176edo: Difference between revisions
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+RTT table |
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== Theory == | == Theory == | ||
176edo is [[consistent]] to the [[11-odd-limit]], tempering out 78732/78125 ([[sensipent comma]]) and | 176edo is [[consistent]] to the [[11-odd-limit]], tempering out 78732/78125 ([[sensipent comma]]) and {{monzo| 41 -20 -4 }} ([[undim comma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and 50421/50000 in the 7-limit; [[441/440]], 3388/3375, 6912/6875, and [[8019/8000]] in the 11-limit, supporting the [[bison]] temperament and the [[commatic]] temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Primes in edo|176}} | {{Primes in edo|176}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 279 -176 }} | |||
| [{{val| 176 279 }}] | |||
| -0.100 | |||
| 0.100 | |||
| 1.47 | |||
|- | |||
| 2.3.5 | |||
| 78732/78125, {{monzo| 41 -20 -4 }} | |||
| [{{val| 176 279 409 }}] | |||
| -0.400 | |||
| 0.432 | |||
| 6.34 | |||
|- | |||
| 2.3.5.7 | |||
| 6144/6125, 10976/10935, 50421/50000 | |||
| [{{val| 176 279 409 494 }}] | |||
| -0.243 | |||
| 0.463 | |||
| 6.79 | |||
|- | |||
| 2.3.5.7.11 | |||
| 441/440, 3388/3375, 6144/6125, 8019/8000 | |||
| [{{val| 176 279 409 494 609 }}] | |||
| -0.250 | |||
| 0.414 | |||
| 6.08 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 351/350, 364/363, 441/440, 2197/2187, 3146/3125 | |||
| [{{val| 176 279 409 494 609 651 }}] | |||
| -0.123 | |||
| 0.473 | |||
| 6.93 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 17\176 | |||
| 115.91 | |||
| 77/72 | |||
| [[Countermiracle]] / countermiraculous (176f) / counterbenediction (176) | |||
|- | |||
| 1 | |||
| 35\176 | |||
| 238.64 | |||
| 147/128 | |||
| [[Tokko]] | |||
|- | |||
| 1 | |||
| 65\176 | |||
| 443.18 | |||
| 162/125 | |||
| [[Sensipent]] | |||
|- | |||
| 1 | |||
| 73\176 | |||
| 497.73 | |||
| 4/3 | |||
| [[Gary]] / [[cotoneum]] | |||
|- | |||
| 1 | |||
| 83\176 | |||
| 565.91 | |||
| 13/9 | |||
| [[Tricot]] / [[trident]] | |||
|- | |||
| 2 | |||
| 23\176 | |||
| 20.45 | |||
| 81/80 | |||
| [[Commatic]] (176f) | |||
|- | |||
| 2 | |||
| 23\176 | |||
| 156.82 | |||
| 35/32 | |||
| [[Bison]] | |||
|- | |||
| 8 | |||
| 83\176<br>(5\176) | |||
| 565.91<br>(34.09) | |||
| 168/121<br>(55/54) | |||
| [[Octowerck]] (176f) | |||
|- | |||
| 11 | |||
| 73\176<br>(7\176) | |||
| 497.73<br>(47.73) | |||
| 4/3<br>(36/35) | |||
| [[Hendecatonic]] | |||
|- | |||
| 22 | |||
| 73\176<br>(1\176) | |||
| 497.73<br>(6.82) | |||
| 4/3<br>(385/384) | |||
| [[Icosidillic]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
Revision as of 12:41, 7 September 2021
The 176 equal divisions of the octave (176edo), or the 176(-tone) equal temperament (176tet, 176et) when viewed from a regular temperament perspective, is the equal division of the octave into 176 parts of 6.8182 cents each.
Theory
176edo is consistent to the 11-odd-limit, tempering out 78732/78125 (sensipent comma) and [41 -20 -4⟩ (undim comma) in the 5-limit; 6144/6125, 10976/10935, and 50421/50000 in the 7-limit; 441/440, 3388/3375, 6912/6875, and 8019/8000 in the 11-limit, supporting the bison temperament and the commatic temperament.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [279 -176⟩ | [⟨176 279]] | -0.100 | 0.100 | 1.47 |
| 2.3.5 | 78732/78125, [41 -20 -4⟩ | [⟨176 279 409]] | -0.400 | 0.432 | 6.34 |
| 2.3.5.7 | 6144/6125, 10976/10935, 50421/50000 | [⟨176 279 409 494]] | -0.243 | 0.463 | 6.79 |
| 2.3.5.7.11 | 441/440, 3388/3375, 6144/6125, 8019/8000 | [⟨176 279 409 494 609]] | -0.250 | 0.414 | 6.08 |
| 2.3.5.7.11.13 | 351/350, 364/363, 441/440, 2197/2187, 3146/3125 | [⟨176 279 409 494 609 651]] | -0.123 | 0.473 | 6.93 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 17\176 | 115.91 | 77/72 | Countermiracle / countermiraculous (176f) / counterbenediction (176) |
| 1 | 35\176 | 238.64 | 147/128 | Tokko |
| 1 | 65\176 | 443.18 | 162/125 | Sensipent |
| 1 | 73\176 | 497.73 | 4/3 | Gary / cotoneum |
| 1 | 83\176 | 565.91 | 13/9 | Tricot / trident |
| 2 | 23\176 | 20.45 | 81/80 | Commatic (176f) |
| 2 | 23\176 | 156.82 | 35/32 | Bison |
| 8 | 83\176 (5\176) |
565.91 (34.09) |
168/121 (55/54) |
Octowerck (176f) |
| 11 | 73\176 (7\176) |
497.73 (47.73) |
4/3 (36/35) |
Hendecatonic |
| 22 | 73\176 (1\176) |
497.73 (6.82) |
4/3 (385/384) |
Icosidillic |