81/80: Difference between revisions

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By tempering [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])3 is equated with [[3/2]], because of being able to be rewritten as 9/8 * 8/7 * 7/6 (this observation can be generalized to define the family of [[ultraparticular]] commas). This is an unusually natural extension, because of a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. This means that as [[81/80|S6/S8]] is already tempered in meantone, it is natural to want [[49/48|49/48 = S7]] (which is bigger than S8 and smaller than S6) to be equated, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)3 = [[1728/1715]] = S6/S7, the orwellisma.

This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6 = S7 = S8 with S9 tempered, we can try S8 = S10 by tempering [[176/175|176/175 = S8/S10 = (11/7)/(5/4)2]] , taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 11/~~8 at~~

This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6 = S7 = S8 with S9 tempered, we can try S8 = S10 by tempering [[176/175|176/175 = S8/S10 = (11/7)/(5/4)2]] , taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])2 = [[36/25]] = ([[3/2]])/([[25/24]]).

=== 31edo as splitting the fifth into two, three and nine ===

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 By tempering [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as 9/8 * 8/7 * 7/6 (this observation can be generalized to define the family of [[ultraparticular]] commas). This is an unusually natural extension, because of a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. This means that as [[81/80|S6/S8]] is already tempered in meantone, it is natural to want [[49/48|49/48 = S7]] (which is bigger than S8 and smaller than S6) to be equated, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] = S6/S7, the orwellisma.
-This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6 = S7 = S8 with S9 tempered, we can try S8 = S10 by tempering [[176/175|176/175 = S8/S10 = (11/7)/(5/4)<sup>2</sup>]] , taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 11/8 at
+This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6 = S7 = S8 with S9 tempered, we can try S8 = S10 by tempering [[176/175|176/175 = S8/S10 = (11/7)/(5/4)<sup>2</sup>]] , taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])<sup>2</sup> = [[36/25]] = ([[3/2]])/([[25/24]]).
 === 31edo as splitting the fifth into two, three and nine ===