Magic family: Difference between revisions
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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 [[19edo|19EDO]] or [[22edo|22EDO]] 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 [[19edo|19EDO]] or [[22edo|22EDO]] is always possible. | ||
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[[Tuning ranges]]: | [[Tuning ranges]]: | ||
* [[ | * [[Diamond monotone]] range: ~5/4 = [360.000, 400.000] (3\10 to 1\3) | ||
* [[ | * [[Diamond tradeoff]] range: ~5/4 = [378.910, 386.314] | ||
* | * Diamond monotone and tradeoff: ~5/4 = [378.910, 386.314] | ||
[[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s. | [[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s. | ||
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[[Badness]]: 0.039163 | [[Badness]]: 0.039163 | ||
=== Seven-limit extensions | === Seven-limit 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. | ||
== Magic | == Magic == | ||
{{main| Magic }} | {{main| Magic }} | ||
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[[Tuning ranges]]: | [[Tuning ranges]]: | ||
* [[ | * [[Diamond monotone]] range: ~5/4 = [378.947, 381.818] (6\19 to 7\22) | ||
* [[ | * [[Diamond tradeoff]] range: ~5/4 = [378.910, 386.314] | ||
* | * Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818] | ||
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents. | [[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents. | ||
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[[Badness]]: 0.018918 | [[Badness]]: 0.018918 | ||
=== 11-limit | === 11-limit === | ||
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.) | Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.) | ||
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Tuning ranges: | Tuning ranges: | ||
* | * Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22) | ||
* | * Diamond tradeoff range: ~5/4 = [378.910, 386.314] | ||
* | * Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818] | ||
Vals: {{Val list| 19, 22, 41, 104, 145c }} | Vals: {{Val list| 19, 22, 41, 104, 145c }} | ||
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Badness: 0.020352 | Badness: 0.020352 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Tuning ranges: | Tuning ranges: | ||
* | * Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22) | ||
* | * Diamond tradeoff range: ~5/4 = [378.617, 386.314] | ||
* | * Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818] | ||
Vals: {{Val list| 19, 22f, 41, 265cdef }} | Vals: {{Val list| 19, 22f, 41, 265cdef }} | ||
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Badness: 0.021509 | Badness: 0.021509 | ||
==== Sorcery | ==== Sorcery ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Tuning ranges: | Tuning ranges: | ||
* | * Diamond monotone range: ~5/4 = 378.947 (6\19) | ||
* | * Diamond tradeoff range: ~5/4 = [359.472, 386.314] | ||
* | * Diamond monotone and tradeoff: ~5/4 = 378.947 | ||
Vals: {{Val list| 19, 22, 41f }} | Vals: {{Val list| 19, 22, 41f }} | ||
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Badness: 0.025829 | Badness: 0.025829 | ||
==== Necromancy | ==== Necromancy ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Tuning ranges: | Tuning ranges: | ||
* | * Diamond monotone range: ~5/4 = [380.488, 380.952] (13\41 to 20\63) | ||
* | * Diamond tradeoff range: ~5/4 = [378.910, 386.314] | ||
* | * Diamond monotone and tradeoff: ~5/4 = [380.488, 380.952] | ||
Vals: {{Val list| 19f, 22, 41, 63, 104 }} | Vals: {{Val list| 19f, 22, 41, 63, 104 }} | ||
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Badness: 0.025275 | Badness: 0.025275 | ||
==== Soothsaying | ==== Soothsaying ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.055443 | Badness: 0.055443 | ||
=== Telepathy | === Telepathy === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.027109 | Badness: 0.027109 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.025522 | Badness: 0.025522 | ||
=== Horcrux | === Horcrux === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.039282 | Badness: 0.039282 | ||
=== Divination | === Divination === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.035864 | Badness: 0.035864 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.034551 | Badness: 0.034551 | ||
=== Witchcraft | === Witchcraft === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.030706 | Badness: 0.030706 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.023547 | Badness: 0.023547 | ||
=== Hocus | === Hocus === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.038519 | Badness: 0.038519 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.030280 | Badness: 0.030280 | ||
== Muggles | == Muggles == | ||
Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo|19EDO]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. | Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo|19EDO]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. | ||
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[[Badness]]: 0.056206 | [[Badness]]: 0.056206 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 316: | Line 316: | ||
Badness: 0.048038 | Badness: 0.048038 | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.030386 | Badness: 0.030386 | ||
== Astrology | == Astrology == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 344: | Line 344: | ||
[[Badness]]: 0.082673 | [[Badness]]: 0.082673 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 357: | Line 357: | ||
Badness: 0.039151 | Badness: 0.039151 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor] | ||
==== Horoscope | ==== Horoscope ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.035284 | Badness: 0.035284 | ||
== Spell | == Spell == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 401: | Line 401: | ||
[[Badness]]: 0.080958 | [[Badness]]: 0.080958 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 414: | Line 414: | ||
Badness: 0.059791 | Badness: 0.059791 | ||
==== 13-limit | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 427: | Line 427: | ||
Badness: 0.045591 | Badness: 0.045591 | ||
==== Cantrip | ==== Cantrip ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 440: | Line 440: | ||
Badness: 0.041603 | Badness: 0.041603 | ||
== Hocum | == Hocum == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 455: | Line 455: | ||
[[Badness]]: 0.107115 | [[Badness]]: 0.107115 | ||
== Trismegistus | == Trismegistus == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 470: | Line 470: | ||
[[Badness]]: 0.098334 | [[Badness]]: 0.098334 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 483: | Line 483: | ||
Badness: 0.045623 | Badness: 0.045623 | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 496: | Line 496: | ||
Badness: 0.033081 | Badness: 0.033081 | ||
== Quadrimage | == Quadrimage == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 511: | Line 511: | ||
[[Badness]]: 0.127422 | [[Badness]]: 0.127422 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 524: | Line 524: | ||
Badness: 0.061572 | Badness: 0.061572 | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||