Hemimean clan: Difference between revisions

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~~<div style="float: right;">~~

[[File:Didacus.png|alt=Didacus.png|thumb|600x560px|A chart of the tuning spectrum of undecimal didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.]]

~~</div>~~

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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== Hemiwürschmidt ==

''[[#Strong extensions|Return to the map]]''

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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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~~{{todo|verify this & get badness|inline=1}}~~

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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== Hemithirds ==

''[[#Strong extensions|Return to the map]]''

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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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== Spell ==

''[[#Strong extensions|Return to the map]]''

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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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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

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Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927

Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}

Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}

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

Badness (~~Dirichlet~~): 0.618

Badness (Sintel): 0.618

==== Roulette ====

Line 1,008:

Line 983:

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~~

~~Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]~~

~~Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}~~

~~: sval mapping generators: ~2, ~28/25~~

~~Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285~~

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

~~Badness (Dirichlet): 0.685~~

~~===== 2.5.7.11.13.17.19 subgroup =====~~

Subgroup: 2.5.7.11.13.17.19

Line 1,035:

Line 995:

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

Line 1,055:

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 1,130:

Line 1,090:

[[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}}
-<div style="float: right;">
-[[File:Didacus.png|alt=Didacus.png|thumb|600x560px|A chart of the tuning spectrum of undecimal didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.]]
-</div>
 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 15: / 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 23: / 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 34: / Line 30: @@
 == Didacus ==
 {{main|Didacus}}
+See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
 [[Subgroup]]: 2.5.7
@@ Line 53: / Line 51: @@
 [[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
-[[Badness]] (Dirichlet): 0.091
+[[Badness]] (Sintel): 0.091
 = Strong extensions =
@@ Line 72: / Line 70: @@
 == Hemiwürschmidt ==
+''[[#Strong extensions|Return to the map]]''
 {{See also| Würschmidt family }}
@@ Line 83: / 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 107: / Line 105: @@
 {{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
-{{todo|verify this & get badness|inline=1}}
+Badness (Sintel): 0.304
 === 11-limit ===
@@ Line 114: / 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 127: / 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 140: / 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 153: / 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 166: / 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 179: / 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 189: / 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 194: / 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 209: / 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 224: / 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 237: / 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 250: / 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 263: / 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 276: / 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 285: / Line 285: @@
 == Hemithirds ==
+''[[#Strong extensions|Return to the map]]''
 {{Main| Hemithirds }}
@@ Line 292: / 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 300: / 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 314: / 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 332: / 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 341: / Line 341: @@
 == Spell ==
+''[[#Strong extensions|Return to the map]]''
 {{See also| Magic family }}
@@ Line 348: / 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 362: / 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 375: / 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 388: / 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 410: / 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 424: / 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 437: / 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 450: / 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 463: / 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 476: / 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 489: / 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 509: / 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 523: / 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 536: / 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 552: / 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 566: / 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 579: / 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 592: / 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 605: / 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 618: / 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 631: / 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 645: / 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 659: / 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 672: / 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 685: / 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 694: / 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 701: / 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 715: / 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 728: / 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 741: / 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 754: / 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 767: / 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 780: / 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 793: / 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 810: / 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 824: / 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 837: / 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 850: / 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 863: / 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 872: / 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 879: / 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 893: / 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 906: / 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 919: / 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 946: / Line 934: @@
 RMS error: 0.5567 cents
-Badness (Dirichlet): 0.195
+Badness (Sintel): 0.195
 === Tridecimal didacus ===
@@ Line 967: / Line 955: @@
 Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
-Badness (Dirichlet): 0.324
+Badness (Sintel): 0.324
 ==== Mediantone ====
@@ Line 974: / Line 962: @@
 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
@@ Line 999: / Line 972: @@
 Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
-Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}
+Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}
+<nowiki />* 117d only appears without prime 19
-Badness (Dirichlet): 0.618
+Badness (Sintel): 0.618
 ==== Roulette ====
@@ Line 1,008: / Line 983: @@
 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
-Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]
-Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}
-: sval mapping generators: ~2, ~28/25
-Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285
-Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
-Badness (Dirichlet): 0.685
-===== 2.5.7.11.13.17.19 subgroup =====
 Subgroup: 2.5.7.11.13.17.19
@@ Line 1,035: / 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 1,055: / 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 1,130: / 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]]