Marvel temperaments: Difference between revisions

Revision as of 15:19, 21 November 2021

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

Temperaments considered in families and clans are:

Sharp, {25/24, 28/27} → Dicot family
Pelogic, {21/20, 135/128} → Pelogic family
August, {36/35, 128/125} → Augmented family
Pajara, {50/49, 64/63} → Diaschismic family
Meantone, {81/80, 126/125} → Meantone family
Magic, {225/224, 245/243} → Magic family
Passive, {225/224, 256/245} → Passion family
Miracle, {225/224, 1029/1024} → Gamelismic clan
Orwell, {225/224, 1728/1715} → Semicomma family
Garibaldi, {225/224, 3125/3087} → Schismatic family
Catakleismic, {225/224, 4375/4374} → Kleismic family
Decic, {225/224, 16807/16384} → Cloudy clan
Escapade, {225/224, 65625/65536} → Escapade family
Fog, {225/224, 156250/151263} → Misty family
Compton, {225/224, 250047/250000} → Compton family
Betic, {225/224, 1071875/1062882} → Sycamore family
Houborizic {225/224, 1250000/1240029} → Amity family
Qintosec, {225/224, 2560000/2470629} → Qintosec family
Quintapole, {225/224, 7812500/7411887} → Quintaleap family
Maquila, {225/224, 30233088/28824005} → Maquila family
Marvo, {225/224, 78125000/78121827} → Gravity family

Considered below are negri, 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.

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

Negri

Main article: Negri

Negri tempers out the negri comma in the 5-limit, 49/48 and 225/224 in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 to the comma list; this will be discussed below under the title of negra.

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

Wedgie: ⟨⟨ 4 -3 -14 ]]

POTE generator: ~16/15 = 125.7549

Template:Val list

Badness: 0.086856

7-limit

Subgroup: 2.3.5.7

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]

Wedgie: ⟨⟨ 4 -3 2 -14 -8 13 ]]

POTE generator: ~15/14 = 125.608

Template:Val list

Badness: 0.026483

Negra

This is the 2.3.5.7.13 extension of negri.

Subgroup: 2.3.5.7.13

Comma list: 49/48, 65/64, 91/90

Sval mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -3]]

Gencom mapping: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3]]

Gencom: [2 14/13; 49/48 65/64 91/90]

POTE generator: ~14/13 = 125.567

Template:Val list

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 56/55

Mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -5]]

POTE generator: ~15/14 = 126.474

@@ Line 20: / Line 20: @@
 * ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
 * ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
+* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
 * ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
 * ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
-Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, gracecordial, and quintapole.
+Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.
 Since (5/4)<sup>2</sup> = 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.
@@ Line 1,046: / Line 1,047: @@
 Badness: 0.052235
-== Quintapole ==
-: ''See also: [[16ed5/2 #Regular temperaments]] or [[42ed11 #Regular temperaments]]''
-Aside from the marvel comma, the ''quintapole'' temperament (12&amp;85) tempers out 7812500/7411887 (sepru-atritriyo). In the 11-limit, it tempers out the ptolemisma (100/99) as well as 85184/84035 (trilo-aquinru-agu). It is so named for the following reasons - it has the same commas as the [[Marvel family #Appolo|apollo temperament]], and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Proper pronunciation of the word "quintapole" is /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". ''Not to be confused with [[Octagar temperaments #Quintupole|quint<u>'''u'''</u>pole]] temperament (12&amp;121).''
-Subgroup: 2.3.5.7
-[[Comma list]]: 225/224, 7812500/7411887
-[[Mapping]]: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]
-[[POTE generator]]: ~21/20 = 98.994
-{{Val list|legend=1| 12, 73c, 85, 97d }}
-[[Badness]]: 0.192498
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 100/99, 225/224, 85184/84035
-Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]
-POTE generator: ~21/20 = 98.954
-Vals: {{Val list| 12, 73ce, 85, 97d }}
-Badness: 0.104353
 [[Category:Regular temperament theory]]

Marvel temperaments: Difference between revisions

Revision as of 15:19, 21 November 2021

Negri

7-limit

Negra

11-limit

13-limit

Negril

13-limit

Negric

13-limit

Negroni

13-limit

Wilsec

13-limit

17-limit

19-limit

Wizard

11-limit

Lizard

17-limit

19-limit

Gizzard

17-limit

19-limit

Mage

Triton

11-limit

Tritonic

11-limit

13-limit

Tritoni

Merman

11-limit

13-limit

Untriton

11-limit

13-limit

Septimin

11-limit

13-limit

Misneb

11-limit

13-limit

Musneb

Slender

11-limit

13-limit

Marvolo

11-limit

13-limit

Amavil

11-limit

13-limit

Enneaportent

11-limit

13-limit

Submajor

7-limit

11-limit

13-limit

Interpental

13-limit

Alphorn

11-limit

Tertiosec

7-limit

11-limit

Gwazy

11-limit

Gracecordial

11-limit

13-limit

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