Bird's eye view of temperaments by accuracy: Difference between revisions
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* 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]]) | * 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]]) | ||
Mohaha is a 2.3.5.11 (no-7's [[11-limit]]) "hemi-meantone" temperament that splits [[#Meantone]]'s fifth into two [[~]][[11/9]]'s by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. It has two main extensions to the full 11-limit; if you accept the [[#Septimal meantone]] mapping of 7 you get [[migration]], but maybe more natural is if you instead equate the flat [[~]][[33/32]] interval with S6 = [[36/35]] = ([[6/5]])/([[7/6]]), which results in [[mohajira]], which finds 7 at a negative number of gens so that composite harmonics of 7 are simpler to find (as primes 3, 5 and 11 are all found at a positive number of gens). Because of this, mohajira is usually the preferred extension as it is more note-efficient, but both extensions merge in [[31edo]], which is a good tuning for both. | Mohaha is a 2.3.5.11 (no-7's [[11-limit]]) "hemi-meantone" temperament that splits [[#Meantone]]'s fifth into two [[~]][[11/9]]'s by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. It has two main extensions to the full 11-limit; if you accept the [[#Septimal meantone]] mapping of 7 you get [[migration]], but maybe more natural is if you instead equate the flat [[~]][[33/32]] interval with S6 = [[36/35]] = ([[6/5]])/([[7/6]]), which results in [[mohajira]], which finds 7 at a negative number of gens so that composite harmonics of 7 are simpler to find (as primes 3, 5 and 11 are all found at a positive number of gens). Because of this, mohajira is usually the preferred extension as it is more note-efficient, but both extensions merge in [[31edo]], which is a good tuning for both. | ||
== ~17-limit focus == | == ~17-limit focus == | ||
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Septimal meantone is an extension of [[#meantone]] that finds 8/7 as the diminished third and 7/6 as the augmented second, so that using an augmented sixth with a major triad forms a harmonic seventh chord. Though 12 notes is about sufficient for achieving its harmony, often one wants to use a 19-note MOS ([[12L 7s]]) for more freedom and availability. | Septimal meantone is an extension of [[#meantone]] that finds 8/7 as the diminished third and 7/6 as the augmented second, so that using an augmented sixth with a major triad forms a harmonic seventh chord. Though 12 notes is about sufficient for achieving its harmony, often one wants to use a 19-note MOS ([[12L 7s]]) for more freedom and availability. | ||
It has two main extensions to prime 11, both similarly complex, discussed in [[meantone vs meanpop]], though the one called [[undecimal meantone]] is arguably more elegant as being the merge of septimal meantone and the no-3's 11-limit temperament [[#Didacus]], which can be seen as every other gen of undecimal meantone. An alternative extension that splits the generator is by interpreting [[~]][[11/9]] as half of the meantone fifth, by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. This leads to [[#Migration]] if you accept the septimal meantone mapping of 7 (which becomes double as complex), or [[mohaha]] if you interpret it as no-7's. | It has two main extensions to prime 11, both similarly complex, discussed in [[meantone vs meanpop]], though the one called [[undecimal meantone]] is arguably more elegant as being the merge of septimal meantone and the no-3's 11-limit temperament [[#Didacus]], which can be seen as every other gen of undecimal meantone. An alternative extension that splits the generator is by interpreting [[~]][[11/9]] as half of the meantone fifth, by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. This leads to [[#Migration]] if you accept the septimal meantone mapping of 7 (which becomes double as complex), or [[mohaha]] if you interpret it as no-7's. | ||
=== [[Mothra]] === | === [[Mothra]] === | ||
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Bound-violating intervals: [[9/8]], [[10/9]] (none in [[7-odd-limit]] or if 9 is omitted) | Bound-violating intervals: [[9/8]], [[10/9]] (none in [[7-odd-limit]] or if 9 is omitted) | ||
Mothra makes a near-just [[8/7]] equal to a third of a meantone fifth ([[~]][[3/2]]) and is most notable for being a surprisingly elegant extension of meantone to the 7-limit (though it splits the generator), as tempering 81/80 makes S6 = [[36/35]] (the distance between [[6/5]] and [[7/6]]) and S8 = [[64/63]] (the distance between [[8/7]] and [[9/8]]) equivalent, so it seems natural to want to equate S6 = S7 = S8, where S7 = [[49/48]] (the distance between 7/6 and 8/7), so that 9/8, 8/7, 7/6, 6/5 are made equidistant. As a result, not only is 8/7 a third of 3/2, but also, because of tempering [[1728/1715|S6/S7]], we have that 7/6 is a third of [[8/5]]. Combining it with [[#Septimal meantone]] (among other things) results in [[31edo]], where it is quite accurate, while combining it with the less accurate [[#Flattone]] results in [[26edo]], where it is quite damaged. | Mothra makes a near-just [[8/7]] equal to a third of a meantone fifth ([[~]][[3/2]]) and is most notable for being a surprisingly elegant extension of meantone to the 7-limit (though it splits the generator), as tempering out 81/80 makes S6 = [[36/35]] (the distance between [[6/5]] and [[7/6]]) and S8 = [[64/63]] (the distance between [[8/7]] and [[9/8]]) equivalent, so it seems natural to want to equate S6 = S7 = S8, where S7 = [[49/48]] (the distance between 7/6 and 8/7), so that 9/8, 8/7, 7/6, 6/5 are made equidistant. As a result, not only is 8/7 a third of 3/2, but also, because of tempering out [[1728/1715|S6/S7]], we have that 7/6 is a third of [[8/5]]. Combining it with [[#Septimal meantone]] (among other things) results in [[31edo]], where it is quite accurate, while combining it with the less accurate [[#Flattone]] results in [[26edo]], where it is quite damaged. | ||
The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5. | The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5. | ||
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* 46 for adding {13} (22L 14s) | * 46 for adding {13} (22L 14s) | ||
Echidna has a generator of [[11/10]] or equivalently [[9/7]] because 11/10 * 9/7 = [[99/70]] is its period of half an octave, and can be seen as splitting the fourth of [[srutal archagall]] into three [[11/10]]'s by tempering [[4000/3993|S10/S11]] = (12/9)/(11/10)<sup>3</sup> = (4/3)/(11/10)<sup>3</sup> so that [[12/11]] and [[10/9]] are made equidistant from 11/10. It can be seen as a high-accuracy version of [[hedgehog]] and as a mild detempering of [[22edo]] that achieves an accurate and distinctly consistent [[11-odd-limit]]. In fact, of the three smallest edos that are distinctly consistent in the 11-odd-limit, which are [[58edo]], [[72edo]] and [[80edo]], echidna is supported by the smallest and third-smallest (so 72edo is in a sense the odd one out, being the one that ''doesn't'' support echidna). The smallest edo consistent in the 11-odd-limit, 22edo, is in fact a trivial tuning of echidna, where the generator is conflated with 12/11 and 10/9. 58edo and 80edo are both interesting options, being the merge of echidna and a variety of other notable temperaments, so depending on preference and tuning needs, though 80edo is the more optimal tuning for it (especially in the full 17-limit). | Echidna has a generator of [[11/10]] or equivalently [[9/7]] because 11/10 * 9/7 = [[99/70]] is its period of half an octave, and can be seen as splitting the fourth of [[srutal archagall]] into three [[11/10]]'s by tempering out [[4000/3993|S10/S11]] = (12/9)/(11/10)<sup>3</sup> = (4/3)/(11/10)<sup>3</sup> so that [[12/11]] and [[10/9]] are made equidistant from 11/10. It can be seen as a high-accuracy version of [[hedgehog]] and as a mild detempering of [[22edo]] that achieves an accurate and distinctly consistent [[11-odd-limit]]. In fact, of the three smallest edos that are distinctly consistent in the 11-odd-limit, which are [[58edo]], [[72edo]] and [[80edo]], echidna is supported by the smallest and third-smallest (so 72edo is in a sense the odd one out, being the one that ''doesn't'' support echidna). The smallest edo consistent in the 11-odd-limit, 22edo, is in fact a trivial tuning of echidna, where the generator is conflated with 12/11 and 10/9. 58edo and 80edo are both interesting options, being the merge of echidna and a variety of other notable temperaments, so depending on preference and tuning needs, though 80edo is the more optimal tuning for it (especially in the full 17-limit). | ||
Echidna is notable as achieving no-13's [[17-limit]] harmony with accuracy in a surprisingly small number of notes. [[13/8]] can be found too but is the most complex, being found at (11/10)<sup>16</sup> plus a half-octave period, octave-reduced. However, as primes 5 and 11 are also found in the same direction, intervals of 13 are common even in the 22-note MOS, so the 36-note MOS is more useful than might be suspected, despite not finding every odd from the same position. | Echidna is notable as achieving no-13's [[17-limit]] harmony with accuracy in a surprisingly small number of notes. [[13/8]] can be found too but is the most complex, being found at (11/10)<sup>16</sup> plus a half-octave period, octave-reduced. However, as primes 5 and 11 are also found in the same direction, intervals of 13 are common even in the 22-note MOS, so the 36-note MOS is more useful than might be suspected, despite not finding every odd from the same position. | ||
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* 22 (or 23) for adding {11, 55(, 69)} ([[3L 19s]] or [[3L 22s]]) | * 22 (or 23) for adding {11, 55(, 69)} ([[3L 19s]] or [[3L 22s]]) | ||
Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)<sup>8</sup>. The rationale for this is that (5/4)<sup>3</sup> falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)<sup>2</sup> = [[25/16]] by [[128/125]] twice we get [[~]][[3/2]]. Therefore, in an optimized tuning, we can expect the fifth to be slightly flat, so that [[25/24]] is sharpened so that it makes sense to equate with a slightly flattened [[~]][[24/23]] by tempering their difference, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|as documented in its interval chain]]. Because of dividing 6/1 into eight, it admits a neutral third at 4 generators so that an extension to prime 11 is also natural by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]] so that [[~]][[11/9]][[~]][[27/22]] is the neutral third. | Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)<sup>8</sup>. The rationale for this is that (5/4)<sup>3</sup> falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)<sup>2</sup> = [[25/16]] by [[128/125]] twice we get [[~]][[3/2]]. Therefore, in an optimized tuning, we can expect the fifth to be slightly flat, so that [[25/24]] is sharpened so that it makes sense to equate with a slightly flattened [[~]][[24/23]] by tempering out their difference, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|as documented in its interval chain]]. Because of dividing 6/1 into eight, it admits a neutral third at 4 generators so that an extension to prime 11 is also natural by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]] so that [[~]][[11/9]][[~]][[27/22]] is the neutral third. | ||
Würschmidt can be seen as something like a [[cluster temperament]] with 3 main clusters, and with [[~]][[128/125]][[~]][[46/45]] as the interval separating intervals in a given cluster. A notable extension to prime 7 is [[#Hemiwürschmidt]] by splitting the generator into two [[~]][[28/25]]'s, which is thus the result of combining würschmidt with [[#Didacus]]. | Würschmidt can be seen as something like a [[cluster temperament]] with 3 main clusters, and with [[~]][[128/125]][[~]][[46/45]] as the interval separating intervals in a given cluster. A notable extension to prime 7 is [[#Hemiwürschmidt]] by splitting the generator into two [[~]][[28/25]]'s, which is thus the result of combining würschmidt with [[#Didacus]]. | ||
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Note count: 23 for {3, 5, 7, 9, 11, 15, 21} ([[10L 11s]] or [[10L 21s]]) | Note count: 23 for {3, 5, 7, 9, 11, 15, 21} ([[10L 11s]] or [[10L 21s]]) | ||
Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], splitting 3/2 into three by tempering [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] and then splitting the [[~]][[8/7]] in two by tempering [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[meantone]] for 31edo and [[garibaldi]] for 41edo. | Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] and then splitting the [[~]][[8/7]] in two by tempering out [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[meantone]] for 31edo and [[garibaldi]] for 41edo. | ||
== ~17-limit focus == | == ~17-limit focus == | ||
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Many extensions to other primes exist, but most are not accurate enough to be microtemperaments, except for the extension to prime 41 by tempering out [[1025/1024]] = ([[41/32]])/([[32/25]]). However, as it is common to want to extend schismic, we will note common extensions here: | Many extensions to other primes exist, but most are not accurate enough to be microtemperaments, except for the extension to prime 41 by tempering out [[1025/1024]] = ([[41/32]])/([[32/25]]). However, as it is common to want to extend schismic, we will note common extensions here: | ||
* [[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth. | * [[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering out [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth. | ||
* Schismic [[tridecapyth comma|tridecapyth]] (which is the 2.3.5.13 version of [[#Cassandra]]) finds 13/4 as (9/8)<sup>10</sup> and demands an approximately Pythagorean tuning. | * Schismic [[tridecapyth comma|tridecapyth]] (which is the 2.3.5.13 version of [[#Cassandra]]) finds 13/4 as (9/8)<sup>10</sup> and demands an approximately Pythagorean tuning. | ||