Marvel temperaments: Difference between revisions

This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.  

* ''[[Pelogic]]'' → [[Pelogic family #Pelogic|Pelogic family]] (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)

* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)

* [[Garibaldi temperament|Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+3125/3087, generated by the fifth with 5/4 mapped to the d4)

* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] (+50/49 or 64/63, generated by the fifth with a semioctave period)

* ''[[Sharp]]'' → [[Dicot family #Sharp|Dicot family]] (+25/24 or 28/27, fifth sliced in two)

* ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] (+781250/750141, twelfth sliced in two)

* ''[[August]]'' → [[Augmented family #August|Augmented family]] (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)

* ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+156250/151263, generated by the fifth with a 1/3-octave period)

* [[Negri]] → [[Slendro clan #Negri|Slendro clan]] (+49/48, fourth sliced in four)

* [[Magic]] → [[Magic family #Magic|Magic family]] (+245/243, twelfth sliced in five)

* ''[[Passive]]'' → [[Passion family #Passive|Passion family]] (+256/245, fourth sliced in five)

* ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five)

* ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five)

* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)

* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)

* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)

* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)

* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)

* ''[[Snipes]]''  → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)

* ''[[Escapade]]'' → [[Escapade family #Escapade|Escapade family]] (+65625/65536, fourth sliced in nine)

* ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period)

* ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten)

* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven)

* ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period)

* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)

* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)

* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty)

Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.  

Since (5/4)² = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.

Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

== Wizard ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''

[[Comma list]]: 225/224, 118098/117649

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

: mapping generators: ~1225/864, ~245/216

 

 

{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}

 

Scales: [[wizard22]]

Comma list: 225/224, 385/384, 4000/3993

Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.232 (~25/22 = 216.768)

{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c }}

Badness: 0.018539

 

==== Lizard ====

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}

Optimal tuning (POTE): ~55/39 = 1\2, ~5/4 = 383.389 (~25/22 = 216.711)

{{Optimal ET sequence|legend=1| 22, 50, 72, 122, 194df }}

Badness: 0.021781

===== 17-limit =====

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.381 (~17/15 = 216.619)

{{Optimal ET sequence|legend=1| 22, 50, 72, 122g, 194dfg }}

Badness: 0.014536

===== 19-limit =====

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.477 (~17/15 = 216.523)

{{Optimal ET sequence|legend=1| 22h, 50, 72, 122g, 194dfg }}

Badness: 0.015702

Subgroup: 2.3.5.7.11.13

Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.170 (~25/22 = 216.830)

{{Optimal ET sequence|legend=1| 72, 166, 238cf }}

Badness: 0.020252

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.175 (~25/22 = 216.825)

{{Optimal ET sequence|legend=1| 72, 166g, 238cfg }}

Badness: 0.013624

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594

Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.138 (~17/15 = 216.862)

{{Optimal ET sequence|legend=1| 72, 94, 166g }}

Badness: 0.014810

=== Mage ===

Comma list: 99/98, 176/175, 1331/1296

Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}

Optimal tuning (POTE): ~77/54 = 1\2, ~5/4 = 383.124 (~55/48 = 216.876)

{{Optimal ET sequence|legend=1| 22, 50e, 72ee, 94ee }}

Badness: 0.057799

== Tritonic ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''

[[Comma list]]: 225/224, 50421/50000

{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}

{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}

 

{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}

[[Badness]]: 0.047578

Comma list: 121/120, 225/224, 441/440

Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}

{{Multival|legend=1| 5 -11 -12 -3 -29 -33 -22 3 31 33 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.267

 

Badness: 0.023659

Comma list: 105/104, 121/120, 196/195, 275/273

Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.108

{{Optimal ET sequence|legend=1| 29, 31, 60e, 151cde }}

Badness: 0.022993

=== Tritoni ===

Comma list: 225/224, 385/384, 27783/27500

Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}

{{Multival|legend=1| 5 -11 -12 28 -29 -33 27 3 103 120 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.389

{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}

Badness: 0.045456

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''

[[Comma list]]: 225/224, 84035/82944

{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}

{{Multival|legend=1| 11 -6 10 -35 -15 40 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 263.632

 

[[Badness]]: 0.054502

Comma list: 225/224, 385/384, 2401/2376

Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.634

{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}

Badness: 0.031309

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.700

{{Optimal ET sequence|legend=1| 41, 91 }}

Badness: 0.023117

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''

[[Comma list]]: 225/224, 2500000/2470629

{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}

{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 585.585

 

[[Badness]]: 0.055078

Comma list: 225/224, 441/440, 1344/1331

Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.606

{{Optimal ET sequence|legend=1| 41, 84, 125e }}

Badness: 0.036383

 

 

 

 

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.657

 

 

 

Slender (31 & 32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.

[[Comma list]]: 225/224, 589824/588245

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

{{Multival|legend=1| 13 -10 6 -46 -27 42 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 38.413

 

[[Badness]]: 0.056934

Comma list: 225/224, 385/384, 1331/1323

Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.387

{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}

Badness: 0.025342

Comma list: 225/224, 275/273, 385/384, 1331/1323

Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.314

{{Optimal ET sequence|legend=1| 31, 63, 94 }}

Badness: 0.025913

== Triton ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}

{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 568.865

{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}

[[Badness]]: 0.059245

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 1029/1000

Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 569.144

{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}

Badness: 0.045675

[[Subgroup]]: 2.3.5

[[Comma list]]: 69198046875/68719476736

{{Mapping|legend=1| 1 4 -1 | 0 -8 11 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10125/8192 = 362.321

{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}

[[Badness]]: 0.130236

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 51200/50421

{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }}

{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255

[[Badness]]: 0.060533

Subgroup: 2.3.5.7.11

Mapping: {{mapping| 1 4 -1 1 11 | 0 -8 11 6 -25 }}

Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101

{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cde }}

Badness: 0.050582

Subgroup: 2.3.5.7.11.13

Mapping: {{mapping| 1 4 -1 1 11 4 | 0 -8 11 6 -25 -1 }}

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105

{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cdef }}

Badness: 0.027689

Comma list: 99/98, 176/175, 51200/50421

Mapping: {{mapping| 1 4 -1 1 -5 | 0 -8 11 6 28 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418

{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}

Badness: 0.051806

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: {{mapping| 1 4 -1 1 -5 4 | 0 -8 11 6 28 -1 }}

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402

{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}

Badness: 0.029680

== Marvolo ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 156250000/155649627

{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}

{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.348

{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}

[[Badness]]: 0.083338

Comma list: 225/224, 441/440, 4000/3993

Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.340

{{Optimal ET sequence|legend=1| 29, 43, 72 }}

Badness: 0.028965

=== 13-limit ===

Comma list: 169/168, 225/224, 364/363, 441/440

Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330

{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}

Badness: 0.021470

=== 17-limit ===

Comma list: 169/168, 221/220, 225/224, 364/363, 441/440

 

 

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330

{{Optimal ET sequence|legend=1| 29g, 43, 72 }}

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440

 

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330

 

{{Multival|legend=1| 18 -9 18 -56 -22 67 }}

[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)

 

[[Badness]]: 0.093679

Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}

Optimal tuning (POTE): ~121/112 = 1\9, ~210/121 = 950.1873 (~99/98 = 16.8540)

{{Optimal ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}

Badness: 0.030426

Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}

Optimal tuning (POTE): ~14/13 = 1\9, ~26/15 = 950.2867 (~105/104 = 16.9534)

{{Optimal ET sequence|legend=1| 9, 63, 72, 279cf }}

Badness: 0.022322

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Gracecordial]].''

{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.824

{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}

[[Badness]]: 0.096279

Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.834

{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}

Badness: 0.089588

Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841

{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}

Badness: 0.052235

Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841

{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}

Badness: 0.038565

Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842

{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}

Badness: 0.028165

Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.843

{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}

Badness: 0.021879

Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842

{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}

Badness: 0.018011

Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.838

{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}

Badness: 0.016007

Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.885

{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}

Badness: 0.066964

Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.890

{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}

Badness: 0.044196

Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.870

{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}

Badness: 0.036637

Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.866

{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}

Badness: 0.027559

{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}

 

{{Multival|legend=1| 16 -5 22 -45 -10 65 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 556.221