Tour of regular temperaments: Difference between revisions
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===[[Archytas clan|Archytas or Ru clan]] (P8, P5)=== | ===[[Archytas clan|Archytas or Ru clan]] (P8, P5)=== | ||
This clan tempers out the Archytas comma, [[64/63]]. It equates 7/4 with 16/9. The clan consists of rank two temperaments, and should not be confused with the [[archytas family]] of rank three temperaments. Its best downward extension is [[superpyth]]. | This clan tempers out the Archytas comma, [[64/63]]. It equates 7/4 with 16/9. The clan consists of rank two temperaments, and should not be confused with the [[archytas family]] of rank three temperaments. Its best downward extension is [[superpyth]]. | ||
===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) === | |||
This clan tempers out the septimal third-tone [[28/27]], a low-accuracy temperament that equates 7/6 with 9/8, and 7/4 with 27/16. | |||
=== [[Harrison's comma|Harrison or Laru clan]] (P8, P5) === | === [[Harrison's comma|Harrison or Laru clan]] (P8, P5) === | ||
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This clan tempers out the [[garischisma]], {{Monzo|25 -14 0 -1}} = 33554432/33480783. It equates 8/7 to two apotomes ({{Monzo|-11 7}} = 2187/2048), and 7/4 to a double-diminished 8ve {{Monzo|23 -14}}. This clan includes [[Vulture family #Vulture|vulture]], [[Breedsmic temperaments #Newt|newt]], [[Schismatic family #Garibaldi|garibaldi]], [[Landscape microtemperaments #Sextile|sextile]], and [[Canousmic temperaments #Satin|satin]]. | This clan tempers out the [[garischisma]], {{Monzo|25 -14 0 -1}} = 33554432/33480783. It equates 8/7 to two apotomes ({{Monzo|-11 7}} = 2187/2048), and 7/4 to a double-diminished 8ve {{Monzo|23 -14}}. This clan includes [[Vulture family #Vulture|vulture]], [[Breedsmic temperaments #Newt|newt]], [[Schismatic family #Garibaldi|garibaldi]], [[Landscape microtemperaments #Sextile|sextile]], and [[Canousmic temperaments #Satin|satin]]. | ||
===[[ | ===[[Slendro clan|Slendro (Semaphore) or Zozo clan]] (P8, P4/2)=== | ||
This clan tempers out the | This clan tempers out the slendro diesis, [[49/48]]. Its generator is ~8/7 or ~7/6. Its best downward extension is [[godzilla]]. See also [[Semaphore]]. | ||
=== Laruru clan (P8/2, P5) === | === Laruru clan (P8/2, P5) === | ||
This clan tempers out the Laruru comma, {{Monzo|-7 8 0 -2}} = 6561/6272. Two ~81/56 periods equal an 8ve. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major 2nd ~9/8 is divided in half, with each half equated to ~28/27. See also the Diaschismatic or Sagugu temperament and the Jubalismic or Biruyo temperament. | This clan tempers out the Laruru comma, {{Monzo|-7 8 0 -2}} = 6561/6272. Two ~81/56 periods equal an 8ve. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major 2nd ~9/8 is divided in half, with each half equated to ~28/27. See also the Diaschismatic or Sagugu temperament and the Jubalismic or Biruyo temperament. | ||
=== Sasa-zozo clan (P8, P5/2) === | === Sasa-zozo clan (P8, P5/2) === | ||
This clan tempers out the Sasa-zozo comma, {{Monzo|15 -13 0 2}} = 1605632/1594323, and includes as a strong extension the [[Hemififths]] temperament. 7/4 is equated to 13 generators minus 3 octaves. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament. | This clan tempers out the Sasa-zozo comma, {{Monzo|15 -13 0 2}} = 1605632/1594323, and includes as a strong extension the [[Hemififths]] temperament. 7/4 is equated to 13 generators minus 3 octaves. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament. | ||
=== | ===[[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)=== | ||
This clan tempers out the | This clan tempers out the gamelisma, {{Monzo|-10 1 0 3}} = 1029/1024. Three ~8/7 generators equals a 5th. 7/4 is equated to an 8ve minus a generator. Five generators is slightly flat of 2/1, making this a [[cluster temperament]]. See also Sawa and Lasepzo. | ||
A particularly noteworthy member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps, thus it's a weak extension. Its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72EDO. | |||
=== Trizo clan (P8, P5/3) === | === Trizo clan (P8, P5/3) === | ||
This clan tempers out the Trizo comma, {{Monzo|-2 -4 0 3}} = 343/324, a low-accuracy temperament. Three ~7/6 generators equals a 5th, and four equal ~7/4. An obvious interpretation of the ~234¢ generator is 8/7, leading to the much more accurate Gamelismic or Latrizo temperament. | This clan tempers out the Trizo comma, {{Monzo|-2 -4 0 3}} = 343/324, a low-accuracy temperament. Three ~7/6 generators equals a 5th, and four equal ~7/4. An obvious interpretation of the ~234¢ generator is 8/7, leading to the much more accurate Gamelismic or Latrizo temperament. | ||
=== | === Triru clan (P8/3, P5) === | ||
This clan tempers out the | This clan tempers out the Triru comma, {{Monzo|-1 6 0 -3}} = 729/686, a low-accuracy temperament. Three ~9/7 periods equals an 8ve. The generator is ~3/2, and two generators minus a period equals ~7/4. An obvious 5-limit interpretation of the ~400¢ period is 5/4, leading to the [[augmented]] temperament. | ||
=== Latriru clan (P8, P11/3) === | === Latriru clan (P8, P11/3) === | ||
This clan tempers out the Latriru comma, {{Monzo|-9 11 0 -3}} = 177147/175616. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. An obvious 2.3.5.7 interpretation of the generator is 7/5, leading to the [[liese]] temperament, which is a weak extension of Meantone. | This clan tempers out the Latriru comma, {{Monzo|-9 11 0 -3}} = 177147/175616. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. An obvious 2.3.5.7 interpretation of the generator is 7/5, leading to the [[liese]] temperament, which is a weak extension of Meantone. | ||
=== | === Saquadru clan (P8, P12/4) === | ||
This clan | This clan tempers out the Saquadru comma, {{Monzo|16 -3 0 -4}} = 65536/64827. Its generator is ~21/16. Four generators makes ~3/1. 7/4 is equated to 2 octaves minus 3 generators. This clan includes as a strong extension the [[Vulture family|vulture]] temperament, which is in the vulture family. | ||
=== Laquadru clan (P8, P11/4) === | === Laquadru clan (P8, P11/4) === | ||
This clan tempers out the Laquadru comma, {{Monzo|-3 9 0 -4}} = 19683/19208. its generator is ~9/7. Four generators equals ~8/3. 7/4 is equated to 4 octaves minus 9 generators. This clan includes as a strong extension the [[squares]] temperament, which is a weak extension of meantone. | This clan tempers out the Laquadru comma, {{Monzo|-3 9 0 -4}} = 19683/19208. its generator is ~9/7. Four generators equals ~8/3. 7/4 is equated to 4 octaves minus 9 generators. This clan includes as a strong extension the [[squares]] temperament, which is a weak extension of meantone. | ||
=== [[Cloudy clan|Cloudy or Laquinzo clan]] (P8/5, P5) === | === [[Cloudy clan|Cloudy or Laquinzo clan]] (P8/5, P5) === | ||
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=== Saquinzo clan (P8, P12/5) === | === Saquinzo clan (P8, P12/5) === | ||
This clan tempers out the Saquinzo comma, {{Monzo|5 -12 0 5}} = 537824/531441. Its generator is ~243/196 = ~380¢. Five generators makes ~3/1. 7/4 is equated to 12 generators minus 3 octaves. An obvious 5-limit interpretation of the generator is 5/4, leading to the [[magic]] temperament, which is in the Magic family. | This clan tempers out the Saquinzo comma, {{Monzo|5 -12 0 5}} = 537824/531441. Its generator is ~243/196 = ~380¢. Five generators makes ~3/1. 7/4 is equated to 12 generators minus 3 octaves. An obvious 5-limit interpretation of the generator is 5/4, leading to the [[magic]] temperament, which is in the Magic family. | ||
===[[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)=== | |||
This clan temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the [[porcupine]] temperament, as does equating the period to ~7/5. | |||
=== Lasepzo clan (P8, P11/7) === | === Lasepzo clan (P8, P11/7) === | ||
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. | 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. | ||
===[[Septiness clan|Septiness or Sasasepru clan]] (P8, P11/7) === | |||
This clan tempers out the ''septiness'' comma {{Monzo|26 -4 0 -7}} = 67108864/66706983. Its generator is ~147/128, four of them gives ~7/4, and seven of them gives ~8/3. Five generators is ~12.5¢ sharp of 2/1, making this a [[cluster temperament]]. | |||
=== Sepru clan (P8, P12/7) === | === Sepru clan (P8, P12/7) === | ||
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===[[Mirwomo temperaments|Mirwomo or Labizoyo temperaments]]=== | ===[[Mirwomo temperaments|Mirwomo or Labizoyo temperaments]]=== | ||
Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768. | Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768. | ||
===[[Catasyc temperaments|Catasyc or Laruquadbiyo temperaments]]=== | |||
Catasyc rank-two temperaments temper out the ''catasyc'' comma, {{Monzo|-11 -3 8 -1}} = 390625/387072. | |||
===[[Compass temperaments|Compass or Quinruyoyo temperaments]]=== | |||
Compass rank-two temperaments temper out the compass comma, {{Monzo|-6 -2 10 -5}} = 9765625/9680832. | |||
===[[Trimyna temperaments|Trimyna or Quinzogu temperaments]]=== | ===[[Trimyna temperaments|Trimyna or Quinzogu 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, {{Monzo|16 2 -1 -6}} = 589824/588245. | Hewuermera rank-two temperaments temper out the ''hewuermera'' comma, {{Monzo|16 2 -1 -6}} = 589824/588245. | ||
===[[Decovulture temperaments|Decovulture or Sasabirugugu temperaments]]=== | |||
Decovulture rank-two temperaments temper out the ''decovulture'' comma, {{Monzo|26 -7 -4 -2}} = 67108864/66976875. | |||
===[[Horwell temperaments|Horwell or Lazoquinyo temperaments]]=== | ===[[Horwell temperaments|Horwell or Lazoquinyo temperaments]]=== | ||