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The '''hemimean clan''' tempers out the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}. The head of this clan is the 2.5.7 [[subgroup temperament]] didacus, generated by a tempered hemithird of [[28/25]]. Two generator steps make [[5/4]] and five make [[7/4]].  
{{Technical data page}}
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].


The second comma of the comma list determines which 7-limit family member we are looking at. These extensions, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.  
The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.  


Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.  
Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.  


Discussed elsewhere are
Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.
* ''[[Passion]]'' → [[Passion family #Septimal passion|Passion family]] (+64/63 or 3125/3087)
* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80, 126/125, 225/224)
* ''[[Mohavila]]'' → [[Pelogic family #Mohavila|Pelogic family]] (+135/128 or 1323/1250)
* ''[[Cohemimabila]]'' → [[Mabila family #Cohemimabila|Mabila family]] (+65536/64827)
* ''[[Sycamore]]'' → [[Sycamore family #Septimal sycamore|Sycamore family]] (+686/675 or 875/864)
* ''[[Bidia]]'' → [[Diaschismic family #Bidia|Diaschismic family]] (+2048/2025)
* ''[[Hexe]]'' → [[Augmented family #Hexe|Augmented family]] (+50/49 or 128/125)
* [[Misty]] → [[Misty family #Septimal misty|Misty family]] (+5120/5103)
* ''[[Bischismic]]'' → [[Schismatic family #Bischismic|Schismatic family]] (+32805/32768)
* ''[[Clyde]]'' → [[Kleismic family #Clyde|Kleismic family]] (+245/243)
* [[Parakleismic]] → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]] (+4375/4374)
* ''[[Arch]]'' → [[Escapade family #Arch|Escapade family]] (+5250987/5242880)
* ''[[Decistearn]]'' → [[Stearnsmic clan #Decistearn|Stearnsmic clan]] (+118098/117649)
* ''[[Quintagar]]'' → [[Quindromeda family #Quintagar|Quindromeda family]] (+33554432/33480783)
* ''[[Rubidium]]'' → [[37th-octave temperaments]] (+4194304/4117715)


Considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita.  
Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Mavila family #Mohavila|Mavila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'' (+2048/2025) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hexe]]'' (+50/49 or 128/125) → [[Augmented family #Hexe|Augmented family]]
* [[Misty]] (+5120/5103) → [[Misty family #Septimal misty|Misty family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]


A notable subgroup extension of didacus is [[Chromatic pairs #Roulette|roulette]].
= 2.5.7 subgroup =
== Didacus ==
{{main|Didacus}}


== Didacus ==
See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
Subgroup: 2.5.7
 
[[Subgroup]]: 2.5.7


[[Comma list]]: 3136/3125
[[Comma list]]: [[3136/3125]]


[[Sval]] [[mapping]]: [{{val| 1 0 -3 }}, {{val| 0 2 5 }}]
{{Mapping|legend=2| 1 0 -3 | 0 2 5 }}


Sval mapping generators: ~2, ~56/25
: sval mapping generators: ~2, ~56/25


Gencom mapping: [{{val| 1 0 0 -3 }}, {{val| 0 0 2 5 }}]
{{Mapping|legend=3| 1 0 0 -3 | 0 0 2 5 }}


[[Gencom]]: [2 56/25; 3136/3125]
: [[gencom]]: [2 56/25; 3136/3125]


[[POTE generator]]: ~28/25 = 193.772
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}


=== Rectified Hebrew ===
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Main| Rectified Hebrew }}


Rectified Hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out 4394/4375 and spliting the generator of didacus in three.
[[Badness]] (Sintel): 0.091


Subgroup: 2.5.7.13
= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|-
! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
|-
! Temperament !! 5-limit generator location
|-
| [[#Hemiwürschmidt|Hemiwürschmidt]] || [[Würschmidt family#Würschmidt|Würschmidt]] || +2 || +16 || ↓ [[31edo|31]]
|-
| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 <br /> ↓ [[25edo|25]]
|-
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
<nowiki />* Defined by intersection with other documented extensions
 
== Hemiwürschmidt ==
''[[#Strong extensions|Return to the map]]''
 
{{See also| Würschmidt family }}
 
'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.
 
[[Subgroup]]: 2.3.5.7


Comma list: 3136/3125, 4394/4375
[[Comma list]]: 2401/2400, 3136/3125


Sval mapping: [{{val| 1 2 2 3 }}, {{val| 0 6 15 13 }}]
{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}


Sval mapping generators: ~2, ~26/25
Mapping generators: ~2, ~25/14


POTE generator: ~26/25 = 64.6086
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898


{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}


== Hemiwürschmidt ==
[[Badness]]: 0.020307
{{See also| Würschmidt family #Hemiwürschmidt }}


'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
=== 2.3.5.7.23 subgroup ===
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.23


[[Comma list]]: 2401/2400, 3136/3125
[[Comma list]]: 576/575, 736/735, 1127/1125


[[Mapping]]: [{{val| 1 15 4 7 }}, {{val| 0 -16 -2 -5 }}]
{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}


{{Multival|legend=1| 16 2 5 -34 -37 6 }}
Mapping generators: ~2, ~25/14


[[POTE generator]]: ~28/25 = 193.898
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901


{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}


[[Badness]]: 0.020307
Badness (Sintel): 0.304


=== 11-limit ===
=== 11-limit ===
Line 84: Line 112:
Comma list: 243/242, 441/440, 3136/3125
Comma list: 243/242, 441/440, 3136/3125


Mapping: [{{val| 1 15 4 7 37 }}, {{val| 0 -16 -2 -5 -40 }}]
Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}


POTE generator: ~28/25 = 193.840
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840


{{Optimal ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}


Badness: 0.021069
Badness: 0.021069
Line 97: Line 125:
Comma list: 243/242, 351/350, 441/440, 3584/3575
Comma list: 243/242, 351/350, 441/440, 3584/3575


Mapping: [{{val| 1 15 4 7 37 -29 }}, {{val| 0 -16 -2 -5 -40 39 }}]
Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}


POTE generator: ~28/25 = 193.829
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829


{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
Line 110: Line 138:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: [{{val| 1 15 4 7 37 -3 }}, {{val| 0 -16 -2 -5 -40 8 }}]
Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}


POTE generator: ~28/25 = 193.918
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918


{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
Line 123: Line 151:
Comma list: 121/120, 176/175, 1375/1372
Comma list: 121/120, 176/175, 1375/1372


Mapping: [{{val| 1 15 4 7 11 }}, {{val| 0 -16 -2 -5 -9 }}]
Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}


POTE generator: ~28/25 = 193.884
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884


{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
Line 136: Line 164:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: [{{val| 1 15 4 7 11 -3 }}, {{val| 0 -16 -2 -5 -9 8 }}]
Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}


POTE generator: ~28/25 = 194.004
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004


{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
Line 149: Line 177:
Comma list: 66/65, 105/104, 121/120, 1375/1372
Comma list: 66/65, 105/104, 121/120, 1375/1372


Mapping: [{{val| 1 15 4 7 11 23 }}, {{val| 0 -16 -2 -5 -9 -23 }}]
Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}


POTE generator: ~28/25 = 193.698
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698


{{Optimal ET sequence|legend=1| 6f, 31 }}
{{Optimal ET sequence|legend=1| 6f, 31 }}
Line 159: Line 187:
=== Quadrawürschmidt ===
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 164: Line 194:
Comma list: 2401/2400, 3025/3024, 3136/3125
Comma list: 2401/2400, 3025/3024, 3136/3125


Mapping: [{{val| 1 15 4 7 24 }}, {{val| 0 -32 -4 -10 -49 }}]
Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}


Mapping generators: ~2, ~147/110
: mapping generators: ~2, ~147/110


POTE generator: ~147/110 = 503.0404
Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404


{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
Line 179: Line 209:
Comma list: 2401/2400, 3136/3125, 9801/9800
Comma list: 2401/2400, 3136/3125, 9801/9800


Mapping: [{{val| 2 14 6 9 -10 }}, {{val| 0 -16 -2 -5 25 }}]
Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}


Mapping generators: ~99/70, ~495/392
: mapping generators: ~99/70, ~495/392


POTE generator: ~28/25 = 193.9021
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
Line 194: Line 224:
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125


Mapping: [{{val| 2 14 6 9 -10 25 }}, {{val| 0 -16 -2 -5 25 -26 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}


POTE generator: ~28/25 = 193.9035
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
Line 207: Line 237:
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625


Mapping: [{{val| 2 14 6 9 -10 25 19 }}, {{val| 0 -16 -2 -5 25 -26 -16 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}


POTE generator: ~28/25 = 193.9112
Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
Line 220: Line 250:
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625


Mapping: [{{val| 2 14 6 9 -10 25 19 20 }}, {{val| 0 -16 -2 -5 25 -26 -16 -17 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}


POTE generator: ~19/17 = 193.9145
Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
Line 228: Line 258:
Badness: 0.021707
Badness: 0.021707


===== Semihemiwürand =====
===== Semihemiwüram =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224


Mapping: [{{val| 2 14 6 9 -10 25 -4 }}, {{val| 0 -16 -2 -5 25 -26 18 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}


POTE generator: ~28/25 = 193.9112
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112


{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
Line 246: Line 276:
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224


Mapping: [{{val| 2 14 6 9 -10 25 -4 -3 }}, {{val| 0 -16 -2 -5 25 -26 18 17 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}


POTE generator: ~19/17 = 193.9428
Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428


{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
Line 255: Line 285:


== Hemithirds ==
== Hemithirds ==
{{main| Hemithirds }}
''[[#Strong extensions|Return to the map]]''
{{see also| Luna family #Hemithirds }}


Subgroup: 2.3.5.7
{{Main| Hemithirds }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 1029/1024, 3136/3125


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
 
[[POTE generator]]: ~28/25 = 193.244


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/25 -2/35 0 1/35 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
: [[Eigenmonzo]]s (unchanged-intervals): 2, 9/7
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
Line 283: Line 314:
Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}


POTE generator: ~28/25 = 193.227
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 0 0 -5/9 5/9 }}, {{monzo| 64/27 0 0 2/27 -2/27 }}, {{monzo| 79/27 0 0 5/27 -5/27 }}, {{monzo| 79/27 0 0 -22/27 22/27 }}]
: Eigenmonzos (unchanged-intervals): 2, 11/7
: Eigenmonzos (unchanged-intervals): 2, 11/7


Line 300: Line 332:
Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 196/195, 352/351, 385/384, 625/624


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}


POTE generator: ~28/25 = 193.166
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166


{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
Line 309: Line 341:


== Spell ==
== Spell ==
''[[#Strong extensions|Return to the map]]''
{{See also| Magic family }}
{{See also| Magic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 3125/3072
[[Comma list]]: 49/48, 3125/3072


[[Mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 10 2 5 }}]
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}


{{Multival|legend=1| 10 2 5 -20 -20 6 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
 
[[POTE generator]]: ~28/25 = 189.927


{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
Line 330: Line 362:
Comma list: 49/48, 56/55, 125/121
Comma list: 49/48, 56/55, 125/121


Mapping: [{{val| 1 0 2 2 3 }}, {{val| 0 10 2 5 3 }}]
Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}


POTE generator: ~11/10 = 190.285
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285


{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
Line 343: Line 375:
Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 49/48, 56/55, 78/77, 125/121


Mapping: [{{val| 1 0 2 2 3 4 }}, {{val| 0 10 2 5 3 -2 }}]
Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}


POTE generator: ~11/10 = 189.928
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928


{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
Line 356: Line 388:
Comma list: 49/48, 56/55, 91/90, 125/121
Comma list: 49/48, 56/55, 91/90, 125/121


Mapping: [{{val| 1 0 2 2 3 1 }}, {{val| 0 10 2 5 3 17 }}]
Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}


POTE generator: ~11/10 = 190.360
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360


{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}


Badness: 0.041603
Badness: 0.041603
= Weak extensions =


== Semisept ==
== Semisept ==
Line 375: Line 409:
{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}
{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}


{{Multival|legend=1| 17 6 15 -30 -24 18 }}
: mapping generators: ~2, ~75/49


[[Optimal tuning]] ([[POTE]]): ~75/49 = 735.155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155


{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
Line 390: Line 424:
Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}


Optimal tuning (POTE): ~55/36 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125


{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
Line 403: Line 437:
Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}


Optimal tuning (POTE): ~55/36 = 735.126
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}
Line 416: Line 450:
Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}


Optimal tuning (POTE): ~26/17 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}
Line 429: Line 463:
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}


Optimal tuning (POTE): ~26/17 = 735.116
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}
Line 442: Line 476:
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}


Optimal tuning (POTE): ~26/17 = 735.106
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106


{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
Line 455: Line 489:
Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}


Optimal tuning (POTE): ~13/10 = 464.980
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980


{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
Line 464: Line 498:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''


[[Emka]] tempers out {{monzo|-50 -8 27}} in the 5-limit. This temperament can be described as 37&amp;50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87&amp;224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 &amp; 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 &amp; 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 84035/82944
[[Comma list]]: 3136/3125, 84035/82944


[[Mapping]]: [{{val| 1 14 6 12 }}, {{val| 0 -27 -8 -20 }}]
{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}


{{Multival|legend=1| 27 8 20 -50 -44 24 }}
: mapping generators: ~2, ~48/35


[[POTE generator]]: ~48/35 = 551.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782


{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
Line 485: Line 519:
Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


Mapping: [{{val| 1 14 6 12 3 }}, {{val| 0 -27 -8 -20 1 }}]
Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}


POTE generator: ~11/8 = 551.765
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
Line 498: Line 532:
Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


Mapping: [{{val| 1 14 6 12 3 6 }}, {{val| 0 -27 -8 -20 1 -5 }}]
Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}


POTE generator: ~11/8 = 551.758
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
Line 507: Line 541:


== Decipentic ==
== Decipentic ==
The generator for the ''decipentic'' temperament (43&amp;56) is tenth root of fifth harmonic ([[5/1]], pentave), 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to 27/23). Aside from the hemimean comma, this temperament tempers out the [[Bronzismic temperaments|bronzisma]], 2097152/2083725 (satriru-agugu). [[99edo|99EDO]] is a good tuning for decipentic, with generator 23\99, and MOS of 9, 13, 17, 30, 43 or 56 notes are available.
The generator for the decipentic temperament (43 &amp; 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 2097152/2083725
[[Comma list]]: 3136/3125, 2097152/2083725


[[Mapping]]: [{{val| 1 6 0 -3 }}, {{val| 0 -19 10 25 }}]
{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
 
{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}


[[POTE generator]]: ~75/64 = 278.800
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800


{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}
Line 528: Line 560:
Comma list: 441/440, 1344/1331, 3136/3125
Comma list: 441/440, 1344/1331, 3136/3125


Mapping: [{{val| 1 6 0 -3 3 }}, {{val| 0 -19 10 25 2 }}]
Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}


POTE generator: ~75/64 = 278.799
Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
Line 541: Line 573:
Comma list: 169/168, 441/440, 832/825, 975/968
Comma list: 169/168, 441/440, 832/825, 975/968


Mapping: [{{val| 1 6 0 -3 3 3 }}, {{val| 0 -19 10 25 2 3 }}]
Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}


POTE generator: ~13/11 = 278.802
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
Line 554: Line 586:
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374


Mapping: [{{val| 1 6 0 -3 3 3 2 }}, {{val| 0 -19 10 25 2 3 9 }}]
Mapping: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}


POTE generator: ~13/11 = 278.798
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
Line 567: Line 599:
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285


Mapping: [{{val| 1 6 0 -3 3 3 2 1 }}, {{val| 0 -19 10 25 2 3 9 14 }}]
Mapping: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}


POTE generator: ~13/11 = 278.790
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
Line 580: Line 612:
Comma list: 3136/3125, 15488/15435, 16384/16335
Comma list: 3136/3125, 15488/15435, 16384/16335


Mapping: [{{val| 1 6 0 -3 3 }}, {{val| 0 -38 20 50 47 }}]
Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}


POTE generator: ~896/825 = 139.403
Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403


{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}
{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}
Line 593: Line 625:
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435


Mapping: [{{val| 1 6 0 -3 3 8 }}, {{val| 0 -38 20 50 47 -37 }}]
Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}


POTE generator: ~13/12 = 139.403
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403


{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}
{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}
Line 602: Line 634:


== Sengagen ==
== Sengagen ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 420175/419904
[[Comma list]]: 3136/3125, 420175/419904


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 29 16 40 }}]
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
 
{{Multival|legend=1| 29 16 40 -42 -18 48 }}


[[POTE generator]]: ~686/675 = 24.217
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217


{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}
{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}
Line 621: Line 651:
Comma list: 540/539, 1344/1331, 3136/3125
Comma list: 540/539, 1344/1331, 3136/3125


Mapping: [{{val| 1 1 2 2 3 }}, {{val| 0 29 16 40 23 }}]
Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}


POTE generator: ~99/98 = 24.235
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235


{{Optimal ET sequence|legend=1| 49, 50, 99e }}
{{Optimal ET sequence|legend=1| 49, 50, 99e }}
Line 634: Line 664:
Comma list: 351/350, 540/539, 975/968, 1344/1331
Comma list: 351/350, 540/539, 975/968, 1344/1331


Mapping: [{{val| 1 1 2 2 3 4 }}, {{val| 0 29 16 40 23 -15 }}]
Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}


POTE generator: ~99/98 = 24.181
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181


{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}
{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}
Line 647: Line 677:
Comma list: 144/143, 196/195, 364/363, 625/624
Comma list: 144/143, 196/195, 364/363, 625/624


Mapping: [{{val| 1 1 2 2 3 3 }}, {{val| 0 29 16 40 23 35 }}]
Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}


POTE generator: ~99/98 = 24.234
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234


{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}
{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}


Badness: 0.037416
Badness: 0.037416
== Subpental ==
{{See also| Sensipent family }}
Subgroup: 2.3.5.7
[[Comma list]]: 3136/3125, 19683/19600
[[Mapping]]: [{{val| 1 6 8 17 }}, {{val| 0 -14 -18 -45 }}]
{{Multival|legend=1| 14 18 45 -4 32 54 }}
[[POTE generator]]: ~56/45 = 378.467
{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
[[Badness]]: 0.054303
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 8019/8000
Mapping: [{{val| 1 6 8 17 -6 }}, {{val| 0 -14 -18 -45 30 }}]
POTE generator: ~56/45 = 378.440
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
Badness: 0.045352
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 3136/3125
Mapping: [{{val| 1 6 8 17 -6 16 }}, {{val| 0 -14 -18 -45 30 -39 }}]
POTE generator: ~56/45 = 378.437
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
Badness: 0.023940


== Mowglic ==
== Mowglic ==
The ''mowglic'' temperament (19&amp;161) is an extension of the [[Syntonic-enneadecal equivalence continuum|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.
The mowglic temperament (19 &amp; 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 177147/175000
[[Comma list]]: 3136/3125, 177147/175000


[[Mapping]]: [{{val| 1 0 0 -3 }}, {{val| 0 15 22 55 }}]
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
 
{{Multival|legend=1| 15 22 55 0 45 66 }}


[[POTE generator]]: ~27/25 = 126.706
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706


{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}
{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}
Line 720: Line 705:
Comma list: 540/539, 3136/3125, 72171/71680
Comma list: 540/539, 3136/3125, 72171/71680


Mapping: [{{val| 1 0 0 -3 8 }}, {{val| 0 15 22 55 -43 }}]
Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}


POTE generator: ~27/25 = 126.711
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711


{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}
{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}
Line 733: Line 718:
Comma list: 351/350, 540/539, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 1701/1690, 3136/3125


Mapping: [{{val| 1 0 0 -3 8 -2 }}, {{val| 0 15 22 55 -43 54 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}


POTE generator: ~14/13 = 126.705
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705


{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}
Line 746: Line 731:
Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125


Mapping: [{{val| 1 0 0 -3 8 -2 10 }}, {{val| 0 15 22 55 -43 54 -56 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}


POTE generator: ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
Line 759: Line 744:
Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690
Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 }}, {{val| 0 15 22 55 -43 54 -56 -45 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}


POTE generator: ~14/13 = 126.705
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
Line 772: Line 757:
Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 6 }}, {{val| 0 15 22 55 -43 54 -56 -45 -14 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}


POTE generator: ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
Line 785: Line 770:
Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 6 0 }}, {{val| 0 15 22 55 -43 54 -56 -45 -14 46 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}


POTE generator: ~14/13 = 126.704
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
Line 798: Line 783:
Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 6 0 2 }}, {{val| 0 15 22 55 -43 54 -56 -45 -14 46 28 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}


POTE generator: ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703


{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}
Line 807: Line 792:


== Tremka ==
== Tremka ==
The name "tremka" is initially used for [[No-sevens subgroup temperaments|no-sevens version]] of 50&amp;111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 &amp; 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.


Subgroup: 2.3.5.7
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 2125764/2100875
[[Comma list]]: 3136/3125, 2125764/2100875


[[Mapping]]: [{{val|1 -4 -2 -8}}, {{val|0 31 24 60}}]
{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}


{{Multival|legend=1|31 24 60 -34 8 72}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
 
[[POTE generator]]: ~4375/3888 = 216.173


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
Line 828: Line 812:
Comma list: 540/539, 3136/3125, 35937/35840
Comma list: 540/539, 3136/3125, 35937/35840


Mapping: [{{val|1 -4 -2 -8 4}}, {{val|0 31 24 60 -3}}]
Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}


POTE generator: ~112/99 = 216.168
Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168


{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}
Line 841: Line 825:
Comma list: 351/350, 540/539, 847/845, 3136/3125
Comma list: 351/350, 540/539, 847/845, 3136/3125


Mapping: [{{val|1 -4 -2 -8 4 1}}, {{val|0 31 24 60 -3 15}}]
Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}


POTE generator: ~112/99 = 216.172
Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
Line 854: Line 838:
Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088
Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088


Mapping: [{{val|1 -4 -2 -8 4 1 -6}}, {{val|0 31 24 60 -3 15 56}}]
Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}


POTE generator: ~17/15 = 216.172
Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
Line 867: Line 851:
Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539
Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539


Mapping: [{{val|1 -4 -2 -8 4 1 -6 -8}}, {{val|0 31 24 60 -3 15 56 68}}]
Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}


POTE generator: ~17/15 = 216.170
Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170


{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}
Line 876: Line 860:


== Undetrita ==
== Undetrita ==
The ''undetrita'' temperament (111&amp;118) tempers out the hemimean comma (3136/3125) and [[skeetsma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt|11EDT]] tuning, and the name ''undetrita'' is a play on the words "undecimus" (Latin for "eleventh") and "[[tritave]]" (third harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111&amp;118.
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''


Subgroup: 2.3.5.7
The undetrita temperament (111 &amp; 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 &amp; 118.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 14348907/14336000
[[Comma list]]: 3136/3125, 14348907/14336000


[[Mapping]]: [{{val|1 0 -2 -8}}, {{val|0 11 30 75}}]
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
 
{{Multival|legend=1|11 30 75 22 88 90}}


[[POTE generator]]: ~448/405 = 172.917
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917


{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}
{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}
Line 897: Line 881:
Comma list: 3025/3024, 3136/3125, 8019/8000
Comma list: 3025/3024, 3136/3125, 8019/8000


Mapping: [{{val|1 0 -2 -8 0}}, {{val|0 11 30 75 24}}]
Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}


POTE generator: ~400/363 = 172.912
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912


{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}
{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}
Line 910: Line 894:
Comma list: 352/351, 729/728, 1001/1000, 3025/3024
Comma list: 352/351, 729/728, 1001/1000, 3025/3024


Mapping: [{{val|1 0 -2 -8 0 5}}, {{val|0 11 30 75 24 -9}}]
Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}


POTE generator: ~72/65 = 172.930
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930


{{Optimal ET sequence|legend=1| 111, 229f }}
{{Optimal ET sequence|legend=1| 111, 229f }}
Line 923: Line 907:
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125


Mapping: [{{val|1 0 -2 -8 0 -11}}, {{val|0 11 30 75 24 102}}]
Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}


POTE generator: ~400/363 = 172.933
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933


{{Optimal ET sequence|legend=1| 111, 229 }}
{{Optimal ET sequence|legend=1| 111, 229 }}


Badness: 0.042744
Badness: 0.042744
= Subgroup extensions =
== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
Subgroup: 2.5.7.11
Comma list: [[176/175]], [[1375/1372]]
Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428
Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}
RMS error: 0.5567 cents
Badness (Sintel): 0.195
=== Tridecimal didacus ===
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 1375/1372
Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}
: sval mapping generators: ~2, ~56/25
Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}
: gencom: [2 28/25; 176/175 1375/1372 640/637]
Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594
Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
Badness (Sintel): 0.324
==== Mediantone ====
Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].
In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])<sup>2</sup>/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.
Subgroup: 2.5.7.11.13.17.19
Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}
<nowiki />* 117d only appears without prime 19
Badness (Sintel): 0.618
==== Roulette ====
{{See also | Chromatic pairs #Roulette }}
Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
Subgroup: 2.5.7.11.13.17.19
Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]
Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}
: sval mapping generators: ~2, ~28/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
Badness (Sintel): 0.676
== Rectified hebrew ==
{{Main| Rectified hebrew }}
Rectified hebrew (37 &amp; 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three. Notably, it is the no-threes restriction of [[Sycamore family#Septimal sycamore|sycamore]].
Subgroup: 2.5.7.13
Comma list: 3136/3125, 4394/4375
Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}
: sval mapping generators: ~2, ~26/25
Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086
{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}


== Isra ==
== Isra ==
Isra results from taking every other generator of [[septimal meantone]]. It is named after the Isrāʾ (''iss-RAH'') night journey in the Qur'an, because it is similar to [[luna]].  
Isra (''iss-RAH'') results from taking every other generator of [[septimal meantone]], or from [[didacus]] if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).


[[Subgroup]]: 2.9.5.7
[[Subgroup]]: 2.9.5.7
Line 947: Line 1,030:


=== Tutone ===
=== Tutone ===
Tutone is every other step of [[Meantone vs meanpop|undecimal meantone]].  
Tutone is every other step of [[Meantone vs meanpop|undecimal meantone]], or undecimal [[didacus]] with the generator interpreted as 9/8.


[[Subgroup]]: 2.9.5.7.11
[[Subgroup]]: 2.9.5.7.11
Line 964: Line 1,047:


[[Badness]]: 0.00536
[[Badness]]: 0.00536
=== Leantone ===
{{See also| Chromatic pairs #Leantone }}
Leantone is every other step of [[vincenzo]].
[[Subgroup]]: 2.9.5.7.11
[[Comma list]]: 45/44, 56/55, 81/80
{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}
{{Mapping|legend=3| 1 3/2 2 2 3 | 0 1/2 2 5 3 }}
: [[gencom]]: [2 9/8; 45/44 56/55 81/80]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500
{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 31e, 56bee, 81beee }}
[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents


=== Deutone ===
=== Deutone ===
{{See also| Chromatic pairs #Deutone }}
{{See also| Chromatic pairs #Deutone }}


=== Leantone ===
Deutone is (also) every other step of [[vincenzo]].
{{See also| Chromatic pairs #Leantone }}
 
[[Subgroup]]: 2.9.5.7.13
 
[[Comma list]]: 65/64, 81/80, 91/90
 
{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}
 
{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}
 
: [[gencom]]: [2 9/8; 65/64 81/80 91/90]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25f, 44df }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean]]
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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