Hemimage temperaments: Difference between revisions

This is a collection of temperaments tempering out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include commatic, chromat, degrees, subfourth, bisupermajor and cotoneum, considered below, as well as the following discussed elsewhere:  

* ''[[quasisuper]]'', {64/63, 2430/2401} → [[Archytas clan #Quasisuper]]

* ''[[liese]]'', {81/80, 686/675} → [[Meantone family #Liese]]

* ''[[unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn]]

* [[magic]], {225/224, 245/243} → [[Magic family #Magic]]

* ''[[guiron]]'', {1029/1024, 10976/10935} → [[Gamelismic clan #Guiron]]

* ''[[echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna]]

* [[hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths]]

* ''[[dodecacot]]'', {3125/3087, 10976/10935} → [[Tetracot family #Dodecacot]]

* [[parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic]]

* ''[[pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto]]

* ''[[hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic]]

* ''[[marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths]]

* ''[[yarman]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman]]

== Commatic ==

The commatic temperament has a period of half octave and a generator of 20.4 cents. It is so named because the generator is a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

Subgroup: 2.3.5.7

[[Comma list]]: 10976/10935, 50421/50000

[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]

[[POTE generator]]: ~81/80 = 20.377

{{Val list|legend=1| 58, 118, 294, 412d, 530d }}

[[Badness]]: 0.084317

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]

POTE generator: ~81/80 = 20.390

Vals: {{Val list| 58, 118, 294, 412d }}

Badness: 0.030461

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[Amity family|amity extension]] with third-octave period.

Subgroup: 2.3.5.7

[[Comma list]]: 10976/10935, 235298/234375

[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]

{{Multival|legend=1| 15 39 48 27 34 2 }}

[[POTE generator]]: ~28/27 = 60.528

{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}

[[Badness]]: 0.057499

== Degrees ==

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.

Subgroup: 2.3.5.7

[[Comma list]]: 10976/10935, 390625/388962

[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]

{{Multival|legend=1| 20 40 60 17 39 27 }}

[[POTE generator]]: ~3/2 = 703.015

{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}

[[Badness]]: 0.106471

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]

POTE generator: ~3/2 = 703.231

Vals: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}

Badness: 0.046770

=== 13-limit ===

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]

POTE generator: ~3/2 = 703.080

Vals: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}

Badness: 0.032718

== Subfourth ==

Subgroup: 2.3.5.7

[[Comma list]]: 10976/10935, 65536/64827

[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]

{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}

[[POTE generator]]: ~21/16 = 475.991

 

=== 11-limit ===

Comma list: 540/539, 896/891, 12005/11979

Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]

POTE generator: ~21/16 = 475.995

Vals: {{Val list| 58, 121, 179e, 300bde }}

Badness: 0.045323

=== 13-limit ===

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]

POTE generator: ~21/16 = 475.996

Vals: {{Val list| 58, 121, 179ef, 300bdef }}

Badness: 0.023800

{{see also| Very high accuracy temperaments #Kwazy }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]

[[POTE generator]]: ~192/175 = 162.8061

{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}

Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]

POTE generators: ~11/10 = 162.7733

Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}

== Cotoneum ==

{{Main| Cotoneum }}

The ''cotoneum'' temperament (41&217, named by [[User:Xenllium|Xenllium]] after the Latin for "quince") tempers out the [[Quince clan|quince comma]], 823543/819200 and the [[garischisma]], 33554432/33480783. This temperament is supported by [[41edo|41]], [[176edo|176]], [[217edo|217]], and [[258edo|258]] EDOs, and can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.

Subgroup: 2.3.5.7

[[Comma list]]: 10976/10935, 823543/819200

[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]

{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}

[[POTE generator]]: ~3/2 = 702.317

[[Minimax tuning]]:

: [[Eigenmonzo]]s: 2, 6/5

* 9-odd-limit: ~3/2 = {{Monzo| 29/51 2/51 -1/51 }}

: [{{Monzo| 1 0 0 0 }}, {{Monzo| 80/51 2/51 -1/51 0 }}, {{Monzo| 160/51 -98/51 49/51 0 }}, {{Monzo| 155/51 -28/51 14/51 0 }}]

* 7-odd-limit [[diamond monotone]]: ~3/2 = [701.5385, 702.8571] (38\65 to 41\70)

* 9-odd-limit diamond monotone: ~3/2 = [701.8868, 702.8571] (31\53 to 41\70)

* [[Diamond tradeoff]] range: ~3/2 = [701.9550, 702.3575]

* Diamond monotone and tradeoff: ~3/2 = [701.9550, 702.3575]

[[Badness]]: 0.105632

Comma list: 441/440, 10976/10935, 16384/16335

Mapping: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]

POTE generator: ~3/2 = 702.303

* 11-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}

Tuning ranges:

* Diamond monotone range: ~3/2 = [702.1277, 702.4390] (55\94 to 24\41)

 

Badness: 0.050966

Comma list: 364/363, 441/440, 3584/3575, 10976/10935

Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]

POTE generator: ~3/2 = 702.306

* 13-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}

: Eigenmonzos: 2, 11/10

* 15-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}

Tuning ranges:

* 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2222, 702.3693]

 

Badness: 0.036951

Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262

Mapping: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]

POTE generator: ~3/2 = 702.307

Minimax tuning:  

* 17-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}

* Diamond monotone range: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)

* Diamond tradeoff range: ~3/2 = [701.9550, 702.3693]

* Diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3693]

Vals: {{Val list| 41, 176, 217 }}

Badness: 0.029495

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728

Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]

POTE generator: ~3/2 = 702.308

Minimax tuning:  

* 19- and 21-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}

* Diamond monotone range: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)

* Diamond tradeoff range: ~3/2 = [701.9550, 702.3771]

* Diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3771]

Vals: {{Val list| 41, 176, 217 }}

Badness: 0.021811

[[Category:Regular temperament theory]]

[[Category:Temperament collection]]

[[Category:Hemimage]]