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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.
== Petrtri ==
== Petrtri ==
Subgroup: 2.9/5.11/5.13/5
Subgroup: 2.11/5.13/5


Period: 1\1
Comma: 2200/2197


Optimal ([[Lp tuning|POL2]]) generator: 458.0950
Svalname: 3&5


EDO generators: [[13edo|5\13]], [[21edo|8\21]], [[34edo|13\34]]
[[Tp_tuning|POT2 generator]]: ~13/10 = 455.012


[[Comma]] list: 100/99, 144/143
[[Gencom|Gencom]]: [2 13/10; 2200/2197]


[[Mapping]] (for 2, 9/5, 11/5, 13/5): [⟨1 2 0 1], ⟨0 -3 3 1]]
Gencom mapping: [<1 0 -1/3 0 -1/3 2/3|, <0 0 -4/3 0 5/3 -1/3|]


Mapping generators: ~2, ~13/10
Mapping: [<1 0 1|, <0 3 1|]


{{Vals|legend=1| 13, 21, 34 }}
EDOs: {{EDOs|21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c}}
 
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
! Degree
! Size in 13edo
! Size in 21edo
! Size in 34edo
! Size in POTE tuning
! Note name on Q
! class="unsortable"| 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
| +6
|-
| 4
| 5\13, 461.54
| 8\21, 457.14
| 13\34, 458.82
| 459.15
| L
| 13/10
| +1
|-
| 5
| 7\13, 646.15
| 11\21, 628.57
| 18\34, 635.294
| 636.60
| M
| 13/9
| +4
|-
| 6
| 9\13, 830.77
| 14\21, 800.00
| 23\34, 811.77
| 814.05
| N
|
| +7
|-
| 7
| 10\13, 923.08
| 16\21, 914.29
| 26\34, 917.65
| 918.30
| O
| 22/13
| +2
|-
| 8
| 12\13, 1107.69
| 19\21, 1085.71
| 31\34, 1094.12
| 1095.75
| P
|
| +5
|}


=== Tridec ===
=== Tridec ===
Line 116: Line 32:
[[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 ====
Line 133: Line 49:
! 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
|-
|-
Line 214: Line 130:
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]]
Line 220: Line 136:
[[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 ====
Line 236: Line 152:
! 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
|-
|-
Line 317: Line 233:
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]]
Line 325: Line 241:
[[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 ===
Line 341: Line 257:
! 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
|-
|-
Line 408: Line 324:
| +5
| +5
|}
|}
<references/>
<nowiki />* The ratio interpretations that are not valid for 18edo are italicized.


== Buzzard ==
== Buzzard ==
Line 421: Line 337:
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
Line 442: Line 356:
! 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
|-
|-
Line 518: Line 432:
|}
|}


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