176edo: Difference between revisions
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Update infobox and expand on theory |
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| Fifth = 103\176 (702.27¢) | | Fifth = 103\176 (702.27¢) | ||
| Major 2nd = 30\176 (205¢) | | Major 2nd = 30\176 (205¢) | ||
| | | Semitones = 17:13 (116¢ : 89¢) | ||
| | | Consistency = 11 | ||
}} | }} | ||
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 [[EDO|equal division of the octave]] into 176 parts of about 6.82 [[cent]]s each, a size close to [[243/242]], the rastma. | 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 [[EDO|equal division of the octave]] into 176 parts of about 6.82 [[cent]]s each, a size close to [[243/242]], the rastma. | ||
== Theory == | == Theory == | ||
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, | 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, [[8019/8000]] and [[9801/9800]] in the 11-limit, supporting the [[bison]] temperament and the [[commatic]] temperament. Using the [[patent val]], [[351/350]], [[364/363]], [[2080/2079]], [[2197/2187]], and [[4096/4095]] in the 13-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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| 20.45 | | 20.45 | ||
| 81/80 | | 81/80 | ||
| [[Commatic]] | | [[Commatic]] | ||
|- | |- | ||
| 2 | | 2 | ||
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| 565.91<br>(34.09) | | 565.91<br>(34.09) | ||
| 168/121<br>(55/54) | | 168/121<br>(55/54) | ||
| [[Octowerck]] (176f) | | [[Octowerck]] (176f) / octowerckis (176) | ||
|- | |- | ||
| 11 | | 11 | ||
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[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Countermiracle]] | |||
Revision as of 12:46, 24 October 2021
| ← 175edo | 176edo | 177edo → |
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 about 6.82 cents each, a size close to 243/242, the rastma.
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, 8019/8000 and 9801/9800 in the 11-limit, supporting the bison temperament and the commatic temperament. Using the patent val, 351/350, 364/363, 2080/2079, 2197/2187, and 4096/4095 in the 13-limit.
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 | Mercy / 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 |
| 2 | 23\176 | 156.82 | 35/32 | Bison |
| 8 | 83\176 (5\176) |
565.91 (34.09) |
168/121 (55/54) |
Octowerck (176f) / octowerckis (176) |
| 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 |