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Syntonic comma

The syntonic comma, also known as the comma of Didymus, is a small musical interval in just intonation theory, defined by the frequency ratio of 81:80, which corresponds to approximately 21.5 cents.^[1] This interval represents the precise difference between the Pythagorean major third, tuned as 81/64 (about 407.8 cents), and the just major third, tuned as 5/4 (about 386.3 cents).^[2] In tuning systems, the syntonic comma plays a central role as the interval that is systematically tempered out to achieve greater consonance in thirds and sixths, particularly in meantone temperaments such as quarter-comma meantone, where each perfect fifth is narrowed by one-fourth of the comma to close the circle of fifths harmoniously.^[3] Originating from ancient Greek music theory, it was first explicitly referenced by the theorist Didymus in the 1st century AD through his adoption of the 5/4 ratio for the major third, distinguishing it from the purely fifth-based Pythagorean scale.^[4] The comma's resolution allows for the inflection of notes in tetrachords, transforming Pythagorean semitones (256:243) into more consonant just semitones (16:15), and it remains a foundational concept in microtonal and historical performance practices.^[3]

Definition and Properties

Definition

The syntonic comma is a small musical interval that measures the discrepancy between the Pythagorean major third, expressed as the frequency ratio 81/64, and the just major third, with a ratio of 5/4.^[5] This difference arises when constructing intervals using pure fifths versus incorporating the simpler harmonic ratio involving the number 5.^[6] The interval itself has a frequency ratio of 81/80.^[5] Also referred to as the chromatic diesis, Ptolemaic comma, or Didymian comma, the syntonic comma measures approximately 21.5 cents.^[7]^[8] This size makes it a subtle but noticeable discrepancy, particularly in melodic lines where it can introduce perceptible tension, though it is less prominent in dense harmonic textures.^[9] In tuning theory, the syntonic comma plays a fundamental role in addressing inconsistencies between interval constructions, enabling more consonant major triads by highlighting and compensating for the offset between Pythagorean and just intonation approaches to thirds.^[5]^[9]

Mathematical Calculation

The syntonic comma arises as the interval between the Pythagorean major third, with ratio \frac{81}{64}, and the just major third, with ratio \frac{5}{4}. To derive its ratio, divide the Pythagorean third by the just third:
\frac{81/64}{5/4} = \frac{81}{64} \times \frac{4}{5} = \frac{81}{80}.
This yields the syntonic comma's frequency ratio of \frac{81}{80} \approx 1.0125.^[10] Equivalently, the comma can be obtained from the ratio between the two types of major seconds in 5-limit just intonation: the Pythagorean major second \frac{9}{8} and the harmonic major second \frac{10}{9}. Their ratio is
\frac{9/8}{10/9} = \frac{9}{8} \times \frac{9}{10} = \frac{81}{80},
where two \frac{10}{9} intervals produce a just major third of \frac{100}{81}, differing from \frac{5}{4} by the comma.^[11] To quantify the comma in cents, use the formula for converting a frequency ratio r to cents: $1200 \times \log_2(r). For r = \frac{81}{80}, first compute the base-2 logarithm:
\log_2\left( \frac{81}{80} \right) = \log_2(81) - \log_2(80) = 4\log_2(3) - (4\log_2(2) + \log_2(5)) = 4\log_2(3) - 4 - \log_2(5),
where \log_2(3) \approx 1.5849625 and \log_2(5) \approx 2.321928. Substituting gives
\log_2\left( \frac{81}{80} \right) \approx 4(1.5849625) - 4 - 2.321928 = 6.33985 - 4 - 2.321928 = 0.017922.
Then,
$1200 \times 0.017922 \approx 21.506
cents.^[9]^[12] In the context of the harmonic series, the syntonic comma emerges from the discrepancy between intervals generated by stacking pure perfect fifths of ratio \frac{3}{2} and those derived from natural harmonic thirds. Four stacked fifths yield \left( \frac{3}{2} \right)^4 = \frac{81}{16}, equivalent to two octaves plus a Pythagorean major third of \frac{81}{64} after reducing by $2^2 = 4. This Pythagorean third exceeds the natural just major third of \frac{5}{4} (from the fifth and third harmonics) by the comma \frac{81}{80}.^[13] The prime factorization of the syntonic comma's ratio provides insight into its tuning properties: $81 = 3^4 and $80 = 2^4 \times 5^1, so \frac{81}{80} = 2^{-4} \times 3^{4} \times 5^{-1}. In Monzo arrow notation, using exponents for the primes 2, 3, and 5, this is represented as |-4\ 4\ -1\rangle.^[13]

Relationships to Intervals and Tunings

Pythagorean and Just Intonation Comparisons

The syntonic comma emerges as a key discrepancy between Pythagorean and just intonation, particularly in the construction of major thirds. In Pythagorean tuning, the major third is formed by stacking two whole tones, each with a frequency ratio of \frac{9}{8}, resulting in a combined ratio of \frac{81}{64}.^[14] By contrast, the just major third derives from the fifth harmonic, yielding a simpler ratio of \frac{5}{4}.^[5] The syntonic comma is precisely this difference, expressed as the ratio \frac{81/64}{5/4} = \frac{81}{80}, representing the amount by which the Pythagorean interval exceeds the just one.^[15] This interval mismatch also manifests in the arithmetic of stacked fifths. A sequence of four just perfect fifths, each at \frac{3}{2}, produces \left( \frac{3}{2} \right)^4 = \frac{81}{16}. This chain spans two octaves plus a major third, equivalent to $2^2 \times \frac{5}{4} = 5. The resulting excess is \frac{81/16}{5} = \frac{81}{80}, again isolating the syntonic comma as the reconciling factor.^[16] These differences have profound implications for diatonic scale construction. Pythagorean tuning maintains pure fifths throughout the scale but produces thirds that are perceptibly sharper than just intonation's consonant \frac{5}{4}, leading to tension in harmonic progressions.^[17] In a full chromatic Pythagorean scale, the circle of 12 fifths fails to close perfectly, necessitating a "wolf fifth"—typically between G and E♭—which is flattened and dissonant to accommodate the accumulated comma.^[18] Just intonation, however, prioritizes purity in both fifths and thirds using small-integer ratios, fostering greater harmonic consonance within a given key, though it limits free modulation across keys without retuning.^[19] A concrete example illustrates this adjustment: the C-to-E interval in Pythagorean tuning measures \frac{81}{64}, while the just version is \frac{5}{4}; narrowing the Pythagorean E by the syntonic comma \frac{81}{80} aligns it precisely with the just intonation.^[20]

Role in Meantone and Equal Temperament

In quarter-comma meantone temperament, the syntonic comma is tempered out by flattening each of the twelve perfect fifths by one quarter of the comma, or approximately 5.38 cents, which yields pure major thirds with a just intonation ratio of 5:4 (386.31 cents).^[21] This adjustment addresses the discrepancies in Pythagorean tuning by distributing the comma's influence evenly across the circle of fifths, prioritizing consonant triads in polyphonic music while introducing a "wolf" fifth between G♯ and E♭ to close the cycle.^[22] In 12-tone equal temperament (12-TET), the syntonic comma is distributed across the entire octave by dividing it into twelve equal semitones of 100 cents each, resulting in major thirds that are approximately 13.69 cents sharp relative to their just intonation value of 386.31 cents.^[23] This tempering equalizes all intervals, eliminating the need for wolf notes and enabling modulation across all keys, though at the cost of slightly wide thirds that produce audible beating in dense harmonies.^[24] The schisma, defined as the difference between the Pythagorean comma (23.46 cents) and the syntonic comma (21.51 cents), measures approximately 1.95 cents and plays a key role in refining meantone variants.^[21] In systems like 31-equal temperament, which approximates quarter-comma meantone, the schisma represents the residual discrepancy in fifth sizes, allowing for near-pure intervals while extending the scale beyond 12 tones without significant comma drift.^[25] The size of the syntonic comma facilitates enharmonic equivalents in keyboard temperaments by enabling close approximations between nominally distinct pitches, such as F♯ and G♭, which differ by a schisma in quarter-comma meantone.^[21] This small offset supports fixed-pitch instruments like harpsichords, where tempering absorbs the comma to make enharmonics functionally interchangeable in most keys, though with subtle timbral variations in extreme registers.^[26]

Historical Development

Ancient and Medieval Contexts

The syntonic comma traces its origins to the 1st century AD with Didymus the Musician, who proposed tuning the major third to the just ratio of 5/4, differing from the Pythagorean major third of 81/64 by the small interval of 81/80, thereby naming and defining this comma as a correction for greater consonance.^[27] This adjustment, known as the comma of Didymus, addressed the sharper Pythagorean third to align better with harmonic principles derived from the natural overtone series.^[28] In the following century, Claudius Ptolemy expanded on these ideas in his Harmonics, establishing the syntonic diatonic scale as a systematic framework where the major third is precisely 5/4, and the syntonic comma functions as the key interval for chromatic semitone adjustments within tetrachords.^[29] Ptolemy's approach positioned the inner notes of tetrachords—such as the second and third degrees—a full syntonic comma higher than their Pythagorean counterparts, promoting a tuning that prioritized pure thirds over strict fifth-based stacking.^[29] During the medieval period, Boethius played a pivotal role in transmitting ancient Greek music theory to the Latin West through his De institutione musica (c. 510 AD), which emphasized Pythagorean tuning while preserving elements of earlier diatonic systems that implicitly acknowledged discrepancies like the syntonic comma.^[30] This transmission influenced subsequent theorists, including Guido d'Arezzo (c. 991–1033), whose solmization system based on the hexachord facilitated practical intonation in monophonic chant, indirectly highlighting the need for adjustments to resolve the dissonant thirds inherent in Pythagorean scales used for Gregorian chant.^[31] Early awareness of the comma emerged in this context as a means to mitigate triadic dissonances when rudimentary polyphony began to incorporate harmonic intervals beyond pure fifths and fourths.^[9]

Renaissance to Modern Periods

During the Renaissance, the syntonic comma played a pivotal role in the evolution of tuning practices for polyphonic music, as composers and theorists sought to achieve more consonant major thirds in vocal and instrumental works. By the mid-16th century, quarter-comma meantone tuning emerged as a standard for keyboard instruments, tempering eleven fifths narrow by one-quarter of the syntonic comma (approximately 5.4 cents each) to produce purer major thirds closer to the just intonation ratio of 5:4, while the remaining fifth (typically between G♯ and E♭) was widened to close the circle of fifths.^[32] This approach addressed the "wolf" interval inherent in earlier Pythagorean tuning, where the comma caused dissonant thirds, and facilitated the rich harmonic textures in polyphony. Works by Josquin des Prez (c. 1450–1521) and Giovanni Pierluigi da Palestrina (c. 1525–1594), such as Josquin's Missa L'Homme Armé and Palestrina's masses, benefited from performances in environments approximating meantone, enhancing chordal consonance through tempered thirds that mitigated comma-induced discrepancies.^[33] In the Baroque era, the syntonic comma influenced irregular well temperaments that distributed its size across multiple intervals, enabling modulation across all keys with varied character. Andreas Werckmeister's temperaments (1681–1698), such as Werckmeister III, partially absorbed the comma by tempering four fifths narrow by one-quarter syntonic comma each, resulting in major thirds that were close to just but varied by key, providing a "circulating" system for organ and harpsichord.^[34] Similarly, Johann Philipp Kirnberger's temperaments (1760s–1770s), particularly Kirnberger III, divided the syntonic comma equally among four fifths (C–G, G–D, D–A, A–E), yielding pure major thirds in C, G, D, and A major while tempering others slightly, reflecting his emphasis on just intonation principles derived from J.S. Bach.^[35] These systems informed Johann Sebastian Bach's The Well-Tempered Clavier (1722, 1742), where the comma's partial tempering allowed exploration of 24 keys with distinct tonal colors, as the unequal fifths created subtle variations in thirds that enhanced contrapuntal expressivity without the extremes of meantone's wolf fifth.^[36] The 19th and 20th centuries saw the widespread adoption of 12-tone equal temperament (12-TET), which evenly distributes the syntonic comma's effect across the 12 semitones, flattening all fifths by approximately 1/12 of a Pythagorean comma (close to the syntonic due to the small schisma difference), resulting in major thirds of 400 cents—14 cents wider than just intonation. This standardization, solidified in orchestral practice by the mid-19th century through figures like Helmholtz and facilitated by piano manufacturing, eliminated comma-specific wolves but approximated just intervals uniformly, influencing composers from Beethoven to Debussy.^[37] Paul Hindemith, in his theoretical works like The Craft of Musical Composition (1937), advocated returning to just intonation for modern composition, emphasizing the syntonic comma's role in deriving a 12-tone scale from 5-limit intervals and critiquing 12-TET's compromises while proposing hybrid systems for acoustic purity in ensemble settings.^[38] In contemporary music, the syntonic comma informs microtonal explorations that extend beyond 12-TET, particularly in just intonation systems addressing historical comma drifts. Harry Partch's 43-tone scale (developed 1923–1940s), an 11-limit just intonation gamut, incorporates the comma as a fundamental interval (81/80, 21.5 cents) within its tonality diamonds, allowing precise adjustments for otonal/utonal symmetries and avoiding Pythagorean comma accumulations in extended progressions, as used in works like Revelation in the Courthouse Park (1960).^[39] Digital tuning software, such as Scala and Alt-Tuner, simulates these comma-based temperaments, enabling composers to model Renaissance meantone, Baroque well systems, or microtonal scales in real-time, facilitating experimental compositions that highlight the comma's acoustic impact through algorithmic retuning of MIDI and synthesizers.

Practical Applications

Comma Pump Mechanism

The comma pump mechanism refers to a repeating sequence of intervals or chords in just intonation that accumulates the syntonic comma (81/80, approximately 21.5 cents), causing a progressive pitch drift that highlights discrepancies between pure intervals like perfect fifths and major thirds.^[40] A representative sequence demonstrating this is an ascending cycle of seven perfect fifths in [Pythagorean tuning](/page/Pythagorean tuning), such as C–G–D–A–E–B–F♯–C♯, where the final C♯ lands sharp by 81/80 relative to the expected just position after octave reductions.^[41] This drift occurs because the Pythagorean fifth (3/2) stacked seven times yields 2187/2048 after reducing four octaves, exceeding the just minor second by the syntonic comma factor.^[40] In 1563, Italian mathematician Giovanni Battista Benedetti illustrated the comma pump in two letters to composer Cipriano de Rore, using a violin to perform a repeating two-measure chord progression that forced adjustments for pure intervals.^[42]^[43] Benedetti's example involved a bottom voice starting on C, tied to an A in the top voice forming a pure major sixth (5/3) or minor third (6/5), requiring subsequent notes like G to be raised for consonance, resulting in an upward drift by one syntonic comma per repetition; the letters, published in 1585, emphasized how this made consistent intonation impossible on violin without fixed tuning.^[42]^[44] Mathematically, each cycle of the pump introduces a factor of 81/80 to the overall pitch, so after n cycles, the cumulative shift is (81/80)^n relative to the initial tuning, equivalent to modulating the sequence by successive syntonic commas.^[43] This accumulation can be expressed in terms of stacked fifths over full loops, where n repetitions of twelve fifths yield a total interval of (3/2)^{12n} / 2^{7n} adjusted by syntonic comma factors to account for third-based deviations in just intonation.^[40] Auditorily, the comma pump reveals tuning inconsistencies when the sequence is played as a melodic or harmonic chain on instruments like the violin, where the gradual sharpening becomes perceptible after several repetitions, producing beats or dissonance that underscore the need for tempered adjustments in extended music.^[42] In Benedetti's violin demonstration, the drift manifests as an unintended crescendo in pitch, challenging performers to either accept the shift or compromise interval purity.^[43]

Notation Systems

In traditional meantone notation, standard accidentals such as the flat (♭) and natural (♮) are used to denote pitch adjustments that incorporate fractions of the syntonic comma, typically lowering or raising notes relative to Pythagorean intonation to achieve consonant major thirds in the tuning system.^[45] For instance, in quarter-comma meantone, the ♭ symbol on E results in a pitch approximately 386 cents above C, tempering the fifths by one-quarter of the syntonic comma to align with the just major third (5:4).^[45] The Helmholtz-Ellis system extends traditional accidentals with upward and downward arrows to specify microtonal deviations, where each arrow alters the pitch by one syntonic comma, approximately ±21.5 cents, from a Pythagorean baseline.^[46] An upward arrow on a sharp (e.g., F♯↑) raises the note by the ratio 81:80, correcting Pythagorean intervals toward just intonation equivalents like the major third (5:4), while a downward arrow (e.g., D↓) lowers it accordingly.^[46] This notation, formalized in the extended Helmholtz-Ellis JI pitch system, supports precise representation in scores for compositions involving just intonation or meantone-derived scales.^[46] Ben Johnston's notation system for just intonation employs plus (+) and minus (-) symbols to indicate alterations by the syntonic comma (81:80), ensuring pitches align with rational ratios in 5-limit tuning.^[47] The + symbol raises a note by 81:80, as in F♯+ to form the just major third above D (5:4), while the - symbol lowers it, such as on B♭- to achieve the perfect fourth (4/3) above F; multiple symbols accumulate the effect for complex harmonies.^[48] Designed for extended just intonation compositions, this system builds on the diatonic scale, using these accidentals to notate triads and intervals without reference to equal temperament.^[47] In modern digital notation, the syntonic comma is encoded via MIDI Tuning Standard (MTS) tables and software like Scala, which define custom scales using the 81:80 ratio for precise microtonal playback.^[49] Scala's .scl files specify note deviations in cents or ratios, such as tempering fifths by fractions of 81/80 for meantone tunings, and export to MIDI for real-time synthesis; for example, a scale file might list the comma as 21.51 cents to integrate just intonation intervals into electronic music production tools like synthesizers or DAWs.^[49] This approach enables composers to apply comma adjustments dynamically across channels, bridging symbolic notation with computational tuning in contemporary works.^[49]

References

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[48]
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... MIDI Tuning Standard real-time single note tuning changes when translating sequence files. ... For instance when the syntonic comma (81/80 = 34.5-1) is given, the ...