736/729

Interval information

Ratio	736/729
Subgroup monzo	2.3.23 [5 -6 1⟩
Size in cents	16.54434¢
Name	23-limit Tenney/Cage comma (HEJI)
Color name	s23o2, satwetho 2nd
FJS name	[math]\text{P1}^{23}[/math]
Special properties	reduced
Tenney height (log2 nd)	19.0333
Weil height (log2 max(n, d))	19.0471
Wilson height (sopfr(nd))	51
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~3.23528 bits
Comma size	small
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736/729, the 23-limit Tenney/Cage comma, is a 2.3.23 subgroup comma. It is the amount by which the octave-reduced 23rd harmonic 23/16 exceeds the Pythagorean augemented fourth (729/512).

Notation

This interval is significant in the Functional Just System and Helmholtz-Ellis notation as the formal comma to translate a Pythagorean interval to a nearby 23-limit (vicesimotertial) interval. The symbols being used in Helmholtz-Ellis notation are virtually identical to up and down arrows, and the authors attribute them to James Tenney and John Cage, who have possibly used them for 1\72.

Sagittal notation

In the Sagittal system, this comma (possibly tempered) is represented by the sagittal ⁠ ⁠ and is called the 23 comma, or 23C for short, because the simplest interval it notates is 23/1 (equiv. 23/16), as for example in F-B⁠ ⁠⁠ ⁠. The downward version is called 1/23C or 23C down and is represented by ⁠ ⁠.