Hemimage comma

Interval information

Ratio	10976/10935
Factorization	25 × 3-7 × 5-1 × 73
Monzo	[5 -7 -1 3⟩
Size in cents	6.478999¢
Name	hemimage comma
Color name	sz3g3, Satrizo-agu comma
FJS name	[math]\displaystyle{ \text{dd3}^{7,7,7}_{5} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	26.8387
Weil norm (log2 max(n, d))	26.8441
Wilson norm (sopfr(nd))	57
Comma size	small
S-expression	S282⋅S29
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The hemimage comma (ratio: 10976/10935), is a small 7-limit comma measuring about 6.48 cents. It marks the difference between a classic diatonic semitone (16/15) and a stack of three septimal major thirds (9/7) octave reduced, or between a classic whole tone (10/9) and a stack of three septimal third tones (28/27), therefore interesting to those who work extensively with third tones.

In terms of other commas, it is the difference between 245/243 and 225/224, the two simplest commas to define the 7-limit magic temperament. It factors into simpler commas as the breedsma and the aberschisma, (2401/2400)⋅(5120/5103), or the hemimean comma and the ragisma, (3136/3125)⋅(4375/4374).

In the 19-limit it factors into (343/342)⋅(1216/1215), and in the 29-limit, (406/405)⋅(784/783), which equals (784/783)²⋅(841/840) in turn, making it a lopsided comma with S-expression S28²⋅S29.

Temperaments

Tempering out this comma alone in the 7-limit leads to the rank-3 hemimage temperament.

See Hemimage family for the rank-3 family where it is tempered out. See Hemimage temperaments for a collection of rank-2 temperaments where it is tempered out.

Etymology

This comma was first named as parahemfi by Gene Ward Smith in 2005 as a contraction of parakleismic and hemififths. It is not clear how it later became hemimage, but the root of hemimage is obvious, being a contraction of hemififths and magic.