81/80: Difference between revisions

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[[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.

== Temperaments ==

If one wants to extend meantone beyond 5-limit, there is a number of ways to do so discussed in the [[meantone family]], usually by decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, a unique opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow — and indeed want — more (flatwards) tempering on the fifth, so may be recommended for this reason. However, as [[9/8]] is typically flat in meantone, we might mention that an opportunity not based on splitting the fifth comes from interpreting the tritone (~9/8)3 as [[7/5]], leading to [[septimal meantone]], a very elegant extension to the [[7-limit]].

=== Splitting the meantone fifth into two (243/242) ===

By tempering [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. (Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)2]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp.) There is also the consideration that tempering [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups.

=== Splitting the meantone fifth into three (1029/1024) ===

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

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

[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]], so that it is very strong in the 2.5.7 subgroup. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is uniquely meantone + valentine. Valentine is a natural [[11-limit]] temperament that tempers [[121/120]] so for this reason might be natural to combine with meantone. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.

== Relations to other superparticular ratios ==

Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.

Names in brackets refer to 7-limit [[Meantone family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas.

Names in brackets refer to 7-limit [[Meantone family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)

{| class="wikitable"

@@ Line 14: / Line 14: @@
 [[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.
+== Temperaments ==
+If one wants to extend meantone beyond 5-limit, there is a number of ways to do so discussed in the [[meantone family]], usually by decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, a unique opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow — and indeed want — more (flatwards) tempering on the fifth, so may be recommended for this reason. However, as [[9/8]] is typically flat in meantone, we might mention that an opportunity not based on splitting the fifth comes from interpreting the tritone (~9/8)<sup>3</sup> as [[7/5]], leading to [[septimal meantone]], a very elegant extension to the [[7-limit]].
+=== Splitting the meantone fifth into two (243/242) ===
+By tempering [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. (Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)<sup>2</sup>]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp.) There is also the consideration that tempering [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups.
+=== Splitting the meantone fifth into three (1029/1024) ===
+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
+=== 31edo as splitting the fifth into two, three and nine ===
+[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]], so that it is very strong in the 2.5.7 subgroup. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is uniquely meantone + valentine. Valentine is a natural [[11-limit]] temperament that tempers [[121/120]] so for this reason might be natural to combine with meantone. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.
 == Relations to other superparticular ratios ==
 Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.
-Names in brackets refer to 7-limit [[Meantone family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas.
+Names in brackets refer to 7-limit [[Meantone family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)
 {| class="wikitable"