715/714

Interval information

Ratio	715/714
Factorization	2-1 × 3-1 × 5 × 7-1 × 11 × 13 × 17-1
Monzo	[-1 -1 1 -1 1 1 -1⟩
Size in cents	2.423001¢
Names	september comma, septembrisma
Color name	17u3o1ory-2, sutholoruyo negative 2nd
FJS name	[math]\displaystyle{ \text{m}{-2}^{5,11,13}_{7,17} }[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	18.9616
Weil height (log2 max(n, d))	18.9636
Wilson height (sopfr(nd))	58
Harmonic entropy (Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])	~1.25375 bits
Comma size	unnoticeable
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715/714, the september comma or septembrisma is a 17-limit unnoticeable superparticular comma with a size of approximately 2.42 cents. It is also described as the septendecimal bridge comma. It is the gap between:

• the subfourths 17/13 and 55/42
• the superminor sixths 21/13 and 55/34
• the semitones 68/65 and 22/21.

Commatic relations

This comma appears as the difference between the following superparticular pairs:

• 120/119 and 144/143
• 154/153 and 196/195
• 273/272 and 441/440
• 325/324 and 595/594
• 385/384 and 833/832
• 442/441 and 1156/1155
• 561/560 and 2601/2600
• 676/675 and 12376/12375

not to mention some nonsuperparticular but useful relations:

• 375/374 and 1575/1573

It factors into the following superparticular pairs:

• 1225/1224 and 1716/1715
• 1089/1088 and 2080/2079
• 1001/1000 and 2500/2499
• 936/935 and 3025/3024
• 729/728 and 37180/37179

not to mention some nonsuperparticular but useful relations:

• 2025/2023 and 2431/2430
• 2200/2197 and 28561/28560

Temperaments

Tempering out this comma in the 17-limit gives the rank-6 september temperament, and enables the related essentially tempered chords.

September

Subgroup: Full 17-limit

Comma basis: 715/714

Mapping: [⟨1 0 0 0 0 0 -1], ⟨0 1 0 0 0 0 -1], ⟨0 0 1 0 0 0 1], ⟨0 0 0 1 0 0 -1], ⟨0 0 0 0 1 0 1], ⟨0 0 0 0 0 1 1]]

Optimal tuning (POTE): ~2 = 1＼1, ~3/2 = 701.9930, ~5/4 = 385.9978, ~7/4 = 969.0337, ~11/8 = 550.6860, ~13/8 = 839.8152

Optimal ET sequence: 19, 26, 31, 41, 46, 68, 72, 103, 111, 121, 140, 183, 224, 270, 364, 436, 547, 843

Etymology

The september comma was named by Scott Dakota in 2021; however, Scott Dakota has disavowed this name.