Gamelismic clan: Difference between revisions

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The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)3 × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.

The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap|3/2 {{=}} (8/7)3 × 1029/1024}}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred.

To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.

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: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''

Rodan tempers out 245/243 and can be described as the 41 & 46 temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.

Rodan tempers out 245/243 and can be described as the {{nowrap|41 & 46}} temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.

[[Subgroup]]: 2.3.5.7

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== Guiron ==

Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as 36 & 41. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap|36 & 41}} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

[[Subgroup]]: 2.3.5.7

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== Mothra ==

Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as 26 & 31. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].

Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap|26 & 31}}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].

Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.

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Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the 16 & 21 temperament.

Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap|16 & 21}} temperament.

If we discard the inaccurate mapping of prime 3, we get [[shoe]], so that the large commas of gorgo are explained practically entirely by the inaccurate 3, meaning that this temperament is much more accurate than its comma list suggests.

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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''

Oncle can be described as the 31 & 36c temperamnet.

Oncle can be described as the {{nowrap|31 & 36c}} temperamnet.

[[Subgroup]]: 2.3.5.7

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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''

Archaeotherium can be described as the 21 & 26 temperamnet.

Archaeotherium can be described as the {{nowrap|21 & 26}} temperamnet.

[[Subgroup]]: 2.3.5.7

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Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the 11 & 16 temperamnet.

Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap|11 & 16}} temperamnet.

[[Subgroup]]: 2.3.5.7

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: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''

Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the 31 & 41 temperament. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.

Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap|31 & 41}} temperament. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

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== Hemiseven ==

Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the 72 & 77 temperament. [[149edo]] is an obvious tuning.

Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap|72 & 77}} temperament. [[149edo]] is an obvious tuning.

[[Subgroup]]: 2.3.5.7

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: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''

Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of ~21/20, three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the {{nowrap|31 & 46}} temperament, and [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)1/9 as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)1/10.

Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (aka 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22*S23*S24]], [[121/120|S11]],) [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]} so that it's the temperament that equalizes the 20::25 segment of the harmonic series.

@@ Line 1: / Line 1: @@
-The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.
+The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap|3/2 {{=}} (8/7)<sup>3</sup> × 1029/1024}}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred.
 To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.
@@ Line 85: / Line 85: @@
 : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''
-Rodan tempers out 245/243 and can be described as the 41 & 46 temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.
+Rodan tempers out 245/243 and can be described as the {{nowrap|41 &amp; 46}} temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.
 [[Subgroup]]: 2.3.5.7
@@ Line 250: / Line 250: @@
 == Guiron ==
-Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as 36 & 41. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
+Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap|36 &amp; 41}} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
 [[Subgroup]]: 2.3.5.7
@@ Line 317: / Line 317: @@
 == Mothra ==
-Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as 26 & 31. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].
+Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap|26 &amp; 31}}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].
 Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
@@ Line 454: / Line 454: @@
 {{ See also | Shoe }}
-Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the 16 & 21 temperament.
+Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap|16 &amp; 21}} temperament.
 If we discard the inaccurate mapping of prime 3, we get [[shoe]], so that the large commas of gorgo are explained practically entirely by the inaccurate 3, meaning that this temperament is much more accurate than its comma list suggests.
@@ Line 565: / Line 565: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''
-Oncle can be described as the 31 & 36c temperamnet.
+Oncle can be described as the {{nowrap|31 &amp; 36c}} temperamnet.
 [[Subgroup]]: 2.3.5.7
@@ Line 586: / Line 586: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''
-Archaeotherium can be described as the 21 & 26 temperamnet.
+Archaeotherium can be described as the {{nowrap|21 &amp; 26}} temperamnet.
 [[Subgroup]]: 2.3.5.7
@@ Line 607: / Line 607: @@
 {{See also| Mavila family }}
-Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the 11 & 16 temperamnet.
+Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap|11 &amp; 16}} temperamnet.
 [[Subgroup]]: 2.3.5.7
@@ Line 646: / Line 646: @@
 : ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''
-Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the 31 & 41 temperament. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.
+Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap|31 &amp; 41}} temperament. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,024: / Line 1,024: @@
 == Hemiseven ==
-Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the 72 & 77 temperament. [[149edo]] is an obvious tuning.
+Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap|72 &amp; 77}} temperament. [[149edo]] is an obvious tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,095: / Line 1,095: @@
 : ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''
-Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of ~21/20, three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the {{nowrap|31 & 46}} temperament, and [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
+Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of ~21/20, three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the {{nowrap|31 &amp; 46}} temperament, and [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
 Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (aka 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22*S23*S24]], [[121/120|S11]],) [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]} so that it's the temperament that equalizes the 20::25 segment of the harmonic series.