Syntonic Comma

The ratio 81/80, in JustMusic prime-factor notation designated as 3⁴5^-1, reducible to 1:1.0125, with an interval size of 0.22 Semitones [= ~21.5062896 (~21¹/₂) cents].

It is the difference between the Pythagorean or 3-Limit "major 3rd" of 4.08 Semitones [= 81/64 = 3⁴ = ~407.8200035 cents] and the 5-limit "just" "major 3rd" of 3.86 Semitones [= 5/4 = 5¹ = ~386.3137139 cents] :

  -- prime-factor vector --
    2  3  5                    ratio    ~cents

  [-6  4  0]  =  2-6 34        81/64   407.8200035
- [-2  0  1]  =  2-2 51         5/4    386.3137139
------------
  [-4  4 -1]  =  2-4 34 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.

Below are some other interval measurements for the syntonic comma:

   ~0.215062896  (~2/9) Semitone
   ~1.290377376  (~1 2/7) 72edo-moria
   ~5.376572399  (~5 3/8) savarts
  ~11.00065477  (~11) grads
  ~13.08299284  (~13 1/12) 730edo Woolhouse-units
  ~17.921908  (~17 59/64) milli8ves
  ~18.99722248  (~18 359/360) Türk cents
  ~21.5062896  (~21 1/2) cents
 ~539.5031964  (~539 1/2) jots
 ~660.0392862  (~660) tuning units
 ~880.8976219  (~880 8/9) cawapus
~3523.590488  (~3523 3/5) midipus

[from Joe Monzo, JustMusic: A New Harmony]

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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:



[from Brian McLaren]


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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).

[from John Chalmers, Divisions of the Tetrachord]

see also comma pump, comma