Revision as of 18:09, 25 February 2021

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.

Comma list: 16875/16384

POTE generator: ~15/14 = 125.7549

Mapping: [⟨1 2 2], ⟨0 -4 3]]

Wedgie: ⟨⟨4 -3 -14]]

@@ Line 521: / Line 521: @@
 = Amavil =
 == 5-limit (mabila) ==
-Comma: 268435456/263671875
+Comma list: 268435456/263671875
 POTE generator: ~512/375 = 529.6849
-Map: [<1 6 1|, <0 -10 3|]
+Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
-EDOs: {{EDOs| 9, 25, 34, 77, 111, 145, 256c }}
+{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
 Badness: 0.2325
 == 7-limit ==
-Commas: 225/224, 17496/16807
+Comma list: 225/224, 17496/16807
 POTE generator: ~48/35 = 529.979
-Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
+Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
-Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
+Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
-EDOs: {{EDOs|9, 34d, 43, 77d}}
+{{Val list|legend=1| 9, 34d, 43, 77d }}
 Badness: 0.1096
 == 11-limit ==
-Commas: 99/98, 176/175, 864/847
+Comma list: 99/98, 176/175, 864/847
 POTE generator: ~15/11 = 529.974
-Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
+Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
-EDOs: {{EDOs| 9, 34d, 43, 77de }}
+{{Val list|legend=1| 9, 34d, 43, 77de }}
 Badness: 0.0426
 == 13-limit ==
-Commas: 78/77, 99/98, 144/143, 176/175
+Comma list: 78/77, 99/98, 144/143, 176/175
 POTE generator: ~15/11 = 529.951
-Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
+Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
-EDOs: {{EDOs| 9, 34d, 43, 77de }}
+{{Val list|legend=1| 9, 34d, 43, 77de }}
 Badness: 0.0258
-=Enneaportent=
+= Enneaportent =
-Commas: 225/224, 40353607/40310784
+Comma list: 225/224, 40353607/40310784
 POTE generator: ~5/4 = 383.165
-Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
+Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
-Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
+Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
-EDOs: {{EDOs|9, 54, 63, 72, 495bcd}}
+{{Val list|legend=1| 9, 54, 63, 72, 495bcd }}
 Badness: 0.0937
-==11-limit==
+== 11-limit ==
-Commas: 225/224, 385/384, 12005/11979
+Comma list: 225/224, 385/384, 12005/11979
 POTE generator: ~5/4 = 383.146
-Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
+Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
-EDOs: {{EDOs|9, 54, 63, 72, 423cd, 495bcd}}
+{{Val list|legend=1| 9, 54, 63, 72, 423cd, 495bcd }}
 Badness: 0.0304
-==13-limit==
+== 13-limit ==
-Commas: 169/168, 225/224, 364/363, 1716/1715
+Comma list: 169/168, 225/224, 364/363, 1716/1715
 POTE generator: ~5/4 = 383.047
-Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
+Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
-EDOs: {{EDOs|9, 54, 63, 72, 279cf}}
+{{Val list|legend=1| 9, 54, 63, 72, 279cf }}
 Badness: 0.0223
@@ Line 603: / Line 603: @@
 = Submajor =
 == 5-limit ==
-Comma: 69198046875/68719476736
+Comma list: 69198046875/68719476736
 POTE generator: ~10125/8192 = 362.321
-Map: [&lt;1 4 -1|, &lt;0 -8 11|]
+Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
-EDOs: {{EDOs|10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
+{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
 Badness: 0.1302
 == 7-limit ==
-Commas: 225/224, 51200/50421
+Comma list: 225/224, 51200/50421
 POTE generator: ~49/40 = 362.255
-Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
+Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
-Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
+Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
-EDOs: {{EDOs|10, 33, 43, 53}}
+{{Val list|legend=1| 10, 33, 43, 53 }}
 Badness: 0.0605
 == 11-limit ==
-Commas: 225/224, 385/384, 6655/6561
+Comma list: 225/224, 385/384, 6655/6561
 POTE generator: ~27/22 = 362.101
-Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
+Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
-EDOs: {{EDOs|10, 53, 116, 169de, 285cde}}
+{{Val list|legend=1| 10, 53, 116, 169de, 285cde }}
 Badness: 0.0506
 === 13-limit ===
-Commas: 169/168, 225/224, 275/273, 385/384
+Comma list: 169/168, 225/224, 275/273, 385/384
 POTE generator: ~16/13 = 362.105
-Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
+Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
-EDOs: {{EDOs|10, 53, 116, 169de, 285cdef}}
+{{Val list|legend=1| 10, 53, 116, 169de, 285cdef }}
 Badness: 0.0277
 == Interpental ==
-Commas: 99/98, 176/175, 51200/50421
+Comma list: 99/98, 176/175, 51200/50421
 POTE generator: ~49/40 = 362.418
-Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
+Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
-EDOs: {{EDOs|43, 53, 96, 149d}}
+{{Val list|legend=1| 43, 53, 96, 149d }}
 Badness: 0.0518
 === 13-limit ===
-Commas: 99/98, 169/168, 176/175, 640/637
+Comma list: 99/98, 169/168, 176/175, 640/637
 POTE generator: ~16/13 = 362.402
-Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
+Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
-EDOs: {{EDOs|43, 53, 96, 149d}}
+{{Val list|legend=1| 43, 53, 96, 149d }}
 Badness: 0.0297
-=Alphorn=
+= Alphorn =
-Commas: 225/224, 5764801/5668704
+Comma list: 225/224, 5764801/5668704
 POTE generator: ~48/35 = 556.221
-Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
+Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
-Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
+Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
-EDOs: 28d, 41, 151cd, 192cd, 233cd
+{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
 Badness: 0.1293
-==11-limit==
+== 11-limit ==
-Commas: 225/224, 385/384, 12250/11979
+Comma list: 225/224, 385/384, 12250/11979
 POTE generator: ~11/8 = 556.144
-Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
+Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
-EDOs: 41, 315cde
+{{Val list|legend=1| 41, 315cde }}
 Badness: 0.0735

Marvel temperaments: Difference between revisions

Revision as of 18:09, 25 February 2021

Negri

5-limit

7-limit

11-limit

13-limit

Negril

13-limit

Negric

13-limit

Negroni

13-limit

Passive

Wizard

5-limit

7-limit

11-limit

Lizard

17-limit

19-limit

Gizzard

17-limit

19-limit

Mage

Triton

11-limit

Tritonic

5-limit

7-limit

11-limit

13-limit

Tritoni

Merman

7-limit

11-limit

13-limit

Septimin

7-limit

11-limit

13-limit

Slender

11-limit

13-limit

Marvo

11-limit

13-limit

Marvolo

11-limit

13-limit

Amavil

5-limit (mabila)

7-limit

11-limit

13-limit

Enneaportent

11-limit

13-limit

Submajor

5-limit

7-limit

11-limit

13-limit

Interpental

13-limit

Alphorn

11-limit

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