176edo: Difference between revisions
Cleanup and +subsets and supersets |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 279 -176 }} | | {{monzo| 279 -176 }} | ||
| {{mapping| 176 279 }} | | {{mapping| 176 279 }} | ||
| | | −0.100 | ||
| 0.100 | | 0.100 | ||
| 1.47 | | 1.47 | ||
| Line 34: | Line 26: | ||
| 78732/78125, {{monzo| 41 -20 -4 }} | | 78732/78125, {{monzo| 41 -20 -4 }} | ||
| {{mapping| 176 279 409 }} | | {{mapping| 176 279 409 }} | ||
| | | −0.400 | ||
| 0.432 | | 0.432 | ||
| 6.34 | | 6.34 | ||
| Line 41: | Line 33: | ||
| 6144/6125, 10976/10935, 50421/50000 | | 6144/6125, 10976/10935, 50421/50000 | ||
| {{mapping| 176 279 409 494 }} | | {{mapping| 176 279 409 494 }} | ||
| | | −0.243 | ||
| 0.463 | | 0.463 | ||
| 6.79 | | 6.79 | ||
| Line 48: | Line 40: | ||
| 441/440, 3388/3375, 6144/6125, 8019/8000 | | 441/440, 3388/3375, 6144/6125, 8019/8000 | ||
| {{mapping| 176 279 409 494 609 }} | | {{mapping| 176 279 409 494 609 }} | ||
| | | −0.250 | ||
| 0.414 | | 0.414 | ||
| 6.08 | | 6.08 | ||
| Line 55: | Line 47: | ||
| 351/350, 364/363, 441/440, 2197/2187, 3146/3125 | | 351/350, 364/363, 441/440, 2197/2187, 3146/3125 | ||
| {{mapping| 176 279 409 494 609 651 }} | | {{mapping| 176 279 409 494 609 651 }} | ||
| | | −0.123 | ||
| 0.473 | | 0.473 | ||
| 6.93 | | 6.93 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 112: | Line 98: | ||
|- | |- | ||
| 4 | | 4 | ||
| 73\176<br>(15\176) | | 73\176<br />(15\176) | ||
| 497.73<br>(102.27) | | 497.73<br />(102.27) | ||
| 4/3<br>(35/33) | | 4/3<br />(35/33) | ||
| [[Undim]] | | [[Undim]] | ||
|- | |- | ||
| 8 | | 8 | ||
| 73\176<br>(7\176) | | 73\176<br />(7\176) | ||
| 497.73<br>(47.73) | | 497.73<br />(47.73) | ||
| 4/3<br>(36/35) | | 4/3<br />(36/35) | ||
| [[Twilight]] | | [[Twilight]] | ||
|- | |- | ||
| 8 | | 8 | ||
| 83\176<br>(5\176) | | 83\176<br />(5\176) | ||
| 565.91<br>(34.09) | | 565.91<br />(34.09) | ||
| 168/121<br>(55/54) | | 168/121<br />(55/54) | ||
| [[Octowerck]] (176f) / octowerckis (176) | | [[Octowerck]] (176f) / octowerckis (176) | ||
|- | |- | ||
| 11 | | 11 | ||
| 73\176<br>(7\176) | | 73\176<br />(7\176) | ||
| 497.73<br>(47.73) | | 497.73<br />(47.73) | ||
| 4/3<br>(36/35) | | 4/3<br />(36/35) | ||
| [[Hendecatonic]] | | [[Hendecatonic]] | ||
|- | |- | ||
| 22 | | 22 | ||
| 73\176<br>(1\176) | | 73\176<br />(1\176) | ||
| 497.73<br>(6.82) | | 497.73<br />(6.82) | ||
| 4/3<br>(385/384) | | 4/3<br />(385/384) | ||
| [[Icosidillic]] / [[major arcana]] | | [[Icosidillic]] / [[major arcana]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
[[Category:Countermiracle]] | [[Category:Countermiracle]] | ||
Revision as of 04:28, 16 November 2024
| ← 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. The equal temperament tempers 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, 9801/9800 and 16384/16335 in the 11-limit. Using the patent val, 351/350, 364/363, 2080/2079, 2197/2187, and 4096/4095 in the 13-limit.
It supports the bison temperament and the commatic temperament, and provides the optimal patent val for countermiracle in the 7- and 11-limit, and countermanna, one of the extensions, in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.32 | +2.32 | -0.64 | +0.95 | -1.89 | -2.68 | +2.49 | -1.00 | -0.03 | +0.42 |
| Relative (%) | +0.0 | +4.7 | +34.1 | -9.4 | +14.0 | -27.7 | -39.3 | +36.5 | -14.7 | -0.5 | +6.1 | |
| Steps (reduced) |
176 (0) |
279 (103) |
409 (57) |
494 (142) |
609 (81) |
651 (123) |
719 (15) |
748 (44) |
796 (92) |
855 (151) |
872 (168) | |
Subsets and supersets
Since 176 factors into 24 × 11, 176edo has subset edos 2, 4, 8, 11, 22, 44, and 88.
Regular temperament properties
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 17\176
| 115.91
| 77/72
| Mercy / countermiracle / counterbenediction / countermanna
|-
| 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
|-
| 4
| 73\176
(15\176)
| 497.73
(102.27)
| 4/3
(35/33)
| Undim
|-
| 8
| 73\176
(7\176)
| 497.73
(47.73)
| 4/3
(36/35)
| Twilight
|-
| 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 / major arcana
Template:Rank-2 end
Template:Orf