5L 3s/Temperaments: Difference between revisions

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[[Oneirotonic]] temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step approximates 9/8 (a very good 9/8 in 12edo), but superpyth has 9/8 ~ 8/7, and meantone has 9/8 ~ 10/9. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it. In A-Team (13&18), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri (13&21) and Tridec (21&29), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R-M2-M3-M5, M5 = major oneirofifth = minor fifth in 13edo) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as the oneirotonic analogue of [[flattone]]:  it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.
{{breadcrumb}}
[[Oneirotonic]] temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step approximates 9/8 (a very good 9/8 in 12edo), but [[superpyth]] has {{nowrap|9/8 ~ 8/7}}, and meantone has {{nowrap|9/8 ~ 10/9}}. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it. In A-Team ({{nowrap|13 & 18}}), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri ({{nowrap|13 & 21}}) and Tridec ({{nowrap|21 & 29}}), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R–M2–M3–M5, {{nowrap|M5 {{=}} major oneirofifth}} {{nowrap|{{=}} minor fifth in 13edo}}) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as the oneirotonic analogue of [[flattone]]—it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.


Vulture/[[Hemifamity_temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only [[harmonic entropy]] minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.
Vulture/[[Hemifamity temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only [[harmonic entropy]] minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.
== Tridec ==
 
== Petrtri ==
Subgroup: 2.11/5.13/5
 
Comma: 2200/2197
 
Svalname: 3&5
 
[[Tp_tuning|POT2 generator]]: ~13/10 = 455.012
 
[[Gencom|Gencom]]: [2 13/10; 2200/2197]
 
Gencom mapping: [<1 0 -1/3 0 -1/3 2/3|, <0 0 -4/3 0 5/3 -1/3|]
 
Mapping: [<1 0 1|, <0 3 1|]
 
EDOs: {{EDOs|21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c}}
 
=== Tridec ===
Subgroup: 2.3.7/5.11/5.13/5
Subgroup: 2.3.7/5.11/5.13/5


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[[Comma]] list: 196/195, 847/845, 1001/1000
[[Comma]] list: 196/195, 847/845, 1001/1000


[[Mapping]] (for 2, 3, 7/5, 11/5, 13/5): [{{val|1 5 2 0 1}}, {{val|0 -9 -4 3 1}}]
[[Mapping]]: [{{val|2: 1, 3: 5, 7/5: 2, 11/5: 0, 13/5: 1}}, {{val|2: 0, 3: -9, 7/5: -4, 11/5: 3, 13/5: 1}}]


Mapping generators: ~2, ~13/10
Mapping generators: ~2, ~13/10


{{Vals|legend=1| 21, 29, 37 }}
{{Optimal ET sequence|legend=1| 21, 29, 37 }}


=== Intervals ===
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Tridec interpretations:
Sortable table of intervals in the Dylathian mode and their Tridec interpretations:


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! Size in POTE tuning
! Size in POTE tuning
! Note name on Q
! Note name on Q
! class="unsortable"| Approximate ratios
! class="unsortable" | Approximate ratios
! #Gens up
! #Gens up
|-
|-
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|}
|}


== Petrtri ==
=== Petrtri extension ===
Subgroup: 2.5.9.11.13.17
Subgroup: 2.5.9.11.13.17


Period: 1\1
Period: 1\1


Optimal ([[POTE]]) generator: 459.1502
Optimal ([[Lp tuning|POL2]]) generator: 459.1502


EDO generators: [[13edo|5\13]], [[21edo|8\21]], [[34edo|13\34]]
EDO generators: [[13edo|5\13]], [[21edo|8\21]], [[34edo|13\34]]
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[[Comma]] list: 100/99, 144/143, 170/169, 221/220
[[Comma]] list: 100/99, 144/143, 170/169, 221/220


[[Mapping]] (for 2, 5, 9, 11, 13, 17): [{{val|1 5 7 5 6 6}}, {{val|0 -7 -10 -4 -6 -5}}]
[[Mapping]] (for 2, 5, 9, 11, 13, 17): [{{val|2: 1, 5: 5, 9: 7, 11: 5, 13: 6, 17: 6}}, {{val|2: 0, 5: -7, 9: -10, 11: -4, 13: -6, 17: -5}}]


Mapping generators: ~2, ~13/10
Mapping generators: ~2, ~13/10


{{Vals|legend=1| 13, 21, 34 }}
{{Optimal ET sequence|legend=1| 13, 21, 34 }}


=== Intervals ===
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
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! Size in POTE tuning
! Size in POTE tuning
! Note name on Q
! Note name on Q
! class="unsortable"| Approximate ratios
! class="unsortable" | Approximate ratios
! #Gens up
! #Gens up
|-
|-
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Period: 1\1
Period: 1\1


Optimal ([[POTE]]) generator: 464.3865
Optimal ([[Lp tuning|POL2]]) generator: 464.3865


EDO generators: [[13edo|5\13]], [[18edo|7\18]], [[31edo|12\31]], [[44edo|17\44]]
EDO generators: [[13edo|5\13]], [[18edo|7\18]], [[31edo|12\31]], [[44edo|17\44]]
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[[Comma]] list: 81/80, 1029/1024
[[Comma]] list: 81/80, 1029/1024


[[Mapping]] (for 2, 5, 9, 21): [{{val|1 0 2 4}}, {{val|0 6 3 1}}]
[[Mapping]]: [{{val|2: 1, 5: 0, 9: 2, 21: 4}}, {{val|2: 0, 5: 6, 9: 3, 21: 1}}]


Mapping generators: ~2, ~21/16
Mapping generators: ~2, ~21/16


{{Vals|legend=1| 13, 18, 31, 44 }}
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}


=== Intervals ===
=== Intervals ===
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! Size in 31edo
! Size in 31edo
! Note name on Q
! Note name on Q
! class="unsortable"| Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
! class="unsortable" | Approximate ratios*
! #Gens up
! #Gens up
|-
|-
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| +5
| +5
|}
|}
<references/>
<nowiki />* The ratio interpretations that are not valid for 18edo are italicized.


== Buzzard ==
== Buzzard ==
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Commas: 1728/1715, 5120/5103
Commas: 1728/1715, 5120/5103


Map: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]
Mapping: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]


Mapping generators: ~2, ~21/16  
Mapping generators: ~2, ~21/16  


Wedgie: &lt;&lt;4 21 -3 24 -16 -66||
{{Optimal ET sequence|legend=1| 48, 53, 111, 164d, 275d}}
 
[[Val]]s: {{Vals| 48, 53, 111, 164d, 275d}}


Badness: 0.0480
Badness: 0.0480
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! Size in POTE tuning
! Size in POTE tuning
! Note name on Q
! Note name on Q
! class="unsortable"| Approximate ratios
! class="unsortable" | Approximate ratios
! #Gens up
! #Gens up
|-
|-
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|}
|}


[[Category:Theory]]
[[Category:Lists of temperaments]]
[[Category:Temperament]]
[[Category:Oneirotonic|T]]
[[Category:Oneirotonic|T]]

Latest revision as of 04:16, 12 June 2025

< 5L 3s

Oneirotonic temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step approximates 9/8 (a very good 9/8 in 12edo), but superpyth has 9/8 ~ 8/7, and meantone has 9/8 ~ 10/9. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it. In A-Team (13 & 18), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri (13 & 21) and Tridec (21 & 29), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R–M2–M3–M5, M5 = major oneirofifth = minor fifth in 13edo) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as the oneirotonic analogue of flattone—it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.

Vulture/Buzzard, in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only harmonic entropy minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.

Petrtri

Subgroup: 2.11/5.13/5

Comma: 2200/2197

Svalname: 3&5

POT2 generator: ~13/10 = 455.012

Gencom: [2 13/10; 2200/2197]

Gencom mapping: [<1 0 -1/3 0 -1/3 2/3|, <0 0 -4/3 0 5/3 -1/3|]

Mapping: [<1 0 1|, <0 3 1|]

EDOs: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c

Tridec

Subgroup: 2.3.7/5.11/5.13/5

Period: 1\1

Optimal (POTE) generator: 455.2178

EDO generators: 8\21, 11\29, 14\37

Comma list: 196/195, 847/845, 1001/1000

Mapping: [⟨2: 1, 3: 5, 7/5: 2, 11/5: 0, 13/5: 1], ⟨2: 0, 3: -9, 7/5: -4, 11/5: 3, 13/5: 1]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 21, 29, 37

Intervals

Sortable table of intervals in the Dylathian mode and their Tridec interpretations:

Degree Size in 21edo Size in 29edo Size in 37edo Size in POTE tuning Note name on Q Approximate ratios #Gens up
1 0\21, 0.00 0\29, 0.00 0\37, 0.00 0.00 Q 1/1 0
2 3\21, 171.43 4\29, 165.52 5\37, 163.16 165.65 J 11/10, 10/9 +3
3 6\21, 342.86 8\29, 331.03 10\37, 324.32 331.31 K 11/9, 6/5 +6
4 8\21, 457.14 11\29, 455.17 14\37, 454.05 455.17 L 13/10, 9/7 +1
5 11\21, 628.57 15\29, 620.69 19\37, 616.22 620.87 M 13/9, 10/7 +4
6 14\21, 800.00 19\29, 786.21 23\37, 778.38 786.52 N 11/7 +7
7 16\21, 914.29 22\29, 910.34 28\37, 908.11 910.44 O 22/13 +2
8 19\21, 1085.71 26\29, 1075.86 33\37, 1070.27 1076.09 P 13/7, 28/15 +5

Petrtri extension

Subgroup: 2.5.9.11.13.17

Period: 1\1

Optimal (POL2) generator: 459.1502

EDO generators: 5\13, 8\21, 13\34

Comma list: 100/99, 144/143, 170/169, 221/220

Mapping (for 2, 5, 9, 11, 13, 17): [⟨2: 1, 5: 5, 9: 7, 11: 5, 13: 6, 17: 6], ⟨2: 0, 5: -7, 9: -10, 11: -4, 13: -6, 17: -5]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 13, 21, 34

Intervals

Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:

Degree Size in 13edo Size in 21edo Size in 34edo Size in POTE tuning Note name on Q Approximate ratios #Gens up
1 0\13, 0.00 0\21, 0.00 0\34, 0.00 0.00 Q 1/1 0
2 2\13, 184.62 3\21, 171.43 5\34, 176.47 177.45 J 10/9, 11/10 +3
3 4\13, 369.23 6\21, 342.86 10\34, 352.94 354.90 K 11/9, 16/13 +6
4 5\13, 461.54 8\21, 457.14 13\34, 458.82 459.15 L 13/10, 17/13, 22/17 +1
5 7\13, 646.15 11\21, 628.57 18\34, 635.294 636.60 M 13/9, 16/11, 23/16 (esp. 21edo) +4
6 9\13, 830.77 14\21, 800.00 23\34, 811.77 814.05 N 8/5 +7
7 10\13, 923.08 16\21, 914.29 26\34, 917.65 918.30 O 17/10 +2
8 12\13, 1107.69 19\21, 1085.71 31\34, 1094.12 1095.75 P 17/9, 32/17, 15/8 +5

A-Team

Subgroup: 2.5.9.21

Period: 1\1

Optimal (POL2) generator: 464.3865

EDO generators: 5\13, 7\18, 12\31, 17\44

Lookalike temperament: Dual-3 A-Team

Comma list: 81/80, 1029/1024

Mapping: [⟨2: 1, 5: 0, 9: 2, 21: 4], ⟨2: 0, 5: 6, 9: 3, 21: 1]]

Mapping generators: ~2, ~21/16

Optimal ET sequence: 13, 18, 31, 44

Intervals

Sortable table of intervals in the Dylathian mode and their A-Team interpretations:

Degree Size in 13edo Size in 18edo Size in 31edo Note name on Q Approximate ratios* #Gens up
1 0\13, 0.00 0\18, 0.00 0\31, 0.00 Q 1/1 0
2 2\13, 184.62 3\18, 200.00 5\31, 193.55 J 9/8, 10/9 +3
3 4\13, 369.23 6\18, 400.00 10\31, 387.10 K 5/4 +6
4 5\13, 461.54 7\18, 466.67 12\31, 464.52 L 21/16, 13/10 +1
5 7\13, 646.15 10\18, 666.66 17\31, 658.06 M 13/9, 16/11 +4
6 9\13, 830.77 13\18, 866.66 22\31, 851.61 N 13/8, 18/11 +7
7 10\13, 923.08 14\18, 933.33 24\31, 929.03 O 12/7 +2
8 12\13, 1107.69 17\18, 1133.33 29\31, 1122.58 P +5

* The ratio interpretations that are not valid for 18edo are italicized.

Buzzard

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: ~21/16 = 475.636

EDO generators: 15\38, 17\43, 19\48, 21\53, 23\58, 25\63

Commas: 1728/1715, 5120/5103

Mapping: [<1 0 -6 4|, <0 4 21 -3|]

Mapping generators: ~2, ~21/16

Optimal ET sequence: 48, 53, 111, 164d, 275d

Badness: 0.0480

Intervals

Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:

Degree Size in 38edo Size in 53edo Size in 63edo Size in POTE tuning Note name on Q Approximate ratios #Gens up
1 0\38, 0.00 0\53, 0.00 0\63, 0.00 0.00 Q 1/1 0
2 7\38, 221.05 10\53, 226.42 12\63, 228.57 227.07 J 8/7 +3
3 14\38, 442.10 20\53, 452.83 24\63, 457.14 453.81 K 13/10, 9/7 +6
4 15\38, 473.68 21\53, 475.47 25\63, 476.19 475.63 L 21/16 +1
5 22\38, 694.73 31\53, 701.89 37\63, 704.76 702.54 M 3/2 +4
6 29\38, 915.78 41\53, 928.30 49\63, 933.33 929.45 N 12/7, 22/13 +7
7 30\38, 947.36 42\53, 950.94 50\63, 952.38 951.27 O 26/15 +2
8 37\38, 1168.42 52\53, 1177.36 62\63, 1180.95 1178.18 P 108/55, 160/81 +5
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