Hemimage temperaments: Difference between revisions
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== Commatic == | == 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. | 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 | Subgroup: 2.3.5.7 | ||
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POTE generator: ~81/80 = 20.390 | POTE generator: ~81/80 = 20.390 | ||
Optimal GPV sequence: {{Val list| 58, 118, 294, 412d }} | |||
Badness: 0.030461 | Badness: 0.030461 | ||
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== Chromat == | == Chromat == | ||
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. | 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 | Subgroup: 2.3.5.7 | ||
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== Degrees == | == 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. | 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 | Subgroup: 2.3.5.7 | ||
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POTE generator: ~3/2 = 703.231 | POTE generator: ~3/2 = 703.231 | ||
Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }} | |||
Badness: 0.046770 | Badness: 0.046770 | ||
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POTE generator: ~3/2 = 703.080 | POTE generator: ~3/2 = 703.080 | ||
Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }} | |||
Badness: 0.032718 | Badness: 0.032718 | ||
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POTE generator: ~21/16 = 475.995 | POTE generator: ~21/16 = 475.995 | ||
Optimal GPV sequence: {{Val list| 58, 121, 179e, 300bde }} | |||
Badness: 0.045323 | Badness: 0.045323 | ||
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POTE generator: ~21/16 = 475.996 | POTE generator: ~21/16 = 475.996 | ||
Optimal GPV sequence: {{Val list| 58, 121, 179ef, 300bdef }} | |||
Badness: 0.023800 | Badness: 0.023800 | ||
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POTE generators: ~11/10 = 162.773 | POTE generators: ~11/10 = 162.773 | ||
Optimal GPV sequence: {{Val list| 22, 74d, 96d, 118, 258e, 376de }} | |||
Badness: 0.032080 | Badness: 0.032080 | ||
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The ''cotoneum'' temperament (41&217, named after the Latin for "[[Wikipedia:quince|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. | The ''cotoneum'' temperament (41&217, named after the Latin for "[[Wikipedia:quince|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 | Subgroup: 2.3.5.7 | ||
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POTE generator: ~3/2 = 702.303 | POTE generator: ~3/2 = 702.303 | ||
Optimal GPV sequence: {{Val list| 41, 135c, 176, 217 }} | |||
Badness: 0.050966 | Badness: 0.050966 | ||
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POTE generator: ~3/2 = 702.306 | POTE generator: ~3/2 = 702.306 | ||
Optimal GPV sequence: {{Val list| 41, 176, 217 }} | |||
Badness: 0.036951 | Badness: 0.036951 | ||
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POTE generator: ~3/2 = 702.307 | POTE generator: ~3/2 = 702.307 | ||
Optimal GPV sequence: {{Val list| 41, 176, 217 }} | |||
Badness: 0.029495 | Badness: 0.029495 | ||
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POTE generator: ~3/2 = 702.308 | POTE generator: ~3/2 = 702.308 | ||
Optimal GPV sequence: {{Val list| 41, 176, 217 }} | |||
Badness: 0.021811 | Badness: 0.021811 | ||
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== Squarschmidt == | == Squarschmidt == | ||
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35. | A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35. | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
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POTE generator: ~44/35 = 396.644 | POTE generator: ~44/35 = 396.644 | ||
Optimal GPV sequence: {{Val list| 118, 239, 357, 596 }} | |||
Badness: 0.038186 | Badness: 0.038186 | ||