[Joe Monzo]

The musical frequency ratio 81/80, in prime-factor notation designated as 2^{-4}3^{4}5^{-1} or in 2,3,5-monzo form as [-4 4, -1>, expressed in decimal form as exactly 1.0125. Its logarithmic interval size is 0.22 Semitones = ~21.5062896 (~21 ^{1}/_{2}) cents.

The syntonic-comma is the difference between the pythagorean (i.e., 3-Limit) "major 3rd" (the "ditone") of 4.08 Semitones [= 81/64 = 3^{4} = ~407.8200035 cents] and the 5-limit "just" "major 3rd" of 3.86 Semitones [= 5/4 = 5^{1} = ~386.3137139 cents] :

2,3,5-monzo ratio ~cents [ -6 4, 0 > = 2^{-6}3^{4}81/64 407.8200035 - [ -2 0, 1 > = 2^{-2}5^{1}5/4 386.3137139 --------------- [ -4 4, -1 > = 2^{-4}3^{4}5^{-1}81/80 21.5062896

Below is a 2-dimensional Monzo lattice illustrating the [3,5] components of the prime-factorization of one example of the syntonic comma:

Also known as the "Comma of Didymus", who was the first theorist to specify the use of 5/4 in music theory; frequently referred to simply as "comma".

The syntonic-comma is the interval tempered out in all tunings of the meantone family. In the most direct example, 1/4-comma meantone narrows each 5th in the chain-of-5ths by 1/4 of the syntonic-comma, so that four 5ths upward, octave-reduced, equal the 5-limit major-3rd. All other varieties of meantone temper the 5th narrow by a different fraction of the syntonic-comma, so that the major-3rd is also tempered -- but the syntonic-comma still vanishes in all these tunings, so that the equation of four 5ths = major-3rd is still true.

Below are some other interval measurements for the syntonic comma:

~0.215062896 (~^{2}/_{9}) Semitone ~1.290377376 (~1^{2}/_{7}) 72-edo-moria ~5.376572399 (~5^{3}/_{8}) savarts ~11.00065477 (~11) grads ~13.08299284 (~13^{1}/_{12}) 730-edo Woolhouse-units ~17.921908 (~17^{59}/_{64}) millioctaves ~18.99722248 (~18^{359}/_{360}) Türk-sents ~21.5062896 (~21^{1}/_{2}) cents ~539.5031964 (~539^{1}/_{2}) jots ~660.0392862 (~660) temperament-units ~880.8976219 (~880^{8}/_{9}) 12mus ~3523.590488 (~3523^{3}/_{5}) 14mus

. . . . . . . . .

[Brian McLaren]

The syntonic comma is the most important of the musical commas for the notation of musical intervals. While the pythagorean comma aids in the construction and analysis of tuning systems, the Syntonic comma defines the way thirds are notated in chords.

The syntonic comma is defined as the difference between a pythagorean major third (made from the difference twixt 4 just 3/2 perfect fifths up and 3 octaves) and a 5/4 major third:

. . . . . . . . .

[John Chalmers, Divisions of the Tetrachord]

A small interval whose ratio is 81/80 (21.5 cents [¢]).

It is the difference between the Ditone and the ratio 5/4, the Major Third of Just Intonation.

Also called the "Comma of Didymos" (latinized as Didymus).

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)