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.  

* [[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]]

* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]

* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty 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 }}

[[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:  

* 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:  

* 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|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:  

* [[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

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:  

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

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

Badness (Sintel): 1.20

Subgroup: 2.3.5.7.11.13