Tour of regular temperaments: Difference between revisions
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This family tempers out the apotome, {{Monzo|-11 7 0}} = 2187/2048, which implies [[7edo|7EDO]]. | This family tempers out the apotome, {{Monzo|-11 7 0}} = 2187/2048, which implies [[7edo|7EDO]]. | ||
===[[ | ===[[Compton family|Pythagorean or Lalawa family]] (P8/12, ^1)=== | ||
The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{Monzo|-19 12 0}}, which implies [[12edo|12EDO]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. | The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{Monzo|-19 12 0}}, which implies [[12edo|12EDO]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. | ||
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This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}};. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7. | This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}};. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7. | ||
=== [[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6) === | ===[[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6) === | ||
This tempers out Ampersand's comma = 34171875/33554432 = {{Monzo|-25 7 6}}. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the [[Miracle]] temperament. | This tempers out Ampersand's comma = 34171875/33554432 = {{Monzo|-25 7 6}}. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the [[Miracle]] temperament. | ||
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===[[Mutt temperament|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)=== | ===[[Mutt temperament|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)=== | ||
This tempers out the [[ | This tempers out the [[mutt comma]], {{Monzo|-44 -3 21}, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is <u>not</u> 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord. | ||
===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)=== | ===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)=== | ||
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=== Lasepzo clan (P8, P11/7) === | === Lasepzo clan (P8, P11/7) === | ||
This clan tempers out the Lasepzo comma | This clan tempers out the Lasepzo comma {{Monzo|-18 -1 0 7}} = 823543/786432. Its generator is ~8/7. Six generators equals ~7/3, and seven generators equals ~8/3. Three generators is ~30¢ sharp of 3/2, and five generators is ~15¢ sharp of 2/1, making this a [[cluster temperament]]. See also Sawa and Latrizo. | ||
=== Sepru clan (P8, P12/7) === | === Sepru clan (P8, P12/7) === | ||
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===[[Landscape microtemperaments|Landscape or Trizogugu temperaments]]=== | ===[[Landscape microtemperaments|Landscape or Trizogugu temperaments]]=== | ||
Lanscape rank-two temperaments temper out the lanscape comma, {{Monzo|-4 6 -6 3}} = 250047/250000. | Lanscape rank-two temperaments temper out the lanscape comma, {{Monzo|-4 6 -6 3}} = 250047/250000. These have a period of 1/3 octave, but ~5/4 is not equated with a period, resulting in small intervals. | ||
===[[Dimcomp temperaments|Dimcomp or Quadruyoyo temperaments]]=== | ===[[Dimcomp temperaments|Dimcomp or Quadruyoyo temperaments]]=== | ||
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===[[Hewuermera temperaments|Hewuermera or Satribiru-agu temperaments]]=== | ===[[Hewuermera temperaments|Hewuermera or Satribiru-agu temperaments]]=== | ||
Hewuermera rank-two temperaments temper out the ''hewuermera'' comma (named by [[User:Xenllium|Xenllium]]), {{Monzo|16 2 -1 -6}} = 589824/588245. | Hewuermera rank-two temperaments temper out the ''hewuermera'' comma (named by [[User:Xenllium|Xenllium]]), {{Monzo|16 2 -1 -6}} = 589824/588245. | ||
===[[Akjaysmic temperaments|Akjaysmic or Trisaseprugu temperaments]]=== | |||
Akjaysmic rank-two temperaments temper out the akjaysma, {{Monzo|47 -7 -7 -7}}. These have a period of 1/7 octave, and five periods equals ~105/64. Unlike the Whitewood or Lawa family, ~3/2 is not equated with four-sevenths of an octave, resulting in small intervals. | |||
= Rank-3 temperaments = | = Rank-3 temperaments = | ||
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Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. | Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. | ||
===[[Garischismic | ===[[Garischismic family|Garischismic or Sasaru family]] (P8, P5, ^1)=== | ||
A garischismic temperament is one which tempers out the garischisma, {{Monzo|25 -14 0 -1}} = 33554432/33480783. | A garischismic temperament is one which tempers out the garischisma, {{Monzo|25 -14 0 -1}} = 33554432/33480783. | ||
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Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, {{Monzo|-10 1 0 3}} = 1029/1024. In the pergen, P5/3 is ~8/7. | Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, {{Monzo|-10 1 0 3}} = 1029/1024. In the pergen, P5/3 is ~8/7. | ||
===[[Stearnsmic | ===[[Stearnsmic family|Stearnsmic or Latribiru family]] (P8/2, P4/3, ^1)=== | ||
Stearnsmic temperaments temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. In the pergen, P8/2 is 343/243 and P4/3 is ~54/49. | Stearnsmic temperaments temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. In the pergen, P8/2 is 343/243 and P4/3 is ~54/49. | ||