136/135: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,444 edits
m draw attention to cancellation of 17/16
Line 10: Line 10:
[[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.
[[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
[[Subgroup]]: 2.3.17/5


Mapping: {{mapping| 1 0 -3 | 0 1 3 }}
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}


[[CTE]] generator: ~3 = 1904.109{{cent}}
: sval mapping generators: ~2, ~3


Patent val EDO tunings with 20/17 and 3/2 off by less than 25% [[relative error]] (contorted in brackets): 5, 12, 17, 22, 29, (34,) 39, 46, (51,) 56, 63, (68,) 80
[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088


See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
 
See also: [[Srutal archagall]] for the rank-2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.


=== Diatic ===
=== Diatic ===
Subgroup: 2.3.5.17
[[Subgroup]]: 2.3.5.17


Mapping: {{mapping| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}


[[CTE]] generators: ~3 = 1904.109{{cent}}, ~5 = 2787.854{{cent}}
: sval mapping generators: ~2, ~3, ~5


Patent val EDO tunings with 20/17, 3/2 and 5/4 off by less than 25% [[relative error]] (contorted in brackets): 12, 22, 34, 46, 56, (68,) 80
[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544


See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}
 
See also: [[Srutal archagall]] for the rank-2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.


=== Diatismic ===
=== 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]].
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


Subgroup: [[17-limit]]
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 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


Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]
[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8


[[CTE]] generators: [same as diatic with purely tuned 7, 11 and 13 added]
{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*


EDO tunings with less than 33% [[relative error]] for all primes in the no-7's no-11's [[17-limit]]: 10, 24, 34, 44, 46, 56, 80, 114
<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]


== Etymology ==
== Etymology ==

Revision as of 09:36, 12 March 2024

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,
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 or fiventeen comma, is a 17-limit small comma. It is equal to (32/27)/(20/17) and therefore (51/40)/(81/64). It is also trivially the difference between 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

Subgroup-val 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 sequence: 5, 12, 17, 46, 63, 143

See also: Srutal archagall for the rank-2 temperament tempering out {S16, S17}.

Diatic

Subgroup: 2.3.5.17

Subgroup-val 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 sequence: 10, 12, 22, 34, 80, 114, 194bc

See also: Srutal archagall for the rank-2 temperament tempering out {S16, S17}.

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

* optimal patent val: 177

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

Retrieved from "https://en.xen.wiki/index.php?title=136/135&oldid=139003"