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.  

* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]

* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]

* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]

* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]

* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]

* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]

* ''[[Snipes]]'' (+6125/5832) → [[Wesley family #Snipes|Wesley family]]

Considered below are wizard, 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 #Wizard]].''

Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.

[[Subgroup]]: 2.3.5.7

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

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

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

* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}

: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}

{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}

[[Badness]] (Sintel): 1.03

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}

{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}

Badness (Sintel): 0.613

==== Lizard ====

Subgroup: 2.3.5.7.11.13

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

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

* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}

{{Optimal ET sequence|legend=0| 22, 50, 72 }}

Badness (Sintel): 0.900

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

Subgroup: 2.3.5.7.11.13.17

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 }}

* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}

{{Optimal ET sequence|legend=0| 22, 50, 72 }}

Badness (Sintel): 0.741

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

Subgroup: 2.3.5.7.11.13.17.19

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 }}

* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}

{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}

Badness (Sintel): 0.955

==== Gizzard ====

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

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

* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}

{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}

Badness (Sintel): 0.837

===== 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 }}

* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}

{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}

Badness (Sintel): 0.694

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

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 }}

* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}

{{Optimal ET sequence|legend=0| 72, 94, 166g }}

Badness (Sintel): 0.901

=== Mage ===

Subgroup: 2.3.5.7.11

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

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

* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}

* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}

{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}