Marvel temperaments: Difference between revisions

This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  

* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]

* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]

* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]

* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]

* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]

* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]

Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].

Since {{nowrap|(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.

The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.

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 {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''

Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}

: mapping generators: ~2, ~10/7

* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}

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

[[Badness]] (Sintel): 1.20

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}

* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}

{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}

Badness (Sintel): 0.782

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}

* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}

{{Optimal ET sequence|legend=0| 29, 31, 60e }}

Badness (Sintel): 0.950

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 273/272

{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}

* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}

{{Optimal ET sequence|legend=0| 29g, 31, 60e }}

Badness (Sintel): 0.973

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272

{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}

* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}

{{Optimal ET sequence|legend=0| 29g, 31, 60e }}

Badness (Sintel): 1.03

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195

{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}

* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}

{{Optimal ET sequence|legend=0| 29g, 31, 60e }}

Badness (Sintel): 1.04

=== Tritoni ===

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}

* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}

{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}

Badness (Sintel): 1.50

== Septimin ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''

Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.  

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}

: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}

: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}

{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}

[[Badness]] (Sintel): 1.38

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 385/384

{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}

Optimal tunings:  

* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}

{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}

Badness (Sintel): 1.04

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}

Optimal tunings:  

* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}

{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}

Badness (Sintel): 0.955

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''

Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.

The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}

: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}

: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}

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

[[Badness]] (Sintel): 1.39

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}

Optimal tunings:  

* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}

{{Optimal ET sequence|legend=0| 41, 84, 125e }}

Badness (Sintel): 1.20

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 225/224, 364/363, 441/440

{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}

Optimal tunings:  

* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}

{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}

Badness (Sintel): 1.14

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 532400/531441

{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}

Optimal tunings:  

* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}

{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}

Badness (Sintel): 1.46

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 10648/10647

{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}

Optimal tunings:  

* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}

{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}

Badness (Sintel): 1.47

Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.

The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~2, ~49/48

[[Optimal tuning]]s:

: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}

{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}

[[Badness]] (Sintel): 1.44

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}

{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}

Badness (Sintel): 0.838

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}

{{Optimal ET sequence|legend=0| 31, 63, 94 }}

Badness (Sintel): 1.07

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''

Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 1029/1000

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

: mapping generators: ~2, ~10/7

* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}

: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}

: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}

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

[[Badness]] (Sintel): 1.50

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}

* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}

{{Optimal ET sequence|legend=0| 2, 17d, 19 }}

Badness (Sintel): 1.51

== Marvolo ==

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~2, ~21/20

* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}

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

[[Badness]] (Sintel): 2.11

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}

* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}

{{Optimal ET sequence|legend=0| 29, 43, 72 }}

Badness (Sintel): 0.958

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}

* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}

{{Optimal ET sequence|legend=0| 29, 43, 72 }}

Badness (Sintel): 0.887

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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

{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}

* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}

{{Optimal ET sequence|legend=0| 29g, 43, 72 }}

Badness (Sintel): 0.760

=== 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

{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}

* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}

{{Optimal ET sequence|legend=0| 29g, 43, 72 }}

Badness (Sintel): 0.895

== Decic ==

Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].  

It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 16807/16384