Xenharmonic Wiki

Hemimean clan: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,231 edits
mNo edit summary
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "
 
(44 intermediate revisions by 9 users not shown)
Line 1: Line 1:
The '''hemimean clan''' [[Tempering out|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 [[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.  
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.  
Line 5: Line 6:
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.  


Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. A notable subgroup extension of didacus is [[Chromatic pairs #Roulette|roulette]]. 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.
 
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]]
* ''[[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]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Pelogic family #Mohavila|Pelogic family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Mavila family #Mohavila|Mavila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
Line 16: Line 19:
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] (+4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Decistearn]]'' (+118098/117649) → [[Trisedodge family #Decistearn|Trisedodge family]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]


= 2.5.7 subgroup =
== Didacus ==
== Didacus ==
{{main|Didacus}}
{{main|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 didacus and its extension to the no-3's 13-limit called roulette only the latter interpretation is relevant.
See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 47: Line 51:
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents


[[Badness]] (Dirichlet): 0.091
[[Badness]] (Sintel): 0.091


=== Undecimal didacus ===
= 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


Subgroup: 2.5.7.11
== Hemiwürschmidt ==
 
''[[#Strong extensions|Return to the map]]''
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 (Dirichlet): 0.195
 
==== Roulette ====
Roulette 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 4{{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 (Dirichlet): 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
 
Comma list: [[176/175]], [[640/637]], [[221/220]], [[1375/1372]]
 
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}
 
: sval mapping generators: ~2, ~56/25
 
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887
 
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d }}
 
Badness (Dirichlet): 0.612
 
===== 2.5.7.11.13.17.19 subgroup =====
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 }}
 
Badness (Dirichlet): 0.618
 
=== 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.
 
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 }}
== Hemiwürschmidt ==
{{See also| Würschmidt family }}
{{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, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
'''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
[[Subgroup]]: 2.3.5.7
Line 151: Line 83:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
Line 175: Line 105:
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}


{{todo|verify this & get badness|inline=1}}
Badness (Sintel): 0.304


=== 11-limit ===
=== 11-limit ===
Line 182: Line 112:
Comma list: 243/242, 441/440, 3136/3125
Comma list: 243/242, 441/440, 3136/3125


Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
{{Mapping|legend=0| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 195: Line 125:
Comma list: 243/242, 351/350, 441/440, 3584/3575
Comma list: 243/242, 351/350, 441/440, 3584/3575


Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
{{Mapping|legend=0| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
Line 208: Line 138:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
{{Mapping|legend=0| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
Line 221: Line 151:
Comma list: 121/120, 176/175, 1375/1372
Comma list: 121/120, 176/175, 1375/1372


Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
{{Mapping|legend=0| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
Line 234: Line 164:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
{{Mapping|legend=0| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
Line 247: Line 177:
Comma list: 66/65, 105/104, 121/120, 1375/1372
Comma list: 66/65, 105/104, 121/120, 1375/1372


Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
{{Mapping|legend=0| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
Line 257: 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 262: Line 194:
Comma list: 2401/2400, 3025/3024, 3136/3125
Comma list: 2401/2400, 3025/3024, 3136/3125


Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
{{Mapping|legend=0| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}


: mapping generators: ~2, ~147/110
: mapping generators: ~2, ~147/110
Line 277: Line 209:
Comma list: 2401/2400, 3136/3125, 9801/9800
Comma list: 2401/2400, 3136/3125, 9801/9800


Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
{{Mapping|legend=0| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}


: mapping generators: ~99/70, ~495/392
: mapping generators: ~99/70, ~495/392
Line 292: Line 224:
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125


Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
{{Mapping|legend=0| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
Line 305: 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: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
{{Mapping|legend=0| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
Line 318: 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: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
{{Mapping|legend=0| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
Line 331: Line 263:
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: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
{{Mapping|legend=0| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
Line 344: 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: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}
{{Mapping|legend=0| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
Line 353: Line 285:


== Hemithirds ==
== Hemithirds ==
''[[#Strong extensions|Return to the map]]''
{{Main| Hemithirds }}
{{Main| Hemithirds }}


Line 360: Line 294:


{{Mapping|legend=1| 1 4 2 2 | 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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
Line 368: Line 300:
* [[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 }}
: {{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 }}
: {{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 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
* [[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 }}
: {{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|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[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 382: Line 314:
Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}
{{Mapping|legend=0| 1 4 2 2 7 | 0 -15 2 5 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
Line 400: Line 332:
Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 196/195, 352/351, 385/384, 625/624


Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}
{{Mapping|legend=0| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
Line 409: Line 341:


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


Line 416: Line 350:


{{Mapping|legend=1| 1 0 2 2 | 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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
Line 430: Line 362:
Comma list: 49/48, 56/55, 125/121
Comma list: 49/48, 56/55, 125/121


Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}
{{Mapping|legend=0| 1 0 2 2 3 | 0 10 2 5 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285
Line 443: Line 375:
Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 49/48, 56/55, 78/77, 125/121


Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}
{{Mapping|legend=0| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928
Line 456: Line 388:
Comma list: 49/48, 56/55, 91/90, 125/121
Comma list: 49/48, 56/55, 91/90, 125/121


Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}
{{Mapping|legend=0| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360
Line 463: Line 395:


Badness: 0.041603
Badness: 0.041603
= Weak extensions =


== Semisept ==
== Semisept ==
Line 476: Line 410:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
Line 490: Line 422:
Comma list: 176/175, 540/539, 1331/1323
Comma list: 176/175, 540/539, 1331/1323


Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
{{Mapping|legend=0| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125
Line 503: Line 435:
Comma list: 176/175, 351/350, 540/539, 1375/1372
Comma list: 176/175, 351/350, 540/539, 1375/1372


Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
{{Mapping|legend=0| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126
Line 516: Line 448:
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714


Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
{{Mapping|legend=0| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125
Line 529: Line 461:
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
{{Mapping|legend=0| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116
Line 542: Line 474:
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
{{Mapping|legend=0| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106
Line 555: Line 487:
Comma list: 144/143, 176/175, 196/195, 275/273
Comma list: 144/143, 176/175, 196/195, 275/273


Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
{{Mapping|legend=0| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980
Line 575: Line 507:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
Line 589: Line 519:
Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}
{{Mapping|legend=0| 1 14 6 12 3 | 0 -27 -8 -20 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765
Line 602: Line 532:
Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}
{{Mapping|legend=0| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758
Line 618: Line 548:


{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
Line 632: Line 560:
Comma list: 441/440, 1344/1331, 3136/3125
Comma list: 441/440, 1344/1331, 3136/3125


Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}
{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799
Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799
Line 645: Line 573:
Comma list: 169/168, 441/440, 832/825, 975/968
Comma list: 169/168, 441/440, 832/825, 975/968


Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}
{{Mapping|legend=0| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802
Line 658: 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: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}
{{Mapping|legend=0| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798
Line 671: 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: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}
{{Mapping|legend=0| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790
Line 684: Line 612:
Comma list: 3136/3125, 15488/15435, 16384/16335
Comma list: 3136/3125, 15488/15435, 16384/16335


Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}
{{Mapping|legend=0| 1 6 0 -3 3 | 0 -38 20 50 47 }}


Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403
Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403
Line 697: Line 625:
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435


Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}
{{Mapping|legend=0| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403
Line 711: Line 639:


{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Multival|legend=1| 29 16 40 -42 -18 48 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
Line 725: Line 651:
Comma list: 540/539, 1344/1331, 3136/3125
Comma list: 540/539, 1344/1331, 3136/3125


Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}
{{Mapping|legend=0| 1 1 2 2 3 | 0 29 16 40 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235
Line 738: Line 664:
Comma list: 351/350, 540/539, 975/968, 1344/1331
Comma list: 351/350, 540/539, 975/968, 1344/1331


Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}
{{Mapping|legend=0| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181
Line 751: Line 677:
Comma list: 144/143, 196/195, 364/363, 625/624
Comma list: 144/143, 196/195, 364/363, 625/624


Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}
{{Mapping|legend=0| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234
Line 760: Line 686:


== 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
Line 767: Line 693:


{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Multival|legend=1| 15 22 55 0 45 66 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
Line 781: Line 705:
Comma list: 540/539, 3136/3125, 72171/71680
Comma list: 540/539, 3136/3125, 72171/71680


Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}
{{Mapping|legend=0| 1 0 0 -3 8 | 0 15 22 55 -43 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711
Line 794: Line 718:
Comma list: 351/350, 540/539, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 1701/1690, 3136/3125


Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Line 807: 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: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Line 820: 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: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Line 833: 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: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Line 846: 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: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704
Line 859: 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: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Line 876: Line 800:


{{Mapping|legend=1| 1 -4 -2 -8 | 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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
Line 890: Line 812:
Comma list: 540/539, 3136/3125, 35937/35840
Comma list: 540/539, 3136/3125, 35937/35840


Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168
Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168
Line 903: Line 825:
Comma list: 351/350, 540/539, 847/845, 3136/3125
Comma list: 351/350, 540/539, 847/845, 3136/3125


Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172
Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172
Line 916: 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: {{mapping| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172
Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172
Line 929: 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: {{mapping| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170
Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170
Line 938: 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]], 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.
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''
 
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
[[Subgroup]]: 2.3.5.7
Line 945: Line 869:


{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Multival|legend=1| 11 30 75 22 88 90 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
Line 959: Line 881:
Comma list: 3025/3024, 3136/3125, 8019/8000
Comma list: 3025/3024, 3136/3125, 8019/8000


Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}
{{Mapping|legend=0| 1 0 -2 -8 0 | 0 11 30 75 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912
Line 972: Line 894:
Comma list: 352/351, 729/728, 1001/1000, 3025/3024
Comma list: 352/351, 729/728, 1001/1000, 3025/3024


Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}
{{Mapping|legend=0| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930
Line 985: Line 907:
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125


Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}
{{Mapping|legend=0| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}


Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933
Line 992: Line 914:


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 1,009: 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 1,027: Line 1,048:
[[Badness]]: 0.00536
[[Badness]]: 0.00536


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


Deutone is every other step of [[meanpop]].  
Leantone is every other step of [[vincenzo]].  


[[Subgroup]]: 2.9.5.7.13
[[Subgroup]]: 2.9.5.7.11


[[Comma list]]: 65/64, 81/80, 91/90
[[Comma list]]: 45/44, 56/55, 81/80


{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}
{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}


{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}
{{Mapping|legend=3| 1 3/2 2 2 3 | 0 1/2 2 5 3 }}


: [[gencom]]: [2 9/8; 65/64 81/80 91/90]
: [[gencom]]: [2 9/8; 45/44 56/55 81/80]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500


{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 44de }}
{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 31e, 56bee, 81beee }}


[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents
[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents


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


Leantone is every other step of [[meanenneadecal]].  
Deutone is (also) every other step of [[vincenzo]].  


[[Subgroup]]: 2.9.5.7.11
[[Subgroup]]: 2.9.5.7.13


[[Comma list]]: 45/44, 56/55, 81/80
[[Comma list]]: 65/64, 81/80, 91/90


{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}
{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}


{{Mapping|legend=3| 1 3/2 2 2 3 | 0 1/2 2 5 3 }}
{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}


: [[gencom]]: [2 9/8; 45/44 56/55 81/80]
: [[gencom]]: [2 9/8; 65/64 81/80 91/90]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059


{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25, 31, 56, 81 }}
{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25f, 44df }}


[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents


[[Category:Temperament clans]]
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Hemimean_clan"