Hemimean clan: Difference between revisions

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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]].

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* ''[[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) → [[~~Pelogic~~ family #Mohavila|~~Pelogic~~ 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]]

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* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic 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]]

* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent 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]]

* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]

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== Didacus ==

See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].

[[Subgroup]]: 2.5.7

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[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

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

[[Badness]] (Sintel): 0.091

= Strong extensions =

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Mapping generators: ~2, ~25/14

~~{{Multival|legend=1| 16 2 5 -34 -37 6 }}~~

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

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Badness (~~Dirichlet~~): 0.304

Badness (Sintel): 0.304

=== 11-limit ===

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Comma list: 243/242, 441/440, 3136/3125

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

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

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

Badness: 0.021069

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Comma list: 243/242, 351/350, 441/440, 3584/3575

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

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

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Comma list: 121/120, 176/175, 196/195, 275/273

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

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

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Comma list: 121/120, 176/175, 1375/1372

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

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

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Comma list: 121/120, 176/175, 196/195, 275/273

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

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

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Comma list: 66/65, 105/104, 121/120, 1375/1372

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

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

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=== Quadrawürschmidt ===

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

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Comma list: 2401/2400, 3025/3024, 3136/3125

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

: mapping generators: ~2, ~147/110

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Comma list: 2401/2400, 3136/3125, 9801/9800

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

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

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Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125

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

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

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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

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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

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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

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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

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~~{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244

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* [[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 }}

: [[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 }}

: {{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

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Comma list: 385/384, 441/440, 3136/3125

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

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

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

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

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

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~~{{Multival|legend=1| 10 2 5 -20 -20 6 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927

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Comma list: 49/48, 56/55, 125/121

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

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

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Comma list: 49/48, 56/55, 78/77, 125/121

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

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

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Comma list: 49/48, 56/55, 91/90, 125/121

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

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

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: mapping generators: ~2, ~75/49

~~{{Multival|legend=1| 17 6 15 -30 -24 18 }}~~

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

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Comma list: 176/175, 540/539, 1331/1323

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

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

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Comma list: 176/175, 351/350, 540/539, 1375/1372

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

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

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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

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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

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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

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Comma list: 144/143, 176/175, 196/195, 275/273

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

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

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: mapping generators: ~2, ~48/35

~~{{Multival|legend=1| 27 8 20 -50 -44 24 }}~~

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

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Comma list: 385/384, 2401/2376, 3136/3125

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

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

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Comma list: 196/195, 364/363, 385/384, 625/624

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

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

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~~{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}~~

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

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Comma list: 441/440, 1344/1331, 3136/3125

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

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

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Comma list: 169/168, 441/440, 832/825, 975/968

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

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

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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 }}

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

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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 }}

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

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Comma list: 3136/3125, 15488/15435, 16384/16335

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

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

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Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

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

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

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~~{{Multival|legend=1| 29 16 40 -42 -18 48 }}~~

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

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Comma list: 540/539, 1344/1331, 3136/3125

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

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

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Comma list: 351/350, 540/539, 975/968, 1344/1331

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

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

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Comma list: 144/143, 196/195, 364/363, 625/624

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

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

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== Mowglic ==

The mowglic temperament (19 & 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 & 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

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~~{{Multival|legend=1| 15 22 55 0 45 66 }}~~

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

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Comma list: 540/539, 3136/3125, 72171/71680

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

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

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Comma list: 351/350, 540/539, 1701/1690, 3136/3125

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

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

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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 }}

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

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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

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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

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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

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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

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~~{{Multival|legend=1| 31 24 60 -34 8 72 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173

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Comma list: 540/539, 3136/3125, 35937/35840

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

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

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Comma list: 351/350, 540/539, 847/845, 3136/3125

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

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

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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 }}

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

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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

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== Undetrita ==

The undetrita temperament (111 & 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 & 118.

: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''

The undetrita temperament (111 & 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 & 118.

[[Subgroup]]: 2.3.5.7

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~~{{Multival|legend=1| 11 30 75 22 88 90 }}~~

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

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Comma list: 3025/3024, 3136/3125, 8019/8000

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

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

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Comma list: 352/351, 729/728, 1001/1000, 3025/3024

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

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

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Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

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

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

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RMS error: 0.5567 cents

Badness (~~Dirichlet~~): 0.195

Badness (Sintel): 0.195

=== Tridecimal didacus ===

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Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}

Badness (~~Dirichlet~~): 0.324

Badness (Sintel): 0.324

==== Mediantone ====

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<nowiki />* 117d only appears without prime 19

Badness (~~Dirichlet~~): 0.618

Badness (Sintel): 0.618

==== Roulette ====

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Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}

Badness (~~Dirichlet~~): 0.676

Badness (Sintel): 0.676

== 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.

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. Notably, it is the no-threes restriction of [[Sycamore family#Septimal sycamore|sycamore]].

Subgroup: 2.5.7.13

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== Isra ==

Isra 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 ~~(''iss-RAH'')~~ night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).

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

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

~~[[Category:Temperament clans]]~~

[[Category:Hemimean clan| ]]

[[Category:Hemimean| ]]

[[Category:~~Rank~~ 2]]

[[Category:Temperament clans]]

[[Category:Catalogs of rank-2 temperaments]]

@@ Line 1: / Line 1: @@
 {{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]].
@@ Line 12: / Line 11: @@
 * ''[[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) → [[Pelogic family #Mohavila|Pelogic 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]]
@@ Line 20: / Line 19: @@
 * ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic 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]]
 * ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent 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]]
 * ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]
@@ Line 31: / Line 30: @@
 == Didacus ==
 {{main|Didacus}}
+See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
 [[Subgroup]]: 2.5.7
@@ Line 50: / Line 51: @@
 [[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
-[[Badness]] (Dirichlet): 0.091
+[[Badness]] (Sintel): 0.091
 = Strong extensions =
@@ Line 82: / Line 83: @@
 Mapping generators: ~2, ~25/14
-{{Multival|legend=1| 16 2 5 -34 -37 6 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
@@ Line 106: / Line 105: @@
 {{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
-Badness (Dirichlet): 0.304
+Badness (Sintel): 0.304
 === 11-limit ===
@@ Line 113: / Line 112: @@
 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 ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
+{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}
 Badness: 0.021069
@@ Line 126: / Line 125: @@
 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
@@ Line 139: / Line 138: @@
 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
@@ Line 152: / Line 151: @@
 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
@@ Line 165: / Line 164: @@
 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
@@ Line 178: / Line 177: @@
 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
@@ Line 188: / Line 187: @@
 === Quadrawürschmidt ===
 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
@@ Line 193: / Line 194: @@
 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
@@ Line 208: / Line 209: @@
 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
@@ Line 223: / Line 224: @@
 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
@@ Line 236: / Line 237: @@
 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
@@ Line 249: / Line 250: @@
 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
@@ Line 262: / Line 263: @@
 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
@@ Line 275: / Line 276: @@
 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
@@ Line 293: / Line 294: @@
 {{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
@@ Line 301: / Line 300: @@
 * [[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 }}
-: [[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 }}
 : {{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 }}
@@ Line 315: / Line 314: @@
 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
@@ Line 333: / Line 332: @@
 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
@@ Line 351: / Line 350: @@
 {{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
@@ Line 365: / Line 362: @@
 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
@@ Line 378: / Line 375: @@
 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
@@ Line 391: / Line 388: @@
 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
@@ Line 413: / Line 410: @@
 : mapping generators: ~2, ~75/49
-{{Multival|legend=1| 17 6 15 -30 -24 18 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
@@ Line 427: / Line 422: @@
 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
@@ Line 440: / Line 435: @@
 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
@@ Line 453: / Line 448: @@
 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
@@ Line 466: / Line 461: @@
 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
@@ Line 479: / Line 474: @@
 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
@@ Line 492: / Line 487: @@
 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
@@ Line 512: / Line 507: @@
 : mapping generators: ~2, ~48/35
-{{Multival|legend=1| 27 8 20 -50 -44 24 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
@@ Line 526: / Line 519: @@
 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
@@ Line 539: / Line 532: @@
 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
@@ Line 555: / Line 548: @@
 {{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
@@ Line 569: / Line 560: @@
 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
@@ Line 582: / Line 573: @@
 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
@@ Line 595: / Line 586: @@
 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
@@ Line 608: / Line 599: @@
 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
@@ Line 621: / Line 612: @@
 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
@@ Line 634: / Line 625: @@
 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
@@ Line 648: / Line 639: @@
 {{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
@@ Line 662: / Line 651: @@
 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
@@ Line 675: / Line 664: @@
 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
@@ Line 688: / Line 677: @@
 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
@@ Line 697: / Line 686: @@
 == 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
@@ Line 704: / Line 693: @@
 {{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
@@ Line 718: / Line 705: @@
 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
@@ Line 731: / Line 718: @@
 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
@@ Line 744: / Line 731: @@
 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
@@ Line 757: / Line 744: @@
 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
@@ Line 770: / Line 757: @@
 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
@@ Line 783: / Line 770: @@
 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
@@ Line 796: / Line 783: @@
 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
@@ Line 813: / Line 800: @@
 {{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
@@ Line 827: / Line 812: @@
 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
@@ Line 840: / Line 825: @@
 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
@@ Line 853: / Line 838: @@
 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
@@ Line 866: / Line 851: @@
 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
@@ Line 875: / Line 860: @@
 == 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
@@ Line 882: / Line 869: @@
 {{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
@@ Line 896: / Line 881: @@
 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
@@ Line 909: / Line 894: @@
 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
@@ Line 922: / Line 907: @@
 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
@@ Line 949: / Line 934: @@
 RMS error: 0.5567 cents
-Badness (Dirichlet): 0.195
+Badness (Sintel): 0.195
 === Tridecimal didacus ===
@@ Line 970: / Line 955: @@
 Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
-Badness (Dirichlet): 0.324
+Badness (Sintel): 0.324
 ==== Mediantone ====
@@ Line 991: / Line 976: @@
 <nowiki />* 117d only appears without prime 19
-Badness (Dirichlet): 0.618
+Badness (Sintel): 0.618
 ==== Roulette ====
@@ Line 1,010: / Line 995: @@
 Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
-Badness (Dirichlet): 0.676
+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.
+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
@@ Line 1,030: / Line 1,015: @@
 == Isra ==
-Isra 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 (''iss-RAH'') night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).
+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
@@ Line 1,105: / Line 1,090: @@
 [[Tp tuning #T2 tuning|RMS error]]: 2.003 cents
-[[Category:Temperament clans]]
 [[Category:Hemimean clan| ]] <!-- main article -->
 [[Category:Hemimean| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament clans]]
+[[Category:Catalogs of rank-2 temperaments]]