g. water splashing, leaves rustling, etc.) and for "musical sounds" (e.g. a piano note, a bird's tweet, etc.). This set of parameters (frequencies, their relative amplitudes, and how the relative amplitudes change over time) are encapsulated by the timbre of the sound. Fourier analysis is the technique that is used to determine these exact timbre parameters from an overall sound signal; conversely, the resulting set of frequencies and amplitudes is called the Fourier series of the original sound signal.

In the case of a musical note, the lowest frequency of its timbre is designated as the sound's fundamental frequency. For simplicity, we often say that the note is playing at that fundamental frequency (e.g. "middle C is 261.6 Hz"), even though the sound of that note consists of many other frequencies as well. The set of the remaining frequencies is called the overtones (or the harmonics, if their frequencies are integer multiples of the fundamental frequency) of the sound. In other words, the fundamental frequency alone is responsible for the pitch of the note, while the overtones define the timbre of the sound. The overtones of a piano playing middle C will be quite different from the overtones of a violin playing the same note; that's what allows us to differentiate the sounds of the two instruments. There are even subtle differences in timbre between different versions of the same instrument (for example, an upright piano vs. a grand piano).

Additive synthesis aims to exploit this property of sound in order to construct timbre from the ground up. By adding together pure frequencies (sine waves) of varying frequencies and amplitudes, we can precisely define the timbre of the sound that we want to create.

Definitions
Harmonic additive synthesis is closely related to the concept of a Fourier series which is a way of expressing a periodic function as the sum of sinusoidal functions with frequencies equal to integer multiples of a common fundamental frequency. These sinusoids are called harmonics, overtones, or generally, partials. In general, a Fourier series contains an infinite number of sinusoidal components, with no upper limit to the frequency of the sinusoidal functions and includes a DC component (one with frequency of 0 Hz). Frequencies outside of the human audible range can be omitted in additive synthesis. As a result, only a finite number of sinusoidal terms with frequencies that lie within the audible range are modeled in additive synthesis.

A waveform or function is said to be periodic if

 * $$ y(t) = y(t+P) $$

for all $$ t $$ and for some period $$ P $$.

The Fourier series of a periodic function is mathematically expressed as:

 * $$ \begin{align}

y(t) &= \frac{a_0}{2} + \sum_{k=1}^{\infty} \left[ a_k \cos(2 \pi k f_0 t ) - b_k \sin(2 \pi k f_0 t ) \right] \\ &= \frac{a_0}{2} + \sum_{k=1}^{\infty} r_k \cos\left(2 \pi k f_0 t + \phi_k \right) \\ \end{align} $$

where

 * $$f_0 = 1/P$$ is the fundamental frequency of the waveform and is equal to the reciprocal of the period,
 * $$a_k = r_k \cos(\phi_k) = 2 f_0 \int_{0}^P y(t) \cos(2 \pi k f_0 t)\, dt, \quad k \ge 0$$
 * $$b_k = r_k \sin(\phi_k) = -2 f_0 \int_{0}^P y(t) \sin(2 \pi k f_0 t)\, dt, \quad k \ge 1$$
 * $$r_k = \sqrt{a_k^2 + b_k^2}$$ is the amplitude of the $$k$$th harmonic,
 * $$\phi_k = \operatorname{atan2}(b_k, a_k)$$ is the phase offset of the $$k$$th harmonic. atan2 is the four-quadrant arctangent function,

Being inaudible, the DC component, $$a_0/2$$, and all components with frequencies higher than some finite limit, $$K f_0$$, are omitted in the following expressions of additive synthesis.

Harmonic form
The simplest harmonic additive synthesis can be mathematically expressed as:

where $$y(t)$$ is the synthesis output, $$r_k$$, $$k f_0$$, and $$\phi_k$$ are the amplitude, frequency, and the phase offset, respectively, of the $$k$$th harmonic partial of a total of $$K$$ harmonic partials, and $$f_0$$ is the fundamental frequency of the waveform and the frequency of the musical note.

Time-dependent amplitudes
More generally, the amplitude of each harmonic can be prescribed as a function of time, $$r_k(t)$$, in which case the synthesis output is

Each envelope $$r_k(t)\,$$ should vary slowly relative to the frequency spacing between adjacent sinusoids. The bandwidth of $$r_k(t)$$ should be significantly less than $$f_0$$.

Inharmonic form
Additive synthesis can also produce inharmonic sounds (which are aperiodic waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency. While many conventional musical instruments have harmonic partials (e.g. an oboe), some have inharmonic partials (e.g. bells). Inharmonic additive synthesis can be described as

 * $$y(t) = \sum_{k=1}^{K} r_k(t) \cos\left(2 \pi f_k t + \phi_k \right),$$

where $$f_k$$ is the constant frequency of $$k$$th partial.

Time-dependent frequencies
In the general case, the instantaneous frequency of a sinusoid is the derivative (with respect to time) of the argument of the sine or cosine function. If this frequency is represented in hertz, rather than in angular frequency form, then this derivative is divided by $$2 \pi$$. This is the case whether the partial is harmonic or inharmonic and whether its frequency is constant or time-varying.

In the most general form, the frequency of each non-harmonic partial is a non-negative function of time, $$f_k(t)$$, yielding

Broader definitions
Additive synthesis more broadly may mean sound synthesis techniques that sum simple elements to create more complex timbres, even when the elements are not sine waves. For example, F. Richard Moore listed additive synthesis as one of the "four basic categories" of sound synthesis alongside subtractive synthesis, nonlinear synthesis, and physical modeling. In this broad sense, pipe organs, which also have pipes producing non-sinusoidal waveforms, can be considered as a variant form of additive synthesizers. Summation of principal components and Walsh functions have also been classified as additive synthesis.

Implementation methods
Modern-day implementations of additive synthesis are mainly digital. (See section Discrete-time equations for the underlying discrete-time theory)

Oscillator bank synthesis
Additive synthesis can be implemented using a bank of sinusoidal oscillators, one for each partial.

Wavetable synthesis
In the case of harmonic, quasi-periodic musical tones, wavetable synthesis can be as general as time-varying additive synthesis, but requires less computation during synthesis. As a result, an efficient implementation of time-varying additive synthesis of harmonic tones can be accomplished by use of wavetable synthesis.

Group additive synthesis
Group additive synthesis  is a method to group partials into harmonic groups (having different fundamental frequencies) and synthesize each group separately with wavetable synthesis before mixing the results.

Inverse FFT synthesis
An inverse fast Fourier transform can be used to efficiently synthesize frequencies that evenly divide the transform period or "frame". By careful consideration of the DFT frequency-domain representation it is also possible to efficiently synthesize sinusoids of arbitrary frequencies using a series of overlapping frames and the inverse fast Fourier transform.

Additive analysis/resynthesis

It is possible to analyze the frequency components of a recorded sound giving a "sum of sinusoids" representation. This representation can be re-synthesized using additive synthesis. One method of decomposing a sound into time varying sinusoidal partials is short-time Fourier transform (STFT)-based McAulay-Quatieri Analysis.

By modifying the sum of sinusoids representation, timbral alterations can be made prior to resynthesis. For example, a harmonic sound could be restructured to sound inharmonic, and vice versa. Sound hybridisation or "morphing" has been implemented by additive resynthesis.

Additive analysis/resynthesis has been employed in a number of techniques including Sinusoidal Modelling, Spectral Modelling Synthesis (SMS), and the Reassigned Bandwidth-Enhanced Additive Sound Model. Software that implements additive analysis/resynthesis includes: SPEAR, LEMUR, LORIS, SMSTools, ARSS.

Products
New England Digital Synclavier had a resynthesis feature where samples could be analyzed and converted into "timbre frames" which were part of its additive synthesis engine. Technos acxel, launched in 1987, utilized the additive analysis/resynthesis model, in an FFT implementation.

Also a vocal synthesizer, Vocaloid have been implemented on the basis of additive analysis/resynthesis: its spectral voice model called Excitation plus Resonances (EpR) model is extended based on Spectral Modeling Synthesis (SMS), and its diphone concatenative synthesis is processed using spectral peak processing (SPP) technique similar to modified phase-locked vocoder (an improved phase vocoder for formant processing). Using these techniques, spectral components (formants) consisting of purely harmonic partials can be appropriately transformed into desired form for sound modeling, and sequence of short samples (diphones or phonemes) constituting desired phrase, can be smoothly connected by interpolating matched partials and formant peaks, respectively, in the inserted transition region between different samples. (See also Dynamic timbres)

Musical instruments
Additive synthesis is used in electronic musical instruments. It is the principal sound generation technique used by Eminent organs.

Speech synthesis
In linguistics research, harmonic additive synthesis was used in 1950s to play back modified and synthetic speech spectrograms.

Later, in early 1980s, listening tests were carried out on synthetic speech stripped of acoustic cues to assess their significance. Time-varying formant frequencies and amplitudes derived by linear predictive coding were synthesized additively as pure tone whistles. This method is called sinewave synthesis. Also the composite sinusoidal modeling (CSM) used on a singing speech synthesis feature on Yamaha CX5M (1984), is known to use a similar approach which was independently developed during 1966–1979. These methods are characterized by extraction and recomposition of a set of significant spectral peaks corresponding to the several resonance modes occurred in the oral cavity and nasal cavity, in a viewpoint of acoustics. This principle was also utilized on a physical modeling synthesis method, called modal synthesis.

History
Harmonic analysis was discovered by Joseph Fourier, who published an extensive treatise of his research in the context of heat transfer in 1822. The theory found an early application in prediction of tides. Around 1876, William Thomson (later ennobled as Lord Kelvin) constructed a mechanical tide predictor. It consisted of a harmonic analyzer and a harmonic synthesizer, as they were called already in the 19th century. The analysis of tide measurements was done using James Thomson's integrating machine. The resulting Fourier coefficients were input into the synthesizer, which then used a system of cords and pulleys to generate and sum harmonic sinusoidal partials for prediction of future tides. In 1910, a similar machine was built for the analysis of periodic waveforms of sound. The synthesizer drew a graph of the combination waveform, which was used chiefly for visual validation of the analysis.

Georg Ohm applied Fourier's theory to sound in 1843. The line of work was greatly advanced by Hermann von Helmholtz, who published his eight years worth of research in 1863. Helmholtz believed that the psychological perception of tone color is subject to learning, while hearing in the sensory sense is purely physiological. He supported the idea that perception of sound derives from signals from nerve cells of the basilar membrane and that the elastic appendages of these cells are sympathetically vibrated by pure sinusoidal tones of appropriate frequencies. Helmholtz agreed with the finding of Ernst Chladni from 1787 that certain sound sources have inharmonic vibration modes.

In Helmholtz's time, electronic amplification was unavailable. For synthesis of tones with harmonic partials, Helmholtz built an electrically excited array of tuning forks and acoustic resonance chambers that allowed adjustment of the amplitudes of the partials. Built at least as early as in 1862, these were in turn refined by Rudolph Koenig, who demonstrated his own setup in 1872. For harmonic synthesis, Koenig also built a large apparatus based on his wave siren. It was pneumatic and utilized cut-out tonewheels, and was criticized for low purity of its partial tones. Also tibia pipes of pipe organs have nearly sinusoidal waveforms and can be combined in the manner of additive synthesis.

In 1938, with significant new supporting evidence, it was reported on the pages of Popular Science Monthly that the human vocal cords function like a fire siren to produce a harmonic-rich tone, which is then filtered by the vocal tract to produce different vowel tones. By the time, the additive Hammond organ was already on market. Most early electronic organ makers thought it too expensive to manufacture the plurality of oscillators required by additive organs, and began instead to build subtractive ones. In a 1940 Institute of Radio Engineers meeting, the head field engineer of Hammond elaborated on the company's new Novachord as having a "subtractive system" in contrast to the original Hammond organ in which "the final tones were built up by combining sound waves". Alan Douglas used the qualifiers additive and subtractive to describe different types of electronic organs in a 1948 paper presented to the Royal Musical Association. The contemporary wording additive synthesis and subtractive synthesis can be found in his 1957 book The electrical production of music, in which he categorically lists three methods of forming of musical tone-colours, in sections titled Additive synthesis, Subtractive synthesis, and Other forms of combinations.

A typical modern additive synthesizer produces its output as an electrical, analog signal, or as digital audio, such as in the case of software synthesizers, which became popular around year 2000.

Timeline
The following is a timeline of historically and technologically notable analog and digital synthesizers and devices implementing additive synthesis.

Discrete-time equations
In digital implementations of additive synthesis, discrete-time equations are used in place of the continuous-time synthesis equations. A notational convention for discrete-time signals uses brackets i.e. $$y[n]\,$$ and the argument $$n\,$$ can only be integer values. If the continuous-time synthesis output $$y(t)\,$$ is expected to be sufficiently bandlimited; below half the sampling rate or $$f_\mathrm{s}/2\,$$, it suffices to directly sample the continuous-time expression to get the discrete synthesis equation. The continuous synthesis output can later be reconstructed from the samples using a digital-to-analog converter. The sampling period is $$T=1/f_\mathrm{s}\,$$.

Beginning with ($$),

 * $$y(t) = \sum_{k=1}^{K} r_k(t) \cos\left(2 \pi \int_0^t f_k(u)\ du + \phi_k \right)$$

and sampling at discrete times $$ t = nT = n/f_\mathrm{s} \,$$ results in

 * $$ \begin{align}

y[n] & = y(nT) = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \int_0^{nT} f_k(u)\ du + \phi_k \right) \\ & = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \sum_{i=1}^{n} \int_{(i-1)T}^{iT} f_k(u)\ du + \phi_k \right) \\ & = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \sum_{i=1}^{n} (T f_k[i]) + \phi_k \right) \\ & = \sum_{k=1}^{K} r_k[n] \cos\left(\frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k[i] + \phi_k \right) \\ \end{align} $$ where

 * $$r_k[n] = r_k(nT) \,$$ is the discrete-time varying amplitude envelope
 * $$f_k[n] = \frac{1}{T} \int_{(n-1)T}^{nT} f_k(t)\ dt \,$$ is the discrete-time backward difference instantaneous frequency.

This is equivalent to

 * $$ y[n] = \sum_{k=1}^{K} r_k[n] \cos\left( \theta_k[n] \right) $$

where

 * $$ \begin{align}

\theta_k[n] &= \frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k[i] + \phi_k \\ &= \theta_k[n-1] + \frac{2 \pi}{f_\mathrm{s}} f_k[n] \\ \end{align} $$ for all $$n>0\,$$ and
 * $$ \theta_k[0] = \phi_k. \,$$

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title: Additive synthesis