Syntonic–chromatic equivalence continuum: Difference between revisions
There's a way to "invert" the continuum |
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We may also invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. The just value of m is 1.2333… | We may also invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. The just value of ''m'' is 1.2333… | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
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The 5-limit 6b&7 temperament. Its name is "porcupine" spelled backwards, because that's what this temperament is – it is porcupine, with the generator sharp of 2\7 such that the major and minor thirds switch places. The fifths are very flat, meaning that this is more of a melodic temperament than a harmonic one. | The 5-limit 6b&7 temperament. Its name is "porcupine" spelled backwards, because that's what this temperament is – it is porcupine, with the generator sharp of 2\7 such that the major and minor thirds switch places. The fifths are very flat, meaning that this is more of a melodic temperament than a harmonic one. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 1125/1024 | [[Comma list]]: 1125/1024 | ||
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[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -3 2 }}] | [[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -3 2 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1/1, ~16/15 = 173.101 | ||
{{Val list|legend=1| 6b, 7 }} | {{Val list|legend=1| 6b, 7 }} | ||
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The 5-limit 7&84 temperament, so named because it truly is an absurd temperament. The generator is 81/80 and the period is 800/729, which is (10/9) / (81/80). This is also part of the syntonic-chromatic equivalence continuum, in this case where (81/80)<sup>5</sup> = 25/24. | The 5-limit 7&84 temperament, so named because it truly is an absurd temperament. The generator is 81/80 and the period is 800/729, which is (10/9) / (81/80). This is also part of the syntonic-chromatic equivalence continuum, in this case where (81/80)<sup>5</sup> = 25/24. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 10460353203/10240000000 | [[Comma list]]: 10460353203/10240000000 | ||
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[[Mapping]]: [{{val| 7 0 -17 }}, {{val| 0 1 3 }}] | [[Mapping]]: [{{val| 7 0 -17 }}, {{val| 0 1 3 }}] | ||
: mapping generators: ~800/729, ~3 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~800/729 = 1\7, ~3/2 = 700.1870 (or ~81/80 = 14.4727) | ||
{{Val list|legend=1| 7, 70, 77, 84, 329 }} | {{Val list|legend=1| 7, 70, 77, 84, 329 }} | ||
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This is a fairly obvious temperament; it just equates 7 10/9's with a 2/1, hence the period is 10/9. One generator from 5\7 puts you at 3/2, two generators from 2\7 puts you at 5/4. | This is a fairly obvious temperament; it just equates 7 10/9's with a 2/1, hence the period is 10/9. One generator from 5\7 puts you at 3/2, two generators from 2\7 puts you at 5/4. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 5000000/4782969 | [[Comma list]]: 5000000/4782969 | ||
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[[Mapping]]: [{{val| 7 0 -6 }}, {{val| 0 1 2 }}] | [[Mapping]]: [{{val| 7 0 -6 }}, {{val| 0 1 2 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 706.288 | ||
{{Val list|legend=1| 7, 42, 49, 56, 119 }} | {{Val list|legend=1| 7, 42, 49, 56, 119 }} | ||
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This is similar to the above, but provides a less complex avenue to 5, but this time at the cost of accuracy. One generator from 5\7 puts you at 3/2, and one generator from 2\7 puts you at 5/4. | This is similar to the above, but provides a less complex avenue to 5, but this time at the cost of accuracy. One generator from 5\7 puts you at 3/2, and one generator from 2\7 puts you at 5/4. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 78125/69984 | [[Comma list]]: 78125/69984 | ||
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[[Mapping]]: [{{val| 7 0 5 }}, {{val| 0 1 1 }}] | [[Mapping]]: [{{val| 7 0 5 }}, {{val| 0 1 1 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~125/108 = 1\7, ~3/2 = 706.410 | ||
{{Val list|legend=1| 7, 35b, 42c, 49c, 56cc, 119cccc }} | {{Val list|legend=1| 7, 35b, 42c, 49c, 56cc, 119cccc }} | ||