Marvel temperaments: Difference between revisions

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This page discusses some of the temperaments tempering out |-5 2 2 -1> =[[225/224| 225/224]], the [[Marvel_family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone_family|meantone]], [[miracle]], [[Magic_family|magic]], [[pajara]], [[orwell]], [[Kleismic_family|catakleismic]], [[Garibaldi_temperament|garibaldi]], [[Augmented_family|august]], [[Pythagorean_family|compton]], [[Dicot_family|sharp]], [[Escapade_family|escapade]] and [[Pelogic_family|mavila]].
This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel_family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]] and [[Pelogic family #Mavila|mavila]].


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.
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.


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.
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.)
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=
=Negri=
More info can be found on the [[Negri|Negri family]] page, although for reference the results are also listed here:
{{main|Negri}}


==5-limit==
==5-limit==
 
Note that this is technically not a Marvel temperament until the 7-limit, but also placed here for reference.
(Note that this is technically not a Marvel temperament until the 7-limit, but also placed here for reference)


[[Comma|Comma]]s: 16875/16384
[[Comma|Comma]]s: 16875/16384
Line 23: Line 22:
[[wedgie|Wedgie]]: &lt;&lt;4 -3 -14||
[[wedgie|Wedgie]]: &lt;&lt;4 -3 -14||


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
EDOs: {{EDOs|9, 10, 19, 86}}


==7-limit==
==7-limit==
Line 34: Line 33:
[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
[[wedgie|Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||


EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
EDOs: {{EDOs|9, 10, 19, 86}}


==11-limit==
==11-limit==
Line 43: Line 42:
Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]


EDOs: 9, 10, 19
EDOs: {{EDOs|9, 10, 19}}


[[Badness|Badness]]: 0.0262
[[Badness|Badness]]: 0.0262


==13-limit==
=== 13-limit ===
Commas: 45/44, 49/48, 56/55, 78/77
Commas: 45/44, 49/48, 56/55, 78/77


Line 54: Line 53:
Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]


EDOs: 9, 10, 19
EDOs: {{EDOs|9, 10, 19}}


=Negril=
== Negril ==
Commas: 49/48, 100/99, 225/224
Commas: 49/48, 100/99, 225/224


Line 63: Line 62:
Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]


EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
EDOs: {{EDOs|19, 29, 48df, 77cdf}}


Badness: 0.0387
Badness: 0.0387


==13-limit==
=== 13-limit ===
Commas: 49/48, 65/64, 91/90, 875/858
Commas: 49/48, 65/64, 91/90, 875/858


Line 74: Line 73:
Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]


EDOs: 9, 10, 19, 29, 48, 77
EDOs: {{EDOs|9, 10, 19, 29, 48, 77}}


Badness: 0.0244
Badness: 0.0244


=Negric=
== Negric ==
Commas: 33/32, 49/48, 77/75
Commas: 33/32, 49/48, 77/75


Line 85: Line 84:
Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]


EDOs: 9, 19, [[85edo|85]]
EDOs: {{EDOs|9, 19, 85}}


Badness: 0.0306
Badness: 0.0306


==13-limit==
=== 13-limit ===
Commas: 33/32, 49/48, 65/64, 91/90
Commas: 33/32, 49/48, 65/64, 91/90


Line 96: Line 95:
Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]


EDOs: 9, 19, 85
EDOs: {{EDOs|9, 19, 85}}


Badness: 0.0202
Badness: 0.0202


=Negroni=
== Negroni ==
Commas: 49/48, 55/54, 225/224
Commas: 49/48, 55/54, 225/224


Line 107: Line 106:
Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]


EDOs: 10, 19e, 29, 77cde
EDOs: {{EDOs|10, 19e, 29, 77cde}}


Badness: 0.0353
Badness: 0.0353


==13-limit==
=== 13-limit ===
Commas: 49/48, 55/54, 65/64, 91/90
Commas: 49/48, 55/54, 65/64, 91/90


Line 118: Line 117:
Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]


EDOs: 10, 19e, 29, 77cdef
EDOs: {{EDOs|10, 19e, 29, 77cdef}}


Badness: 0.0216
Badness: 0.0216
Line 129: Line 128:
Map: [&lt;2 1 5|, &lt;0 6 -1|]
Map: [&lt;2 1 5|, &lt;0 6 -1|]


EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
EDOs: {{EDOs|22, 50, 72, 166, 238c, 310c, 548bc}}


Badness: 0.3864
Badness: 0.3864
Line 142: Line 141:
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||


EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
EDOs: {{EDOs|22, 50, 72, 94, 166, 238c, 310c, 382c}}


Badness: 0.0408
Badness: 0.0408
Line 153: Line 152:
Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]


EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
EDOs: {{EDOs|22, 50, 72, 94, 166, 238c, 310c}}


Badness: 0.0185
Badness: 0.0185


==Lizard==
=== Lizard ===
Commas: 225/224, 351/350, 364/363, 385/384
Commas: 225/224, 351/350, 364/363, 385/384


Line 164: Line 163:
Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]


EDOs: 22, 50, 72, 122, 194df
EDOs: {{EDOs|22, 50, 72, 122, 194df}}


Badness: 0.0218
Badness: 0.0218


===17-limit===
==== 17-limit ====
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


Line 175: Line 174:
Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]


EDOs: 22, 50, 72, 122g, 194dfg
EDOs: {{EDOs|22, 50, 72, 122g, 194dfg}}


Badness: 0.0145
Badness: 0.0145


===19-limit===
==== 19-limit ====
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Line 186: Line 185:
Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]


EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
EDOs: {{EDOs|22h, 50, 72, 122g, 194dfg}}


Badness: 0.0157
Badness: 0.0157


==Gizzard==
=== Gizzard ===
Commas: 225/224, 385/384, 325/324, 1573/1568
Commas: 225/224, 385/384, 325/324, 1573/1568


Line 197: Line 196:
Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]


EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
EDOs: {{EDOs|22, 72, 94, 166, 238cf}}


Badness: 0.0203
Badness: 0.0203


===17-limit===
==== 17-limit ====
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


Line 208: Line 207:
Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]


EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
EDOs: {{EDOs|22f, 72, 94, 166g, 238cfg}}


Badness: 0.0136
Badness: 0.0136


===19-limit===
==== 19-limit ====
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


Line 219: Line 218:
Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]


EDOs: 22f, 72, 94, 166g
EDOs: {{EDOs|22f, 72, 94, 166g}}


Badness: 0.0148
Badness: 0.0148


==Mage==
== Mage ==
Commas: 99/98, 176/175, 1331/1296
Commas: 99/98, 176/175, 1331/1296


Line 230: Line 229:
Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]


EDOs: 22, 72e, 94e
EDOs: {{EDOs|22, 72e, 94e}}


Badness: 0.0578
Badness: 0.0578
Line 243: Line 242:
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||


EDOs: 17d, 19, 78bd, 97bd
EDOs: {{EDOs|17d, 19, 78bd, 97bd}}


Badness: 0.0592
Badness: 0.0592
Line 254: Line 253:
Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]


EDOs: 19, 59bde, 78bde, 97bde
EDOs: {{EDOs|19, 59bde, 78bde, 97bde}}


Badness: 0.0457
Badness: 0.0457


=Tritonic=
= Tritonic =
== 5-limit ==
Comnma: 553584375/536870912
Comnma: 553584375/536870912


Line 265: Line 265:
Map: [&lt;1 4 -3|, &lt;0 -5 11|]
Map: [&lt;1 4 -3|, &lt;0 -5 11|]


EDOs: 29, 31, 60, 91
EDOs: {{EDOs|29, 31, 60, 91}}


Badness: 0.3785
Badness: 0.3785


==7-limit==
== 7-limit ==
Commas: 225/224, 50421/50000
Commas: 225/224, 50421/50000


Line 278: Line 278:
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||


EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
EDOs: {{EDOs|29, 31, 60, 91, 122, 213bcd}}


==11-limit==
== 11-limit ==
Commas: 121/120, 225/224, 441/440
Commas: 121/120, 225/224, 441/440


Line 287: Line 287:
Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]


EDOs: 29, 31, 60e
EDOs: {{EDOs|29, 31, 60e}}


Badness: 0.0237
Badness: 0.0237


==13-limit==
=== 13-limit ===
Commas: 105/104, 121/120, 196/195, 275/273
Commas: 105/104, 121/120, 196/195, 275/273


Line 298: Line 298:
Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]


EDOs: 29, 31, 60e, 151cde
EDOs: {{EDOs|29, 31, 60e, 151cde}}


Badness: 0.0230
Badness: 0.0230


==Tritoni==
== Tritoni ==
Commas: 225/224, 385/384, 27783/27500
Commas: 225/224, 385/384, 27783/27500


Line 309: Line 309:
Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]


EDOs: 31, 91, 122
EDOs: {{EDOs|31, 91, 122}}


Badness: 0.0455
Badness: 0.0455
Line 320: Line 320:
Map: [&lt;1 5 -5|, &lt;0 -7 15|]
Map: [&lt;1 5 -5|, &lt;0 -7 15|]


EDOs: 41, 84, 125, 209, 334c, 543bc
EDOs: {{EDOs|41, 84, 125, 209, 334c, 543bc}}


Badness: 0.4538
Badness: 0.4538
Line 333: Line 333:
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||


EDOs: 41, 84, 125
EDOs: {{EDOs|41, 84, 125}}


Badness: 0.0551
Badness: 0.0551
Line 344: Line 344:
Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]


EDOs: 41, 84, 125e
EDOs: {{EDOs|41, 84, 125e}}


Badness: 0.0364
Badness: 0.0364
Line 355: Line 355:
Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]


EDOs: 41, 84, 125e, 209ef, 293ef
EDOs: {{EDOs|41, 84, 125e, 209ef, 293ef}}


Badness: 0.0275
Badness: 0.0275
Line 366: Line 366:
Map: [&lt;1 4 1|, &lt;0 -11 6|]
Map: [&lt;1 4 1|, &lt;0 -11 6|]


EDOs: 9, 32, 41, 91, 132, 223c, 355bc
EDOs: {{EDOs|9, 32, 41, 91, 132, 223c, 355bc}}


Badness: 0.6709
Badness: 0.6709
Line 379: Line 379:
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||


EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
EDOs: {{EDOs|9, 32, 41, 91, 132}}


==11-limit==
==11-limit==
Line 388: Line 388:
Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]


EDOs: 9, 32, 41, 91, 223cdef
EDOs: {{EDOs|9, 32, 41, 91, 223cdef}}


==13-limit==
==13-limit==
Line 397: Line 397:
Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]


EDOs: 9, 41, 91
EDOs: {{EDOs|9, 41, 91}}


=Slender=
=Slender=
Line 408: Line 408:
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||


EDOs: 31, 94, 125
EDOs: {{EDOs|31, 94, 125}}


Badness: 0.0569
Badness: 0.0569
Line 419: Line 419:
Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]


EDOs: 31, 63, 94, 125
EDOs: {{EDOs|31, 63, 94, 125}}


Badness: 25.342
Badness: 25.342
Line 430: Line 430:
Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]


EDOs: 31, 63, 94
EDOs: {{EDOs|31, 63, 94}}


Badness: 25.913
Badness: 25.913
Line 443: Line 443:
Wedgie: &lt;&lt;6 17 46 13 56 59||
Wedgie: &lt;&lt;6 17 46 13 56 59||


EDOs: 7d, 65d, 72, 137, 209, 281, 569bcc
EDOs: {{EDOs|7d, 65d, 72, 137, 209, 281, 569bcc}}


Badness: 0.0976
Badness: 0.0976
Line 454: Line 454:
Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]


EDOs: 7d, 65d, 72, 281, 353c, 425bc, 497bc
EDOs: {{EDOs|7d, 65d, 72, 281, 353c, 425bc, 497bc}}


Badness: 0.0317
Badness: 0.0317
Line 465: Line 465:
Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]


EDOs: 72, 137, 209, 281f, 490bcf
EDOs: {{EDOs|72, 137, 209, 281f, 490bcf}}


Badness: 0.0269
Badness: 0.0269
Line 478: Line 478:
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||


EDOs: 29, 43, 72, 619bcd, 691bcd
EDOs: {{EDOs|29, 43, 72, 619bcd, 691bcd}}


Badness: 0.0833
Badness: 0.0833
Line 489: Line 489:
Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]


EDOs: 29, 43, 72
EDOs: {{EDOs|29, 43, 72}}


Badness: 0.0290
Badness: 0.0290
Line 500: Line 500:
Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]


EDOs: 29, 43, 72, 115f
EDOs: {{EDOs|29, 43, 72, 115f}}


Badness: 0.0215
Badness: 0.0215
Line 513: Line 513:
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||


EDOs: 9, 34d, 43, 77d
EDOs: {{EDOs|9, 34d, 43, 77d}}


Badness: 0.1096
Badness: 0.1096
Line 524: Line 524:
Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]


EDOs: 9, 34d, 43, 77de
EDOs: {{EDOs|9, 34d, 43, 77de}}


Badness: 0.0426
Badness: 0.0426
Line 535: Line 535:
Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]


EDOs: 9, 34d, 43, 77de
EDOs: {{EDOs|9, 34d, 43, 77de}}


Badness: 0.0258
Badness: 0.0258
Line 548: Line 548:
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||


EDOs: 9, 54, 63, 72, 495bcd
EDOs: {{EDOs|9, 54, 63, 72, 495bcd}}


Badness: 0.0937
Badness: 0.0937
Line 559: Line 559:
Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]


EDOs: 9, 54, 63, 72, 423cd, 495bcd
EDOs: {{EDOs|9, 54, 63, 72, 423cd, 495bcd}}


Badness: 0.0304
Badness: 0.0304
Line 570: Line 570:
Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]


EDOs: 9, 54, 63, 72, 279cf
EDOs: {{EDOs|9, 54, 63, 72, 279cf}}


Badness: 0.0223
Badness: 0.0223


=Submajor=
= Submajor =
== 5-limit ==
Comma: 69198046875/68719476736
Comma: 69198046875/68719476736


Line 581: Line 582:
Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Map: [&lt;1 4 -1|, &lt;0 -8 11|]


EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
EDOs: {{EDOs|10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}


Badness: 0.1302
Badness: 0.1302


==7-limit==
== 7-limit ==
Commas: 225/224, 51200/50421
Commas: 225/224, 51200/50421


Line 594: Line 595:
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||


EDOs: 10, 33, 43, 53
EDOs: {{EDOs|10, 33, 43, 53}}


Badness: 0.0605
Badness: 0.0605


==11-limit==
== 11-limit ==
Commas: 225/224, 385/384, 6655/6561
Commas: 225/224, 385/384, 6655/6561


Line 605: Line 606:
Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]


EDOs: 10, 53, 116, 169de, 285cde
EDOs: {{EDOs|10, 53, 116, 169de, 285cde}}


Badness: 0.0506
Badness: 0.0506


==13-limit==
=== 13-limit ===
Commas: 169/168, 225/224, 275/273, 385/384
Commas: 169/168, 225/224, 275/273, 385/384


Line 616: Line 617:
Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]


EDOs: 10, 53, 116, 169de, 285cdef
EDOs: {{EDOs|10, 53, 116, 169de, 285cdef}}


Badness: 0.0277
Badness: 0.0277


==Interpental==
== Interpental ==
Commas: 99/98, 176/175, 51200/50421
Commas: 99/98, 176/175, 51200/50421


Line 627: Line 628:
Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]


EDOs: 43, 53, 96, 149d
EDOs: {{EDOs|43, 53, 96, 149d}}


Badness: 0.0518
Badness: 0.0518


===13-limit===
=== 13-limit ===
Commas: 99/98, 169/168, 176/175, 640/637
Commas: 99/98, 169/168, 176/175, 640/637


Line 638: Line 639:
Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]


EDOs: 43, 53, 96, 149d
EDOs: {{EDOs|43, 53, 96, 149d}}


Badness: 0.0297
Badness: 0.0297

Revision as of 13:56, 28 October 2020

This page discusses some of the temperaments tempering out [-5 2 2 -1⟩ = 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila.

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

5-limit

Note that this is technically not a Marvel temperament until the 7-limit, but also placed here for reference.

Commas: 16875/16384

POTE temperament: ~15/14 = 125.7549

Map: [<1 2 2|, <0 -4 3|]

Wedgie: <<4 -3 -14||

EDOs: 9, 10, 19, 86

7-limit

Commas: 49/48, 225/224

POTE temperament: ~15/14 = 125.608

Map: [<1 2 2 3|, <0 -4 3 -2|]

Wedgie: <<4 -3 2 -14 -8 13||

EDOs: 9, 10, 19, 86

11-limit

Commas: 45/44, 49/48, 56/55

POTE generator: ~15/14 = 126.474

Map: [<1 2 2 3 4|, <0 -4 3 -2 -5|]

EDOs: 9, 10, 19

Badness: 0.0262

13-limit

Commas: 45/44, 49/48, 56/55, 78/77

POTE generator: ~14/13 = 126.431

Map: [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]

EDOs: 9, 10, 19

Negril

Commas: 49/48, 100/99, 225/224

POTE generator: ~15/14 = 124.767

Map: [<1 2 2 3 2|, <0 -4 3 -2 14|]

EDOs: 19, 29, 48df, 77cdf

Badness: 0.0387

13-limit

Commas: 49/48, 65/64, 91/90, 875/858

POTE generator: ~15/14 = 124.716

Map: [<1 2 2 3 2 4|, <0 -4 3 -2 14 -3|]

EDOs: 9, 10, 19, 29, 48, 77

Badness: 0.0244

Negric

Commas: 33/32, 49/48, 77/75

POTE generator: ~15/14 = 127.039

Map: [<1 2 2 3 3|, <0 -4 3 -2 4|]

EDOs: 9, 19, 85

Badness: 0.0306

13-limit

Commas: 33/32, 49/48, 65/64, 91/90

POTE generator: ~15/14 = 127.039

Map: [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|]

EDOs: 9, 19, 85

Badness: 0.0202

Negroni

Commas: 49/48, 55/54, 225/224

POTE generator: ~15/14 = 124.539

Map: [<1 2 2 3 5|, <0 -4 3 -2 -15|]

EDOs: 10, 19e, 29, 77cde

Badness: 0.0353

13-limit

Commas: 49/48, 55/54, 65/64, 91/90

POTE generator: ~15/14 = 124.545

Map: [<1 2 2 3 5 4|, <0 -4 3 -2 -15 -3|]

EDOs: 10, 19e, 29, 77cdef

Badness: 0.0216

Wizard

Comma: 2197265625/2147483648

POTE generator: ~5/4 = 383.212

Map: [<2 1 5|, <0 6 -1|]

EDOs: 22, 50, 72, 166, 238c, 310c, 548bc

Badness: 0.3864

7-limit

Commas: 225/224, 118098/117649

POTE temperament: 383.256

Map: [<2 1 5 2|, <0 6 -1 10|]

Wedgie: <<12 -2 20 -31 -2 52||

EDOs: 22, 50, 72, 94, 166, 238c, 310c, 382c

Badness: 0.0408

11-limit

Commas: 225/224, 385/384, 4000/3993

POTE temperament: 383.232

Map: [<2 1 5 2 8|, <0 6 -1 10 -3|]

EDOs: 22, 50, 72, 94, 166, 238c, 310c

Badness: 0.0185

Lizard

Commas: 225/224, 351/350, 364/363, 385/384

POTE temperament: 383.389

Map: [<2 1 5 2 8 11|, <0 6 -1 10 -3 -10|]

EDOs: 22, 50, 72, 122, 194df

Badness: 0.0218

17-limit

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

POTE temperament: 383.381

Map: [<2 1 5 2 8 11 6|, <0 6 -1 10 -3 -10 6|]

EDOs: 22, 50, 72, 122g, 194dfg

Badness: 0.0145

19-limit

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

POTE generator: ~5/4 = 383.477

Map: [<2 1 5 2 8 11 6 2|, <0 6 -1 10 -3 -10 6 18|]

EDOs: 22h, 50, 72, 122g, 194dfg

Badness: 0.0157

Gizzard

Commas: 225/224, 385/384, 325/324, 1573/1568

POTE generator: ~5/4 = 383.170

Map: [<2 1 5 2 8 -2|, <0 6 -1 10 -3 26|]

EDOs: 22, 72, 94, 166, 238cf

Badness: 0.0203

17-limit

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

POTE generator: ~5/4 = 383.175

Map: [<2 1 5 2 8 -2 6|, <0 6 -1 10 -3 26 6|]

EDOs: 22f, 72, 94, 166g, 238cfg

Badness: 0.0136

19-limit

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

POTE generator: ~5/4 = 383.138

Map: [<2 1 5 2 8 -2 6 15|, <0 6 -1 10 -3 26 6 -18|]

EDOs: 22f, 72, 94, 166g

Badness: 0.0148

Mage

Commas: 99/98, 176/175, 1331/1296

POTE generator: ~55/48 = 216.876

Map: [<2 1 5 2 4|, <0 6 -1 10 8|]

EDOs: 22, 72e, 94e

Badness: 0.0578

Triton

Commas: 225/224, 1029/1000

POTE temperament: ~7/5 = 568.865

Map: [<1 0 6 7|, <0 3 -7 -8|]

Wedgie: <<3 -7 -8 -18 -21 1||

EDOs: 17d, 19, 78bd, 97bd

Badness: 0.0592

11-limit

Commas: 45/44, 56/55, 1029/1000

POTE generator: ~7/5 = 569.144

Map: [<1 0 6 7 4|, <0 3 -7 -8 -1|]

EDOs: 19, 59bde, 78bde, 97bde

Badness: 0.0457

Tritonic

5-limit

Comnma: 553584375/536870912

POTE generator: ~45/32 = 580.219

Map: [<1 4 -3|, <0 -5 11|]

EDOs: 29, 31, 60, 91

Badness: 0.3785

7-limit

Commas: 225/224, 50421/50000

POTE temperament: 580.286

Map: [<1 4 -3 -3|, <0 -5 11 12|]

Wedgie: <<5 -11 -12 -29 -33 3||

EDOs: 29, 31, 60, 91, 122, 213bcd

11-limit

Commas: 121/120, 225/224, 441/440

POTE generator: ~7/5 = 580.267

Map: [<1 4 -3 -3 2|, <0 -5 11 12 3|]

EDOs: 29, 31, 60e

Badness: 0.0237

13-limit

Commas: 105/104, 121/120, 196/195, 275/273

POTE generator: ~7/5 = 580.108

Map: [<1 4 -3 -3 2 -5|, <0 -5 11 12 3 18|]

EDOs: 29, 31, 60e, 151cde

Badness: 0.0230

Tritoni

Commas: 225/224, 385/384, 27783/27500

POTE generator: ~7/5 = 580.389

Map: [<1 4 -3 -3 17|, <0 -5 11 12 -28|]

EDOs: 31, 91, 122

Badness: 0.0455

Merman

Comma: 1121008359375/1099511627776

POTE generator: ~45/32 = 585.634

Map: [<1 5 -5|, <0 -7 15|]

EDOs: 41, 84, 125, 209, 334c, 543bc

Badness: 0.4538

7-limit

Commas: 225/224, 2500000/2470629

POTE generator: ~7/5 = 585.585

Map: [<1 5 -5 -5|, <0 -7 15 16|]

Wedgie: <<7 -15 -16 -40 -45 5||

EDOs: 41, 84, 125

Badness: 0.0551

11-limit

Commas: 225/224, 441/440, 1344/1331

POTE generator: ~7/5 = 585.606

Map: [<1 5 -5 -5 2|, <0 -7 15 16 3]]

EDOs: 41, 84, 125e

Badness: 0.0364

13-limit

Commas: 144/143, 225/224, 364/363, 441/440

POTE generator: ~7/5 = 585.657

Map: [<1 5 -5 -5 2 12|, <0 -7 15 16 3 -17|]

EDOs: 41, 84, 125e, 209ef, 293ef

Badness: 0.0275

Septimin

Comma: 35595703125/34359738368

POTE generator: ~75/64 = 263.658

Map: [<1 4 1|, <0 -11 6|]

EDOs: 9, 32, 41, 91, 132, 223c, 355bc

Badness: 0.6709

7-limit

Commas: 225/224, 84035/82944

POTE temperament: 263.632

Map: [<1 4 1 5|, <0 -11 6 -10|]

Wedgie: <<11 -6 10 -35 -15 40||

EDOs: 9, 32, 41, 91, 132

11-limit

Commas: 225/224, 385/384, 2401/2376

POTE temperament: 263.634

Map: [<1 4 1 5 5|, <0 -11 6 -10 -7|]

EDOs: 9, 32, 41, 91, 223cdef

13-limit

Commas: 105/104, 144/143, 196/195, 245/242

POTE temperament: 263.700

Map: [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|]

EDOs: 9, 41, 91

Slender

Commas: 225/224, 589824/588245

POTE temperament: ~49/48 = 38.413

Map: [<1 2 2 3|, <0 -13 10 -6|]

Wedgie: <<13 -10 6 -46 -27 42||

EDOs: 31, 94, 125

Badness: 0.0569

11-limit

Commas: 225/224, 385/384, 1331/1323

POTE temperament: ~49/48 = 38.387

Map: [<1 2 2 3 4|, <0 -13 10 -6 -17|]

EDOs: 31, 63, 94, 125

Badness: 25.342

13-limit

Commas: 225/224, 275/273, 385/384, 1331/1323

POTE generator: ~49/48 = 38.314

Map: [<1 2 2 3 4 3|, <0 -13 10 -6 -17 22|]

EDOs: 31, 63, 94

Badness: 25.913

Marvo

Commas: 225/224, 78125000/78121827

POTE generator: ~27/20 = 516.694

Map: [<1 5 12 29|, <0 -6 -17 -46|]

Wedgie: <<6 17 46 13 56 59||

EDOs: 7d, 65d, 72, 137, 209, 281, 569bcc

Badness: 0.0976

11-limit

Commas: 225/224, 243/242, 4000/3993

POTE generator: ~27/20 = 516.699

Map: [<1 5 12 29 12|, <0 -6 -17 -46 -15|]

EDOs: 7d, 65d, 72, 281, 353c, 425bc, 497bc

Badness: 0.0317

13-limit

Commas: 225/224, 243/242, 351/350, 1625/1617

POTE generator: ~27/20 = 516.730

Map: [<1 5 12 29 12 39|, <0 -6 -17 -46 -15 -62|]

EDOs: 72, 137, 209, 281f, 490bcf

Badness: 0.0269

Marvolo

Commas: 225/224, 156250000/155649627

POTE generator: ~21/20 = 83.348

Map: [<1 2 1 1|, <0 -6 19 26|]

Wedgie: <<6 -19 -26 -44 -58 -7||

EDOs: 29, 43, 72, 619bcd, 691bcd

Badness: 0.0833

11-limit

Commas: 225/224, 441/440, 4000/3993

POTE generator: ~21/20 = 83.340

Map: [<1 2 1 1 2|, <0 -6 19 26 21|]

EDOs: 29, 43, 72

Badness: 0.0290

13-limit

Commas: 169/168, 225/224, 364/363, 441/440

POTE generator: ~21/20 = 83.330

Map: [<1 2 1 1 2 3|, <0 -6 19 26 21 10|]

EDOs: 29, 43, 72, 115f

Badness: 0.0215

Amavil

Commas: 225/224, 17496/16807

POTE generator: ~48/35 = 529.979

Map: [<1 6 1 9|, <0 -10 3 -14|]

Wedgie: <<10 -3 14 -28 -6 41||

EDOs: 9, 34d, 43, 77d

Badness: 0.1096

11-limit

Commas: 99/98, 176/175, 864/847

POTE generator: ~15/11 = 529.974

Map: [<1 6 1 9 7|, <0 -10 3 -14 -8|]

EDOs: 9, 34d, 43, 77de

Badness: 0.0426

13-limit

Commas: 78/77, 99/98, 144/143, 176/175

POTE generator: ~15/11 = 529.951

Map: [<1 6 1 9 7 9|, <0 -10 3 -14 -8 -12|]

EDOs: 9, 34d, 43, 77de

Badness: 0.0258

Enneaportent

Commas: 225/224, 40353607/40310784

POTE generator: ~5/4 = 383.165

Map: [<9 0 28 11|, <0 2 -1 2|]

Wedgie: <<18 -9 18 -56 -22 67||

EDOs: 9, 54, 63, 72, 495bcd

Badness: 0.0937

11-limit

Commas: 225/224, 385/384, 12005/11979

POTE generator: ~5/4 = 383.146

Map: [<9 0 28 11 24|, <0 2 -1 2 1|]

EDOs: 9, 54, 63, 72, 423cd, 495bcd

Badness: 0.0304

13-limit

Commas: 169/168, 225/224, 364/363, 1716/1715

POTE generator: ~5/4 = 383.047

Map: [<9 0 28 11 24 19|, <0 2 -1 2 1 2|]

EDOs: 9, 54, 63, 72, 279cf

Badness: 0.0223

Submajor

5-limit

Comma: 69198046875/68719476736

POTE generator: ~10125/8192 = 362.321

Map: [<1 4 -1|, <0 -8 11|]

EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc

Badness: 0.1302

7-limit

Commas: 225/224, 51200/50421

POTE generator: ~49/40 = 362.255

Map: [<1 4 -1 1|, <0 -8 11 6|]

Wedgie: <<8 -11 -6 -36 -32 17||

EDOs: 10, 33, 43, 53

Badness: 0.0605

11-limit

Commas: 225/224, 385/384, 6655/6561

POTE generator: ~27/22 = 362.101

Map: [<1 4 -1 1 11|, <0 -8 11 6 -25|]

EDOs: 10, 53, 116, 169de, 285cde

Badness: 0.0506

13-limit

Commas: 169/168, 225/224, 275/273, 385/384

POTE generator: ~16/13 = 362.105

Map: [<1 4 -1 1 11 4|, <0 -8 11 6 -25 -1|]

EDOs: 10, 53, 116, 169de, 285cdef

Badness: 0.0277

Interpental

Commas: 99/98, 176/175, 51200/50421

POTE generator: ~49/40 = 362.418

Map: [<1 4 -1 1 -5|, <0 -8 11 6 28|]

EDOs: 43, 53, 96, 149d

Badness: 0.0518

13-limit

Commas: 99/98, 169/168, 176/175, 640/637

POTE generator: ~16/13 = 362.402

Map: [<1 4 -1 1 -5 4|, <0 -8 11 6 28 -1|]

EDOs: 43, 53, 96, 149d

Badness: 0.0297

Alphorn

Commas: 225/224, 5764801/5668704

POTE generator: ~48/35 = 556.221

Map: [<1 9 0 13|, <0 -16 5 -22|]

Wedgie: <<16 -5 22 -45 -10 65||

EDOs: 28d, 41, 151cd, 192cd, 233cd

Badness: 0.1293

11-limit

Commas: 225/224, 385/384, 12250/11979

POTE generator: ~11/8 = 556.144

Map: [<1 9 0 13 3|, <0 -16 5 -22 1|]

EDOs: 41, 315cde

Badness: 0.0735

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