136/135

Interval information

Ratio	136/135
Factorization	23 × 3-3 × 5-1 × 17
Monzo	[3 -3 -1 0 0 0 1⟩
Size in cents	12.77669¢
Names	diatisma, diatic comma, fiventeen comma
Color name	17og2, Sogu 2nd, Sogu comma
FJS name	[math]\displaystyle{ \text{d2}^{17}_{5} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	14.1643
Weil norm (log2 max(n, d))	14.1749
Wilson norm (sopfr(nd))	37
Comma size	small
S-expression	S16⋅S17
Open this interval in xen-calc

136/135, the diatisma, diatic comma or fiventeen comma, is a small 17-limit comma. It is the interval that separates 17/10 and 27/16 (or their octave complements 20/17 and 32/27) and that separates 30/17 and 16/9 (or their octave complements 17/15 and 9/8). It is also the difference between 16/15 and 18/17 with an S-expression of S16⋅S17 or ((16/15)⋅(17/16))/((17/16)⋅(18/17)).

Temperaments

Tempering out this comma in the full 17-limit results in the rank-6 diatismic temperament, or in the 2.3.5.17 subgroup, the rank-3 diatic temperament.

Since 136/135 = (225/224)⋅(256/255), it would make sense to temper out both 256/255 (S16) and 289/288 (S17), thereby tempering diatic to srutal archagall, which is equivalently described as "charic semitonic". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth.

Fiventeen

In fiventeen, 17/15 is equated with 9/8, so it implies a supersoft pentic pentad of ~30:34:40:45:51. 17edo makes a good tuning especially for its size, which gives a supersoft pentic scale corresponding approximately to a just 20/17 tuning, although 80edo might be preferred for an approximately just 51/40 to optimize plausibility slightly more, and 97edo (= 80 + 17) and 114edo (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the optimal ET sequence is much more characteristic of optimized tunings, finding 34edo, then 80edo, then 114edo (= 34 + 80) and even 194bc-edo (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting 63edo and 143edo (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

Subgroup: 2.3.17/5

Subgroup-val mapping: [⟨1 0 -3], ⟨0 1 3]]

mapping generators: ~2, ~3

Optimal tunings:

• Subgroup WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢
• Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢

Optimal ET sequence: 5, 12, 17, 46, 63, 143

Diatic

Subgroup: 2.3.5.17

Subgroup-val mapping: [⟨1 0 0 -3], ⟨0 1 0 3], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~5

Optimal tunings:

• WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢, ~5/4 = 389.0228 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢, ~5/4 = 388.6162 ¢

Optimal ET sequence: 10, 12, 22, 34, 80, 114, 194bc

Badness (Sintel): 0.139

Diatismic

The only edo tuning that has less than 25% relative error for all primes in the 17-limit tempering out 136/135 is 46edo, which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then 80edo makes for a good and more accurate tuning. Alternatively, if you do not care as much about prime 11, 68edo makes for a great tuning.

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨	1	0	0	0	0	0	-3	],
⟨	0	1	0	0	0	0	3	],
⟨	0	0	1	0	0	0	1	],
⟨	0	0	0	1	0	0	0	],
⟨	0	0	0	0	1	0	0	],
⟨	0	0	0	0	0	1	0	]]

mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

• WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢, ~5/4 = 389.0228 ¢, ~7/4 = 970.2512 ¢, ~11/8 = 553.4578 ¢, ~13/8 = 842.6669 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢, ~5/4 = 388.6162 ¢, ~7/4 = 969.9161 ¢, ~11/8 = 552.6614 ¢, ~13/8 = 841.9647 ¢

Optimal ET sequence: 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef *

* optimal patent val: 177

Badness (Sintel): 1.15

Etymology

The name of this comma was formerly diatonisma, suggested by Xenllium in 2023, but this name would imply a problematic "diatonic" subgroup temperament. Therefore diatisma, a shortenage of diatonisma, and fiventeenisma a portmanteau of five and seventeen for its relation to a chord involving primes 5 and 17, were proposed by Godtone in 2024. The name fiventeen was soon given to the rank-2 2.3.17/5-subgroup temperament, and hence the name fiventeenisma became just fiventeen comma.

See also

• List of superparticular intervals