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Temperament family page revision proposal: Difference between revisions

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The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.
The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.


Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 3 5 1 }} for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10/9]] interval, and that 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.
Its [[monzo]] is {{monzo| 1 -5 3 }}. The [[generator]] is a minor whole tone, the [[10/9]] interval, and that 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.


Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.  
Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.  
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{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
{{Multival|legend=1| 3 5 -6 1 -18 -28 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
{{Multival|legend=1| 3 5 9 1 6 7 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
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{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
{{Multival|legend=1| 3 5 16 1 17 23 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
{{Multival|legend=1| 3 5 -13 1 -29 -44 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
{{Multival|legend=1| 3 5 1 1 -7 -12 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
{{Multival|legend=1| 3 5 2 1 -5 -9 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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: mapping generators: ~7/5, ~9/7
: mapping generators: ~7/5, ~9/7
{{Multival|legend=1| 6 10 10 2 -1 -5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20
{{Multival|legend=1| 6 10 3 2 -12 -21 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 9 15 19 3 5 2 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35
{{Multival|legend=1| 9 15 4 3 -19 -33 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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