Porcupine family: Difference between revisions
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Restore the layout. We list the canonical extension first. |
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{{Main| Porcupine }} | {{Main| Porcupine }} | ||
The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = (4/3)⋅(250/243), and (10/9)<sup>5</sup> = (8/5)⋅(250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities. | The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = (4/3)⋅(250/243), and (10/9)<sup>5</sup> = (8/5)⋅(250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities. | ||
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[[Badness]] (Smith): 0.030778 | [[Badness]] (Smith): 0.030778 | ||
=== Overview to extensions === | |||
==== 7-limit extensions ==== | |||
The third comma defines which [[7-limit]] family member we are looking at. That means | |||
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]], | |||
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]], | |||
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and | |||
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]]. | |||
Temperaments discussed elsewhere include [[7th-octave temperaments #Jamesbond|jamesbond]]. | |||
==== Subgroup extensions ==== | |||
Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|S11]] }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below. | Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|S11]] }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below. | ||
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Badness (Smith): 0.0305 | Badness (Smith): 0.0305 | ||
== Septimal porcupine == | == Septimal porcupine == | ||
{{Main| Porcupine }} | {{Main| Porcupine }} | ||
Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator | Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. Here, we share the same mapping of 7/4 in terms of fifths as [[archy]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator. | ||
=== 7-limit === | === 7-limit === | ||
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Badness (Smith): 0.021562 | Badness (Smith): 0.021562 | ||
==== 13-limit ( | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 40/39, 55/54, 64/63, 66/65 | |||
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.000, ~11/10 = 163.442 | |||
* POTE: ~2 = 1200.000, ~11/10 = 162.708 | |||
Minimax tuning: | |||
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }} | |||
: eigenmonzo (unchanged-interval) basis: 2.11 | |||
Tuning ranges: | |||
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22) | |||
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22) | |||
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404] | |||
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }} | |||
Badness (Smith): 0.021276 | |||
==== Porcupinefish ==== | |||
{{See also| The Biosphere }} | {{See also| The Biosphere }} | ||
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Badness (Smith): 0.025314 | Badness (Smith): 0.025314 | ||
==== Pourcup ==== | ==== Pourcup ==== | ||