136/135

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Interval information
Ratio 136/135
Factorization 23 × 3-3 × 5-1 × 17
Monzo [3 -3 -1 0 0 0 1
Size in cents 12.776693¢
Names diatisma,
diatic comma,
fiventeen comma
Color name 17og2, Sogu 2nd,
Sogu comma
FJS name [math]\text{d2}^{17}_{5}[/math]
Special properties superparticular,
reduced
Tenney height (log2 nd) 14.1643
Weil height (log2 max(n, d)) 14.1749
Wilson height (sopfr (nd)) 37
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~3.05028 bits
Comma size small
S-expression S16 × S17
open this interval in xen-calc

136/135, the diatisma, diatic comma or fiventeen comma, is a 17-limit small comma. It is equal to (32/27)/(20/17) and also (51/40)/(81/64). It is also trivially the difference between 16/15 and 18/17 and therefore the difference between (17/16)(16/15) = 17/15 and (18/17)(17/16) = 9/8 as the two 17/16's cancel.

Temperaments

Fiventeen

17edo makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a supersoft pentic pentad of 30:34:40:45:51:60 (because as aforementioned 17/15 is equated with 9/8) although 80edo might be preferred for a more accurate 51/40 and it and 46edo might be preferred for more accurate fifths. The same is true of the related rank 3 temperament diatic, described below.

Subgroup: 2.3.17/5

Sval mapping[1 0 -3], 0 1 3]]

sval mapping generators: ~2, ~3

Optimal tuning (subgroup CTE): ~2 = 1\1, ~3/2 = 704.1088

Optimal ET sequence5, 12, 17, 46, 63, 143

Diatic

Subgroup: 2.3.5.17

Sval mapping[1 0 0 -3], 0 1 0 3], 0 0 1 1]]

sval mapping generators: ~2, ~3, ~5

Optimal tuning (subgroup CTE): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544

Optimal ET sequence10, 12, 22, 34, 80, 114, 194bc

Diatismic

The only edo tuning that has less than 25% relative error for all primes in the 17-limit tempering 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 a good and more accurate tuning that extends to the 23-limit. Alternatively, if you don't care (as much) about prime 11, 68edo makes a great tuning in the no-11's 19-limit and no-11's no-29's 31-limit.

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

Optimal tuning (subgroup CTE): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8

Optimal ET sequence22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef*

* optimal patent val: 177

Srutal archagall

Srutal archagall is an efficient rank-2 temperament tempering out both S16 and S17, which is equivalently described as charic semitonic due to the fact that {S16 × S17 , S16/S17} = {S16, S17}

Etymology

The name was formerly diatonisma, suggested by User:Xenllium in 2023, but this name has strong reasons against it due to implying an ambiguously-named "diatonic" subgroup temperament. Therefore fiventeenisma and diatisma were proposed. However, due to the need for a separate name for the rank 2 2.3.17/5 subgroup temperament and due to its relation to the chord (see Talk:136/135), the name "fiventeen" was given to the temperament and hence due to the lack of a need for "-ic/-ismic/-isma" (as that can apply to the already-short name of diatisma, itself a rename & shortenage of diatonisma) the name was shortened to just "fiventeen".

See also