36/35

Interval information

Ratio	36/35
Factorization	22 × 32 × 5-1 × 7-1
Monzo	[2 2 -1 -1⟩
Size in cents	48.77038¢
Names	septimal quartertone, mint comma
Color name	rg1, rugu unison
FJS name	[math]\displaystyle{ \text{P1}_{5,7} }[/math]
Special properties	square superparticular, reduced
Tenney norm (log2 nd)	10.2992
Weil norm (log2 max(n, d))	10.3399
Wilson norm (sopfr(nd))	22
Comma size	medium
S-expressions	S6, S8⋅S9
[sound info]
Open this interval in xen-calc

English Wikipedia has an article on:

Septimal quarter tone

36/35, the septimal quartertone is a 7-limit interval with a size of about 48.8 cents. It arises as the difference between various 7-limit qualities of seconds, thirds, sixths, and sevenths: between supermajor and classical major, and between subminor and classical minor. These are the pairs of intervals separated by 36/35:

• 28/27 and 16/15
• 10/9 and 8/7
• 7/6 and 6/5
• 5/4 and 9/7
• 14/9 and 8/5
• 5/3 and 12/7
• 7/4 and 9/5
• 15/8 and 27/14

It has a numerator which is both the sixth square number and the eighth triangular number, leading to it being the product of two superparticular commas both as (64/63)⋅(81/80) and as (66/65)⋅(78/77); it is also (45/44)⋅(176/175), (51/50)⋅(120/119), (128/125)⋅(225/224), (50/49)⋅(126/125) and (56/55)⋅(99/98).

Temperaments

When treated as a generator, it is almost a fourth of the septimal whole tone 28/25, differing by a wizma. This allows it to be used as a generator for a temperament of hemimean called sengagen, where 5/4, 7/5 and 7/4 map to 8, 12 and 20 quartertones respectively.

When treated as a comma to be tempered out, it is known as the mint comma, and tempering it out leads to the mint temperament. See Mint family for the family of rank-3 temperaments where it is tempered out, and Mint temperaments for the collection of rank-2 temperaments where it is tempered out.

Etymology

The name mint comma was given by Mike Battaglia in 2012, for minor third because "it mixes 7/6 and 6/5 together into one minty interval". Before that, it had been known as the quartonic comma, which refers to another comma today.

Notation

Ben Johnston's notation

In Ben Johnston's notation, this interval is denoted with "7" (a turned "7"), and the reciprocal 35/36 with an ordinary "7". If the base note is C, then 7/4 is reprented by C–Bb7.

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal  and is called the 35 medium diesis, or 35M for short, because the simplest interval it notates is 35/1 = 5×7 (equiv. 35/16), as for example in C-D⁠ ⁠. The upward version is called 1/35M or 35M up and is represented by .

See also

• 35/18 – its octave complement
• 35/24 – its fifth complement
• Gallery of just intervals
• List of superparticular intervals