Magic family: Difference between revisions
Cmloegcmluin (talk | contribs) adding parenthetical "(unchanged intervals)" to eigenmonzo lists |
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The most prominent deficiency of magic temperaments is that they lack [[Rothenberg propriety|proper]] or nearly-proper MOS scales in the 5 to 10 note "diatonic" region. | The most prominent deficiency of magic temperaments is that they lack [[Rothenberg propriety|proper]] or nearly-proper MOS scales in the 5 to 10 note "diatonic" region. | ||
== Five limit magic == | == Five-limit magic == | ||
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 5 1 -10 }} for the [[wedgie]]. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense and using [[19 EDO]] or [[22 EDO]] is always possible. | The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 5 1 -10 }} for the [[wedgie]]. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense and using [[19 EDO]] or [[22 EDO]] is always possible. | ||
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[[Badness]]: 0.039163 | [[Badness]]: 0.039163 | ||
=== | === Extensions === | ||
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator. | The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator. | ||
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Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 225/224, 245/243 | Comma list: 100/99, 225/224, 245/243 | ||
Mapping: [{{val| 1 0 2 -1 6 }}, {{val| 0 5 1 12 -8 }}] | Mapping: [{{val| 1 0 2 -1 6 }}, {{val| 0 5 1 12 -8 }}] | ||
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Badness: 0.033317 | Badness: 0.033317 | ||
=== | === Witchcraft === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 225/224, 245/243, 441/440 | ||
Mapping: [{{val| 2 | Mapping: [{{val| 1 0 2 -1 -7 }}, {{val| 0 5 1 12 33 }}] | ||
POTE generator: ~5/4 = 380. | POTE generator: ~5/4 = 380.232 | ||
Vals: {{Val list| | Vals: {{Val list| 41, 60e, 101cd, 243cde }} | ||
Badness: 0. | Badness: 0.030706 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 105/104 | Comma list: 105/104, 196/195, 245/243, 275/273 | ||
Mapping: [{{val| 2 | Mapping: [{{val| 1 0 2 -1 -7 -2 }}, {{val| 0 5 1 12 33 18 }}] | ||
POTE generator: ~5/4 = | POTE generator: ~5/4 = 380.189 | ||
Vals: {{Val list| | Vals: {{Val list| 41, 60e, 101cd }} | ||
Badness: 0. | Badness: 0.023547 | ||
=== | === Divination === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 225/224, 245/243 | Comma list: 121/120, 225/224, 245/243 | ||
Mapping: [{{val| | Mapping: [{{val| 2 0 4 -2 5 }}, {{val| 0 5 1 12 3 }}] | ||
POTE generator: ~5/4 = 380. | POTE generator: ~5/4 = 380.233 | ||
Vals: {{Val list| | Vals: {{Val list| 22, 38d, 60e, 142cde }} | ||
Badness: 0. | Badness: 0.035864 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 105/104, 196/195, 245/243 | Comma list: 105/104, 121/120, 196/195, 245/243 | ||
Mapping: [{{val| | Mapping: [{{val| 2 0 4 -2 5 -4 }}, {{val| 0 5 1 12 3 18 }}] | ||
POTE generator: ~5/4 = | POTE generator: ~5/4 = 379.920 | ||
Vals: {{Val list| | Vals: {{Val list| 22f, 60e }} | ||
Badness: 0. | Badness: 0.034551 | ||
=== Hocus === | === Hocus === | ||
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[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Magic family| ]] <!-- main article --> | [[Category:Magic family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Magic]] | |||
[[Category:Listen]] | [[Category:Listen]] | ||