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.

[[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 ==

Subgroup: 2.11/5.13/5

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

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

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

=== Tridec ===

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Mapping generators: ~2, ~13/10

==== Intervals ====

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! Size in POTE tuning

! Note name on Q

! class="unsortable"| Approximate ratios

! class="unsortable" | Approximate ratios

! #Gens up

|-

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Mapping generators: ~2, ~13/10

==== Intervals ====

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! Size in POTE tuning

! Note name on Q

! class="unsortable"| Approximate ratios

! class="unsortable" | Approximate ratios

! #Gens up

|-

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Mapping generators: ~2, ~21/16

=== Intervals ===

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! Size in 31edo

! 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

|-

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| +5

|}

<~~references~~/>

<nowiki />* The ratio interpretations that are not valid for 18edo are italicized.

== Buzzard ==

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Mapping generators: ~2, ~21/16

~~Wedgie: <<4 21 -3 24 -16 -66||~~

~~[[Val]]s:~~ {{~~Vals~~| 48, 53, 111, 164d, 275d}}

Badness: 0.0480

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! Size in POTE tuning

! Note name on Q

! class="unsortable"| Approximate ratios

! class="unsortable" | Approximate ratios

! #Gens up

|-

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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 &amp; 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 &amp; 21}}) and Tridec ({{nowrap|21 &amp; 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 ==
 Subgroup: 2.11/5.13/5
@@ Line 17: / Line 19: @@
 Mapping: [&lt;1 0 1|, &lt;0 3 1|]
-EDOs: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c
+EDOs: {{EDOs|21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c}}
 === Tridec ===
@@ Line 34: / Line 36: @@
 Mapping generators: ~2, ~13/10
-{{Vals|legend=1| 21, 29, 37 }}
+{{Optimal ET sequence|legend=1| 21, 29, 37 }}
 ==== Intervals ====
@@ Line 47: / Line 49: @@
 ! Size in POTE tuning
 ! Note name on Q
-! class="unsortable"| Approximate ratios
+! class="unsortable" | Approximate ratios
 ! #Gens up
 |-
@@ Line 138: / Line 140: @@
 Mapping generators: ~2, ~13/10
-{{Vals|legend=1| 13, 21, 34 }}
+{{Optimal ET sequence|legend=1| 13, 21, 34 }}
 ==== Intervals ====
@@ Line 150: / Line 152: @@
 ! Size in POTE tuning
 ! Note name on Q
-! class="unsortable"| Approximate ratios
+! class="unsortable" | Approximate ratios
 ! #Gens up
 |-
@@ Line 243: / Line 245: @@
 Mapping generators: ~2, ~21/16
-{{Vals|legend=1| 13, 18, 31, 44 }}
+{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
 === Intervals ===
@@ Line 255: / Line 257: @@
 ! Size in 31edo
 ! 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
 |-
@@ Line 322: / Line 324: @@
 | +5
 |}
-<references/>
+<nowiki />* The ratio interpretations that are not valid for 18edo are italicized.
 == Buzzard ==
@@ Line 339: / Line 341: @@
 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
@@ Line 356: / Line 356: @@
 ! Size in POTE tuning
 ! Note name on Q
-! class="unsortable"| Approximate ratios
+! class="unsortable" | Approximate ratios
 ! #Gens up
 |-