List of regular temperaments: Difference between revisions

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|monocot|| 7, 12  
|monocot|| 7, 12  
|softer diatonic, m-chromatic|| Common historical temperament for 5-limit diatonic harmony. || 81/80
|softer diatonic, m-chromatic|| Common historical temperament for 5-limit diatonic harmony. || 81/80
|Medium
|4 Medium
|692-697c
|692-697c
|-
|-
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|monocot|| 19, 31  
|monocot|| 19, 31  
|softer diatonic, m-chromatic||  Canonical extension of the above to 2...7. || 81/80, 225/224
|softer diatonic, m-chromatic||  Canonical extension of the above to 2...7. || 81/80, 225/224
|Medium
|4 Medium
|695-697c
|695-697c
|-
|-
Line 32: Line 32:
|Less accurate extension, more melodic intuition, easily extends to higher limits. 7-form version of Meantone.
|Less accurate extension, more melodic intuition, easily extends to higher limits. 7-form version of Meantone.
|525/512, 81/80
|525/512, 81/80
|Low
|6 Low
|692-694c
|692-694c
|-
|-
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|Adds a 600c tritone representing 7/5 to meantone.
|Adds a 600c tritone representing 7/5 to meantone.
|81/80, 50/49
|81/80, 50/49
|Medium-low
|5 Medium-low
|92-96c
|92-96c
|-
|-
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|onyx, pine|| Moderate-accuracy 2.3.5.11 temperament with a ~160c generator and a heptatonic MOS.  
|onyx, pine|| Moderate-accuracy 2.3.5.11 temperament with a ~160c generator and a heptatonic MOS.  
|250/243, 100/99
|250/243, 100/99
|Medium-low
|5 Medium-low
|161-166c
|161-166c
|-
|-
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|onyx, pine|| Extension of the above to the full 11-limit.  
|onyx, pine|| Extension of the above to the full 11-limit.  
|250/243, 100/99, 64/63
|250/243, 100/99, 64/63
|Medium-low
|5 Medium-low
|161-163c
|161-163c
|-
|-
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|mosh, dicoid|| 5-limit exotemperament equating 5/4 and 6/5 to the same interval.  
|mosh, dicoid|| 5-limit exotemperament equating 5/4 and 6/5 to the same interval.  
|25/24
|25/24
|Very low
|7 Very low
|670-680c, 720-730c
|670-680c, 720-730c
|-
|-
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|archeotonic, 7L6s|| Interprets (3/2)^(1/4) as 10/9.  
|archeotonic, 7L6s|| Interprets (3/2)^(1/4) as 10/9.  
|100/99, 243/242
|100/99, 243/242
|High (2.3.5), Medium (extensions)
|2 High (2.3.5), Medium (extensions)
|175-180c
|175-180c
|-
|-
Line 84: Line 84:
|Sets 6/5 to 300 cents; a step up from 600 cents is 3/2.
|Sets 6/5 to 300 cents; a step up from 600 cents is 3/2.
|648/625
|648/625
|Medium-low
|5 Medium-low
|685-700c
|685-700c
|-
|-
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|Exotempered extension of the above that sets 7/6 to 300 cents.
|Exotempered extension of the above that sets 7/6 to 300 cents.
|36/35, 50/49
|36/35, 50/49
|Very low
|7 Very low
|685-700c
|685-700c
|-
|-
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|Sets 5/4 to 400 cents, a step down from 800c is 3/2 and 2 steps up is 7/4.
|Sets 5/4 to 400 cents, a step down from 800c is 3/2 and 2 steps up is 7/4.
|128/125
|128/125
|Low
|6 Low
|705-715c
|705-715c
|-
|-
Line 116: Line 116:
|5-limit interpretation of Pythagorean tuning, best tuned when the fifth is flattened by a fraction of a cent.
|5-limit interpretation of Pythagorean tuning, best tuned when the fifth is flattened by a fraction of a cent.
|schisma
|schisma
|Very high
|1 Very high
|701-702c
|701-702c
|-
|-
Line 126: Line 126:
|7-limit interpretation of Pythagorean tuning. It is most accurate when the fifth is tuned slightly sharp.
|7-limit interpretation of Pythagorean tuning. It is most accurate when the fifth is tuned slightly sharp.
|schisma, 225/224
|schisma, 225/224
|Medium-high
|3 Medium-high
|702-703c
|702-703c
|-
|-
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|5/4 is 4 times the difference between 3/2 and 800c.
|5/4 is 4 times the difference between 3/2 and 800c.
|misty comma
|misty comma
|Medium-high
|3 Medium-high
|701-708c
|701-708c
|-
|-
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|Temperament characterized by a perfect semioctave and a sharpened fifth or semitone generator. Two generators down reaches 5/4.
|Temperament characterized by a perfect semioctave and a sharpened fifth or semitone generator. Two generators down reaches 5/4.
|diaschisma, 136/135
|diaschisma, 136/135
|Medium
|4 Medium
|100-111c
|100-111c
|-
|-
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|Rather complex 7-limit extension of the above.
|Rather complex 7-limit extension of the above.
|diaschisma, 126/125, 136/135
|diaschisma, 126/125, 136/135
|Medium
|4 Medium
|103-104c
|103-104c
|-
|-
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|Jubilic archytas diaschismic temperament. Contains jubilic chord structure and is strongly associated with 22edo.
|Jubilic archytas diaschismic temperament. Contains jubilic chord structure and is strongly associated with 22edo.
|diaschisma, 50/49, 136/135
|diaschisma, 50/49, 136/135
|Medium-low
|5 Medium-low
|109-111c
|109-111c
|-
|-
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|Stacks five flattened major thirds to form a perfect twelfth. 5/4-generated analog of Meantone.
|Stacks five flattened major thirds to form a perfect twelfth. 5/4-generated analog of Meantone.
|magisma
|magisma
|Medium
|4 Medium
|378-382c
|378-382c
|-
|-
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|Eight 5/4s stack to 3/2 and three 25/24s stack to two 16/15s. 5/4-generated analog of Schismic. Bad for MOSes.
|Eight 5/4s stack to 3/2 and three 25/24s stack to two 16/15s. 5/4-generated analog of Schismic. Bad for MOSes.
|576/575, 12167/12150
|576/575, 12167/12150
|Medium-high
|3 Medium-high
|386-389c
|386-389c
|-
|-
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|Extremely inaccurate exotemperament which equates 5/4 with 4/3.
|Extremely inaccurate exotemperament which equates 5/4 with 4/3.
|16/15
|16/15
|Extremely low
|8 Extremely low
|720-800c
|720-800c
|-
|-
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|A highly accurate temperament equating a stack of six slightly sharp 6/5's to one 3/1 and three 6/5's to one 26/15.
|A highly accurate temperament equating a stack of six slightly sharp 6/5's to one 3/1 and three 6/5's to one 26/15.
|kleisma, 325/324
|kleisma, 325/324
|High
|2 High
|317c
|317c
|-
|-
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|Generator is a sharp minor second of 16/15 which stacks 3 times to 5/4, and can be seen as a counterpart of Porcupine.
|Generator is a sharp minor second of 16/15 which stacks 3 times to 5/4, and can be seen as a counterpart of Porcupine.
|16875/16384
|16875/16384
|Medium-low
|5 Medium-low
|124-128c
|124-128c
|-
|-
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|Weak Jubilismic or Buzzard extension of Negri allowing intervals of 7 to be reached with a 600-cent offset.
|Weak Jubilismic or Buzzard extension of Negri allowing intervals of 7 to be reached with a 600-cent offset.
|16875/16384, 50/49, 243/242
|16875/16384, 50/49, 243/242
|Low
|6 Low
|124-128c
|124-128c
|-
|-
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|Semaphore extension of negri.
|Semaphore extension of negri.
|16875/16384, 49/48
|16875/16384, 49/48
|Very low
|7 Very low
|124-128c
|124-128c
|}
|}
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|soft pentic, harder diatonic, p-chromatic|| 2.3.7 counterpart of Meantone, which sharpens the fifth.  
|soft pentic, harder diatonic, p-chromatic|| 2.3.7 counterpart of Meantone, which sharpens the fifth.  
|64/63
|64/63
|Medium-low
|5 Medium-low
|709-720c
|709-720c
|-
|-
Line 273: Line 273:
|soft pentic, harder diatonic, p-chromatic|| Extension of the above to 2...7, favoring flatter tunings.  
|soft pentic, harder diatonic, p-chromatic|| Extension of the above to 2...7, favoring flatter tunings.  
|64/63, 245/243
|64/63, 245/243
|Medium-low
|5 Medium-low
|709-711c
|709-711c
|-
|-
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|Splits the fifth in 3 parts, each of which is 8/7. Little relation to actual [[Equipentatonic#Slendro|Slendro tuning]].  
|Splits the fifth in 3 parts, each of which is 8/7. Little relation to actual [[Equipentatonic#Slendro|Slendro tuning]].  
|gamelisma
|gamelisma
|Medium-high
|3 Medium-high
|231-234c
|231-234c
|-
|-
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|Meantone extension of the above.
|Meantone extension of the above.
|81/80, gamelisma
|81/80, gamelisma
|Medium
|4 Medium
|231c
|231c
|-
|-
Line 304: Line 304:
|More accurate extension of the above.
|More accurate extension of the above.
|245/243, gamelisma
|245/243, gamelisma
|Medium
|4 Medium
|234c
|234c
|-
|-
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|Generated by a 15/14~16/15 semitone, two of which reach a slendric 8/7.
|Generated by a 15/14~16/15 semitone, two of which reach a slendric 8/7.
|225/224, 243/242, gamelisma
|225/224, 243/242, gamelisma
|Medium-high
|3 Medium-high
|117c
|117c
|-
|-
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|Scale with small steps strongly associated with Carlos Alpha.
|Scale with small steps strongly associated with Carlos Alpha.
|126/125, gamelisma
|126/125, gamelisma
|Medium-high
|3 Medium-high
|78c
|78c
|-
|-
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|Sharpens the 21/16 so that four of them stacks to the 3/1.
|Sharpens the 21/16 so that four of them stacks to the 3/1.
|buzzardsma
|buzzardsma
|Medium
|4 Medium
|474-478c
|474-478c
|-
|-
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|Equipentatonic, inaccurate 7-limit temperament.
|Equipentatonic, inaccurate 7-limit temperament.
|49/48
|49/48
|Low
|6 Low
|240-250c
|240-250c
|-
|-
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|No-fives temperament generated by a flattened 9/7 equated with 14/11.
|No-fives temperament generated by a flattened 9/7 equated with 14/11.
|99/98, 243/242
|99/98, 243/242
|Medium
|4 Medium
|424-426c
|424-426c
|}
|}
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|mosh, dicoid|| Maps 11/9 and its fifth complement to a perfect neutral third.  
|mosh, dicoid|| Maps 11/9 and its fifth complement to a perfect neutral third.  
|243/242
|243/242
|Medium-high
|3 Medium-high
|345-355c
|345-355c
|-
|-
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|mosh, dicoid|| Meantone extension of the above. Can optionally be extended to set 7/4 equal to the semiflat minor seventh, or via a strong meantone extension.  
|mosh, dicoid|| Meantone extension of the above. Can optionally be extended to set 7/4 equal to the semiflat minor seventh, or via a strong meantone extension.  
|243/242, 81/80
|243/242, 81/80
|Medium-low
|5 Medium-low
|347-350c
|347-350c
|-
|-
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|Maps 16/13 and its fifth complement to a perfect neutral third.
|Maps 16/13 and its fifth complement to a perfect neutral third.
|512/507
|512/507
|Medium
|4 Medium
|350-360c
|350-360c
|-
|-
Line 404: Line 404:
|Archy extension of the above.
|Archy extension of the above.
|512/507, 64/63
|512/507, 64/63
|Medium-low
|5 Medium-low
|355-360c
|355-360c
|}
|}
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|Sets the spacing of the thirds to 7/6-#-6/5-#-#-5/4-#-9/7; mostly appears as a structural element of EDOs rather than an independent structure.
|Sets the spacing of the thirds to 7/6-#-6/5-#-#-5/4-#-9/7; mostly appears as a structural element of EDOs rather than an independent structure.
|4375/4374, 5120/5103
|4375/4374, 5120/5103
|High
|2 High
|338-340c
|338-340c
|-
|-
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|Acts as a closed circle of 12 fifths (see [[12edo]]), but with 5/4 flattened by a diesis from 400c, and 7/4 flattened by twice that amount from 1000c.
|Acts as a closed circle of 12 fifths (see [[12edo]]), but with 5/4 flattened by a diesis from 400c, and 7/4 flattened by twice that amount from 1000c.
|pythagorean comma
|pythagorean comma
|Medium-high
|3 Medium-high
|385c
|385c
|-
|-
Line 450: Line 450:
|Sets 6/5 and 7/6 the same distance from the 300c period, so that four ~25c generators stack to the semitone separating 3/2 from 600c.
|Sets 6/5 and 7/6 the same distance from the 300c period, so that four ~25c generators stack to the semitone separating 3/2 from 600c.
|50/49, 875/864
|50/49, 875/864
|Medium-low
|5 Medium-low
|325c
|325c
|-
|-
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|Sets 25/24 equal to twice 36/35.
|Sets 25/24 equal to twice 36/35.
|126/125, 1728/1715
|126/125, 1728/1715
|Medium-high
|3 Medium-high
|309-311c
|309-311c
|-
|-
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|Generator is a sharpened 7/6. Interval chain finds 7/6, 11/8, 8/5, 15/8, 11/10, 9/7, 3/2.  
|Generator is a sharpened 7/6. Interval chain finds 7/6, 11/8, 8/5, 15/8, 11/10, 9/7, 3/2.  
|99/98, 121/120, 176/175
|99/98, 121/120, 176/175
|Medium-high
|3 Medium-high
|270-273c
|270-273c
|-
|-
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|Divides the octave into nine equal parts representing 27/25 and half of 7/6.
|Divides the octave into nine equal parts representing 27/25 and half of 7/6.
|2401/2400, 4375/4374
|2401/2400, 4375/4374
|Very high
|1 Very high
|44-53c
|44-53c
|-
|-
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|Generator is a neutral second, but places primes at high complexity, preferring ratios between them.
|Generator is a neutral second, but places primes at high complexity, preferring ratios between them.
|126/125, 2430/2401
|126/125, 2430/2401
|Medium
|4 Medium
|154-155c
|154-155c
|}
|}
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|alpha-triseph<sup>[a]</sup>|| 7, 9  
|alpha-triseph<sup>[a]</sup>|| 7, 9  
|antidiatonic, armotonic, 9L 7s|| Basic antidiatonic temperament with no 3. || mabilisma
|antidiatonic, armotonic, 9L 7s|| Basic antidiatonic temperament with no 3. || mabilisma
|Medium
|4 Medium
|668-680c
|668-680c
|-
|-
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|alpha-triseph|| 16, 25  
|alpha-triseph|| 16, 25  
|antidiatonic, armotonic, 9L 7s|| High-accuracy but high complexity extension of prime 3. || gamelisma, magisma  
|antidiatonic, armotonic, 9L 7s|| High-accuracy but high complexity extension of prime 3. || gamelisma, magisma  
|Medium
|4 Medium
|672-675c
|672-675c
|-
|-
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|antidiatonic, armotonic, 9L 7s|| Combination of Mabilic and Semaphore.  
|antidiatonic, armotonic, 9L 7s|| Combination of Mabilic and Semaphore.  
|49/48, 28672/28125
|49/48, 28672/28125
|Low
|6 Low
|668-672c
|668-672c
|-
|-
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|antidiatonic, armotonic, 7L 9s|| Exotemperament serving as an antidiatonic analog of meantone.  
|antidiatonic, armotonic, 7L 9s|| Exotemperament serving as an antidiatonic analog of meantone.  
|36/35, 135/128
|36/35, 135/128
|Very low
|7 Very low
|675-680c
|675-680c
|-
|-
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|A high-accuracy rank-2 temperament generated by a tempered 77/64.
|A high-accuracy rank-2 temperament generated by a tempered 77/64.
|65536/65219
|65536/65219
|Medium-high
|3 Medium-high
|320-325c
|320-325c
|-
|-
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|Every other step of septimal meantone.
|Every other step of septimal meantone.
|3136/3125
|3136/3125
|High
|2 High
|192-196c
|192-196c
|}
|}
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|Basic tritave temperament that stacks 9/7 twice to reach 5/3. Generates the lambda (4L5s{{angbr|3/1}}) MOS scale, or can be used with octaves as Sensamagic.2.
|Basic tritave temperament that stacks 9/7 twice to reach 5/3. Generates the lambda (4L5s{{angbr|3/1}}) MOS scale, or can be used with octaves as Sensamagic.2.
|245/243
|245/243
|Medium-high
|3 Medium-high
|435-440c
|435-440c
|-
|-
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|Very sharp extension of Sensamagic, which finds the octave at 125/63.
|Very sharp extension of Sensamagic, which finds the octave at 125/63.
|91/90, 126/125, 169/168
|91/90, 126/125, 169/168
|Medium
|4 Medium
|440-445c
|440-445c
|}
|}

Revision as of 00:16, 2 April 2026

Main article: Regular temperament

Rank-2

2.3.5.x families

Rank-2 Temperaments
Form Family Name Subgroup Ploidacot ETs Usual Scale Type Description Commas Accuracy (Vector) Generator size
7 Syntonic Meantone 2.3.5 monocot 7, 12 softer diatonic, m-chromatic Common historical temperament for 5-limit diatonic harmony. 81/80 4 Medium 692-697c
Septimal Meantone 2.3.5.7 monocot 19, 31 softer diatonic, m-chromatic Canonical extension of the above to 2...7. 81/80, 225/224 4 Medium 695-697c
Flattone 2.3.5.7 monocot 19, 26 softer diatonic, m-chromatic Less accurate extension, more melodic intuition, easily extends to higher limits. 7-form version of Meantone. 525/512, 81/80 6 Low 692-694c
12 Injera 2.3.5.7 diploid monocot 12, 26 thalassic Adds a 600c tritone representing 7/5 to meantone. 81/80, 50/49 5 Medium-low 92-96c
7 Porcupine Porcupine 2.3.5.11 omega-tricot 15, 22 onyx, pine Moderate-accuracy 2.3.5.11 temperament with a ~160c generator and a heptatonic MOS. 250/243, 100/99 5 Medium-low 161-166c
Septimal Porcupine 2.3.5.7.11 omega-tricot 15, 22 onyx, pine Extension of the above to the full 11-limit. 250/243, 100/99, 64/63 5 Medium-low 161-163c
Interclassical Interclassical, Dicot 2.3.5 dicot 7, 10 mosh, dicoid 5-limit exotemperament equating 5/4 and 6/5 to the same interval. 25/24 7 Very low 670-680c, 720-730c
Tetracot [rename] Tetracot [rename] 2.3.5.11 tetracot 34, 41 archeotonic, 7L6s Interprets (3/2)^(1/4) as 10/9. 100/99, 243/242 2 High (2.3.5), Medium (extensions) 175-180c
12 Diminished Diminished 2.3.5 tetraploid monocot 12, 16 tetrawood, 4L 8s Sets 6/5 to 300 cents; a step up from 600 cents is 3/2. 648/625 5 Medium-low 685-700c
Dimisept 2.3.5.7 tetraploid monocot 12, 16 tetrawood, 4L 8s Exotempered extension of the above that sets 7/6 to 300 cents. 36/35, 50/49 7 Very low 685-700c
Augmented Augmented 2.3.5.7 triploid monocot 12, 15 triwood, tcherepnin Sets 5/4 to 400 cents, a step down from 800c is 3/2 and 2 steps up is 7/4. 128/125 6 Low 705-715c
Schismic Schismic 2.3.5 monocot 41, 53 harder diatonic, p-chromatic 5-limit interpretation of Pythagorean tuning, best tuned when the fifth is flattened by a fraction of a cent. schisma 1 Very high 701-702c
Garibaldi 2.3.5.7 monocot 41, 53 harder diatonic, p-chromatic 7-limit interpretation of Pythagorean tuning. It is most accurate when the fifth is tuned slightly sharp. schisma, 225/224 3 Medium-high 702-703c
Misty Misty 2.3.5 triploid monocot 12, 51 5/4 is 4 times the difference between 3/2 and 800c. misty comma 3 Medium-high 701-708c
Diaschismic Diaschismic 2.3.5.17 diploid monocot 12, 34 jaric, 10L 2s Temperament characterized by a perfect semioctave and a sharpened fifth or semitone generator. Two generators down reaches 5/4. diaschisma, 136/135 4 Medium 100-111c
Septimal Diaschismic 2.3.5.7.17 diploid monocot 12, 34 jaric, 10L 2s Rather complex 7-limit extension of the above. diaschisma, 126/125, 136/135 4 Medium 103-104c
10 Pajara 2.3.5.7.17 diploid monocot 12, 22 jaric, 10L 2s Jubilic archytas diaschismic temperament. Contains jubilic chord structure and is strongly associated with 22edo. diaschisma, 50/49, 136/135 5 Medium-low 109-111c
3 Magic Magic 2.3.5 alpha-pentacot 19, 22 mosh, sephiroid Stacks five flattened major thirds to form a perfect twelfth. 5/4-generated analog of Meantone. magisma 4 Medium 378-382c
Wurschmidt Wurschmidt 2.3.5.11.23 beta-octacot 31, 34 - Eight 5/4s stack to 3/2 and three 25/24s stack to two 16/15s. 5/4-generated analog of Schismic. Bad for MOSes. 576/575, 12167/12150 3 Medium-high 386-389c
8 Father Father 2.3.5 monocot 3, 5 antipentic Extremely inaccurate exotemperament which equates 5/4 with 4/3. 16/15 8 Extremely low 720-800c
4 Kleismic Kleismic, Cata 2.3.5.13 alpha-hexacot 19, 34 smitonic, 4L7s, 4L11s A highly accurate temperament equating a stack of six slightly sharp 6/5's to one 3/1 and three 6/5's to one 26/15. kleisma, 325/324 2 High 317c
10 Negri Negri 2.3.5 omega-tetracot 10, 19 Generator is a sharp minor second of 16/15 which stacks 3 times to 5/4, and can be seen as a counterpart of Porcupine. 16875/16384 5 Medium-low 124-128c
Semibuzzard 2.3.5.7.11 10, 28 taric Weak Jubilismic or Buzzard extension of Negri allowing intervals of 7 to be reached with a 600-cent offset. 16875/16384, 50/49, 243/242 6 Low 124-128c
Negrisept 2.3.5.7 10, 19 Semaphore extension of negri. 16875/16384, 49/48 7 Very low 124-128c

2.3.7.x families

Form Family Name Subgroup Ploidacot ETs Usual Scale Type Description Commas Accuracy Generator size
5 Archy Archy 2.3.7 monocot 5, 22 soft pentic, harder diatonic, p-chromatic 2.3.7 counterpart of Meantone, which sharpens the fifth. 64/63 5 Medium-low 709-720c
Superpyth 2.3.5.7 monocot 22, 27 soft pentic, harder diatonic, p-chromatic Extension of the above to 2...7, favoring flatter tunings. 64/63, 245/243 5 Medium-low 709-711c
Gamelic Slendric, Wonder 2.3.7 tricot 5, 31 1L 4s, machinoid, 5L 6s Splits the fifth in 3 parts, each of which is 8/7. Little relation to actual Slendro tuning. gamelisma 3 Medium-high 231-234c
Mothra 2.3.5.7 tricot 26, 31 1L 4s, machinoid, 5L 6s Meantone extension of the above. 81/80, gamelisma 4 Medium 231c
Rodan 2.3.5.7 tricot 41, 46 1L 4s, machinoid, 5L 6s More accurate extension of the above. 245/243, gamelisma 4 Medium 234c
Miracle 2.3.5.7.11 hexacot 31, 41 antisinatonic, 10L 1s Generated by a 15/14~16/15 semitone, two of which reach a slendric 8/7. 225/224, 243/242, gamelisma 3 Medium-high 117c
Valentine 2.3.5.7 enneacot 15, 16 15L 1s, Carlos Alpha Scale with small steps strongly associated with Carlos Alpha. 126/125, gamelisma 3 Medium-high 78c
Buzzard Buzzard 2.3.5.7.13 alpha-tetracot 53, 58 Sharpens the 21/16 so that four of them stacks to the 3/1. buzzardsma 4 Medium 474-478c
Interseptimal Interseptimal, Semaphore 2.3.7 alpha-dicot 5, 19 4L 1s, semiquartal Equipentatonic, inaccurate 7-limit temperament. 49/48 6 Low 240-250c
13 Squares Squares 2.3.7.11 beta-tetracot 14, 17 3L 5s, 3L 8s, 3L 11s No-fives temperament generated by a flattened 9/7 equated with 14/11. 99/98, 243/242 4 Medium 424-426c

2.3.x families

Form Family Name Subgroup Ploidacot ETs Usual Scale Type Description Commas Accuracy Generator size
7 Rastmatic Rastmatic 2.3.11 dicot 7, 10 mosh, dicoid Maps 11/9 and its fifth complement to a perfect neutral third. 243/242 3 Medium-high 345-355c
Mohajira 2.3.5.11 dicot 24, 31 mosh, dicoid Meantone extension of the above. Can optionally be extended to set 7/4 equal to the semiflat minor seventh, or via a strong meantone extension. 243/242, 81/80 5 Medium-low 347-350c
Intratridecimal Intratridecimal 2.3.13 dicot 27, 10 mosh, dicoid Maps 16/13 and its fifth complement to a perfect neutral third. 512/507 4 Medium 350-360c
(To be named) 2.3.7.13 dicot 27, 10 mosh, dicoid Archy extension of the above. 512/507, 64/63 5 Medium-low 355-360c

2.3.5.7.x families

Form Family Name Subgroup Ploidacot ETs Usual Scale Type Description Commas Accuracy Generator size
7 Amity Amity 2.3.5.7 gamma-pentacot 46, 53 7L 18s, 7L 25s Sets the spacing of the thirds to 7/6-#-6/5-#-#-5/4-#-9/7; mostly appears as a structural element of EDOs rather than an independent structure. 4375/4374, 5120/5103 2 High 338-340c
12 Compton Compton 2.3.5.7 dodecaploid acot 12, 60 dodecawood Acts as a closed circle of 12 fifths (see 12edo), but with 5/4 flattened by a diesis from 400c, and 7/4 flattened by twice that amount from 1000c. pythagorean comma 3 Medium-high 385c
4 Doublewide Doublewide 2.3.5.7 22, 48 Sets 6/5 and 7/6 the same distance from the 300c period, so that four ~25c generators stack to the semitone separating 3/2 from 600c. 50/49, 875/864 5 Medium-low 325c
Myna Myna 2.3.5.7 beta-decacot 27, 31 - Sets 25/24 equal to twice 36/35. 126/125, 1728/1715 3 Medium-high 309-311c
9 Orwell Orwell 2.3.5.7.11 alpha-heptacot 22, 31 gramitonic Generator is a sharpened 7/6. Interval chain finds 7/6, 11/8, 8/5, 15/8, 11/10, 9/7, 3/2. 99/98, 121/120, 176/175 3 Medium-high 270-273c
Ennealimmal Ennealimmal 2.3.5.7 enneaploid dicot 27, 45 enneawood Divides the octave into nine equal parts representing 27/25 and half of 7/6. 2401/2400, 4375/4374 1 Very high 44-53c
8 Nusecond Nusecond 2.3.5.7.11 31, 70 onyx, pine Generator is a neutral second, but places primes at high complexity, preferring ratios between them. 126/125, 2430/2401 4 Medium 154-155c

No-threes families

Form Family Name Subgroup Ploidacot ETs Usual Scale Type Description Commas Generator size
7 Mabilic Mabilic 2.5.7 alpha-triseph[a] 7, 9 antidiatonic, armotonic, 9L 7s Basic antidiatonic temperament with no 3. mabilisma 4 Medium 668-680c
Trismegistus 2.3.5.7 alpha-triseph 16, 25 antidiatonic, armotonic, 9L 7s High-accuracy but high complexity extension of prime 3. gamelisma, magisma 4 Medium 672-675c
Semabila 2.3.5.7 alpha-triseph 9, 25 antidiatonic, armotonic, 9L 7s Combination of Mabilic and Semaphore. 49/48, 28672/28125 6 Low 668-672c
Mavila 2.3.5.7 monocot 7, 9 antidiatonic, armotonic, 7L 9s Exotemperament serving as an antidiatonic analog of meantone. 36/35, 135/128 7 Very low 675-680c
11 Orgonismic Orgone, Orgonic 2.7.11 trimech[b] 15, 26 4L7s A high-accuracy rank-2 temperament generated by a tempered 77/64. 65536/65219 3 Medium-high 320-325c
6 Hemimean Didacus 2.5.7 diseph 6, 25 1L 5s, 6L 1s Every other step of septimal meantone. 3136/3125 2 High 192-196c

Non-octave families

Rank-2 Temperaments
Form Family Name Subgroup Ploidacot ETs Usual Scale Type Description Commas Accuracy Generator size
b13 Sensamagic Sensamagic 3.5.7 monogem[c] b4, b9[d] lambda Basic tritave temperament that stacks 9/7 twice to reach 5/3. Generates the lambda (4L5s⟨3/1⟩) MOS scale, or can be used with octaves as Sensamagic.2. 245/243 3 Medium-high 435-440c
8 Sensi 2.3.5.7.13 beta-heptacot 19, 27 3L 2s, checkertonic Very sharp extension of Sensamagic, which finds the octave at 125/63. 91/90, 126/125, 169/168 4 Medium 440-445c

[a] seph = divisions of 5/4

[b] mech = divisions of 7/4

[c] gem = divisions of 7/3 in a perfect twelfth (tritave) equivalent context

[d] A "b" prefixed to an equal temperament indicates the equal division of 3/1.

Rank-3

Name Commas Subgroup ETs Description Generators
Marvel 225/224, 385/384 2.3.5.7.11 19, 22, 31 16/15 and 15/14 are equated, or equivalently 32/25 and 9/7 are equated. ~3/2, ~81/80
Hemifamity / Aberschismic 5120/5103 2.3.5.7 41, 46, 53 81/80 and 64/63 are equated. Sometimes used in aberrismic theory to interpret ternary scales as 2.3.5.7, and associated with argent tuning. ~3/2, ~81/80
Parapyth(ic) 352/351, 896/891 2.3.7.11.13 41, 46, 63 Based on Margo Schulter's regular tuning construct called "parapyth". ~3/2, ~28/27

See also

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