Bird's eye view of temperaments by accuracy: Difference between revisions

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[[#Generator tunings|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224

Schismic is ~~a very~~ accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])4, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].

Schismic is an extremely accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])4, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].

In schismic, (9/8)6 overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.

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* 82: see [[#Gary]]

* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25

* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25}

Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2)

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Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]].

Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period).

Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). This structure is practically identical to that of [[cassaschismic]], only that the minicomma is not an independent generator and is instead found in the deep in the diploid chain of fifths.

It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]).

It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure. It isn't a true generator; were it independent, the temperament would be [[cassaschismic]].

It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].

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Note counts:

* 41 for {3, 9~~, 7~~, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])

* 41 for {3, 7, 9, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])

Generator tunings: 24\41, 55\94, 79\135, 498\851

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* the most accurate is [[#Sendai]] which finds primes 23 and 29

* the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23

* the simplest but least accurate is [[#Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament.

* the simplest but least accurate is [[Sensipent family#Sensor|Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament, for which the best tuning is [[46edo]].

==== [[Würschmidt]] ====

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[[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94

Garibaldi is ~~a very natural and very efficient (for its accuracy)~~ way ~~of effectively bestowing~~ prime 7 upon [[#Schismic]], at the cost of accuracy ~~as needing~~ a slightly sharper fifth tunes the 5-limit worse ~~so that~~ it is no longer a microtemperament. This is done by interpreting ([[9/8]])3 as [[~]][[10/7]] by tempering out [[5120/5103~~|S8/S9~~]] so that 8/7 and 10/9 are equidistant from 9/8, ~~with the step being a convenient tempered comma-sized interval that simultaneously not only represents not only~~ [[64/63]] = ~~S8 and~~ [[81/80]] ~~= S9 but~~ also the [[Pythagorean comma]] ~~(as per schismic)~~, equal to (9/8)6 / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 53 = [[147edo]] also ~~supports~~ it but with yet more inconsistencies due to the finer gamut.

Garibaldi is arguably the best way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)3 as [[~]][[10/7]] by tempering out S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is equal to (9/8)6 / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.

~~Extensions include~~:

Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:

* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]), corresponding to [[tetracot]] in 2.3.5.11 (by tempering out (S9/S11 = [[243/242]],) S10 = [[100/99]] and S10/(S9/S11) = [[2200/2187]] respectively) which splits the halved fifth into two small major seconds of [[~]][[11/10]][[~]][[10/9]] around 175.6 cents. However, there is significant damage to 15/13 and 13/10.

* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]], corresponding to reaching 13/4 through (9/8)10 (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.)

* Cassandra (41 & 53), tempering out 385/384 and 352/351, finds prime 11 at 23 fifths, and prime 13 at 20 fifths. It is the best extension, with support ~~up to the 23-limit as seen in 94edo.~~

Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.)

* Andromeda (41f & 53), tempering out 385/384 and 351/350, finds prime 11 at 23 fifths, and prime 13 at -33 fifths.

* Helenus (41ef & 53), tempering out 176/175 and 351/350, finds prime 11 at -30 fifths, and prime 13 at -33 fifths.

Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about primes 7 and 11 or prime 5 and 13:

* For primes 7 and 11, [[~~41edo~~]] ~~is better~~, ~~as it~~ finds ~~[[11/9]]~~[[~]][[27/~~22]] as [[Sqrt(3/2)|half of the fifth~~]] ~~and~~ as ~~a comma above [[~]][[6/5]] or~~ a comma below [[~]][[5/4~~]]; This also corresponds to being the intersection of [[cassandra~~]] and ~~[[andromeda]]~~ (~~respectively~~)~~. Primes 5 and 13 are worse, as these mappings prefer an ever so slightly flat fifth.~~

* For primes 5 and 13, [[53edo]] is better, as it has a close to just fifth that benefits schismic, tempering out 325/~~324 too so that [[~]][[~~16/13~~]] is a comma below~~ [[~]][[5/4]]. This ~~corresponds~~ to [[~~Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic~~]]~~. Primes 7 and 11 are worse, as their~~ cassandra ~~mappings prefer a slightly sharper fifth~~.

=== 11-limit focus ===

@@ Line 90: / Line 90: @@
 [[#Generator tunings|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224
-Schismic is a very accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].
+Schismic is an extremely accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].
 In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.
@@ Line 125: / Line 125: @@
 * 82: see [[#Gary]]
-* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25
+* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25}
 Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2)
@@ Line 131: / Line 131: @@
 Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]].
-Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period).
+Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). This structure is practically identical to that of [[cassaschismic]], only that the minicomma is not an independent generator and is instead found in the deep in the diploid chain of fifths.
-It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]).
+It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure. It isn't a true generator; were it independent, the temperament would be [[cassaschismic]].
 It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].
@@ Line 156: / Line 156: @@
 Note counts:
-* 41 for {3, 9, 7, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])
+* 41 for {3, 7, 9, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])
 Generator tunings: 24\41, 55\94, 79\135, 498\851
@@ Line 202: / Line 202: @@
 * the most accurate is [[#Sendai]] which finds primes 23 and 29
 * the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23
-* the simplest but least accurate is [[#Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament.
+* the simplest but least accurate is [[Sensipent family#Sensor|Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament, for which the best tuning is [[46edo]].
 ==== [[Würschmidt]] ====
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 [[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94
-Garibaldi is a very natural and very efficient (for its accuracy) way of effectively bestowing prime 7 upon [[#Schismic]], at the cost of accuracy as needing a slightly sharper fifth tunes the 5-limit worse so that it is no longer a microtemperament. This is done by interpreting ([[9/8]])<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103|S8/S9]] so that 8/7 and 10/9 are equidistant from 9/8, with the step being a convenient tempered comma-sized interval that simultaneously not only represents not only [[64/63]] = S8 and [[81/80]] = S9 but also the [[Pythagorean comma]] (as per schismic), equal to (9/8)<sup>6</sup> / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 53 = [[147edo]] also supports it but with yet more inconsistencies due to the finer gamut.
+Garibaldi is arguably the best way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is equal to (9/8)<sup>6</sup> / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.
-Extensions include:
+Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
+* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]), corresponding to [[tetracot]] in 2.3.5.11 (by tempering out (S9/S11 = [[243/242]],) S10 = [[100/99]] and S10/(S9/S11) = [[2200/2187]] respectively) which splits the halved fifth into two small major seconds of [[~]][[11/10]][[~]][[10/9]] around 175.6 cents. However, there is significant damage to 15/13 and 13/10.
+* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.)
-* Cassandra (41 & 53), tempering out 385/384 and 352/351, finds prime 11 at 23 fifths, and prime 13 at 20 fifths. It is the best extension, with support up to the 23-limit as seen in 94edo.
+Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.)
-* Andromeda (41f & 53), tempering out 385/384 and 351/350, finds prime 11 at 23 fifths, and prime 13 at -33 fifths.
-* Helenus (41ef & 53), tempering out 176/175 and 351/350, finds prime 11 at -30 fifths, and prime 13 at -33 fifths.
-Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about primes 7 and 11 or prime 5 and 13:
-* For primes 7 and 11, [[41edo]] is better, as it finds [[11/9]][[~]][[27/22]] as [[Sqrt(3/2)|half of the fifth]] and as a comma above [[~]][[6/5]] or a comma below [[~]][[5/4]]; This also corresponds to being the intersection of [[cassandra]] and [[andromeda]] (respectively). Primes 5 and 13 are worse, as these mappings prefer an ever so slightly flat fifth.
-* For primes 5 and 13, [[53edo]] is better, as it has a close to just fifth that benefits schismic, tempering out 325/324 too so that [[~]][[16/13]] is a comma below [[~]][[5/4]]. This corresponds to [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]]. Primes 7 and 11 are worse, as their cassandra mappings prefer a slightly sharper fifth.
 === 11-limit focus ===