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| '''A-Team''' is a 2.9.21 temperament generated by a tempered zo fourth ([[21/16]]) with a size ranging from 461.54¢ (5\13) to 470.78¢ (the pure value for 21/16). It can be viewed as every other note of 2.3.7 latrizo or "[[slendric]]" temperament. Any EDO that has an interval within the range 461.54¢ to 470.78¢ will support A-Team. Three 21/16's are equated to one 9/8, which means that the [[color notation|latrizo]] comma (1029/1024) is tempered out. Its name is a pun on the 18 notes in its proper scale, which is a [[13L_5s|13L 5s]] MOS.
| | #redirect [[Subgroup temperaments #A-team]] |
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| It's natural to consider A-Team a 2.9.21.5 temperament by also equating two 9/8's with one 5/4, tempering out 81/80. Generators optimized for tempering out 81/80 also tend to generate the melodically best scales. This temperament generates [[3L 2s]], [[5L 3s]], and [[5L 8s]] MOSes, most notably the 8-note "oneirotonic" MOS; see also [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale]]. Any EDO with an interval between 461.54¢ and 466.67¢ can be reasonably said to support 2.9.21.5 A-Team.
| | [[Category:Rank-2 temperaments]] |
| | | [[Category:Subgroup temperaments]] |
| [[13edo]], [[18edo]], [[31edo]], and [[44edo]] (with generators 5\13, 7\18, 12\31, and 17\44 respectively) all support 2.9.21.5 A-Team with their closest approximations to 9/8 and 21/16. 13edo, while representing the 2.9.21.5 subgroup less accurately, gives some chords extra interpretations; for example, the chord P1-d4-m5-m7 (O#-J-K-M in Kentaku notation) represents both 8:10:11:13 and 13:16:18:21 in 13edo. 31edo gives the [[optimal patent val]] for 2.9.21.5 A-Team and tunes the 13:17:19 chord to within 1.1 cents. 44edo is similar to 31edo but better approximates 11, 13, 17, and 19 as harmonics with the generator chain, and additionally provides the 23th harmonic.
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| ==Notation==
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| There are several ways to notate A-Team in a JI-agnostic way:
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| #The octatonic notation described by Cryptic Ruse (Dylathian = CDEFGHABC with C = 261.62 Hz) and Kentaku (Ilarnekian = JKLMNOPQJ with J ≈ 180 Hz), based on the oneirotonic 5L 3s scale.
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| #Using the [[pergen]] (P8, M9/3). Though the tuning lacks perfect fifths, three 21/16 generators are equal to twice a perfect fifth (i.e. a conventional major ninth).
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| #As every other note of the third-fifths pergen (P8, P5/3). This is backwards compatible with a notation that has fifths.
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| ==A-Team tuning spectrum==
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| ==="Meantone" A-Team tunings===
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| If you optimize the tuning for tempering out [[81/80]], the generator size ranges from 13edo's 461.54 cents to 18edo's 466.67 cents. This range offers the best ratio of large to small melodic steps too: ranging from 13edo's L:s = 2:1 to 18edo's L:s = 3:1; 18edo's oneirotonic is analogous to 17edo's diatonic scale in that L/s = 3, while 13edo is analogous to 12edo.
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| [[18edo]] (466.67 cents) is an edge case, as it tempers out 81/80 but fails to approximate other JI intervals like 13edo and 31edo do.
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| ===="19-limit A-Team": The 13edo-to-31edo range====
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| '''tl;dr:''' [[44edo]] good
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| The generator ccupies the flat end of the spectrum, from 13edo's 461.54 cents to 31edo's 464.52 cents.
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| Surprisingly, A-Team tunings in this range can approximate some high-limit JI chords, owing to the generator being close to the 17/13 ratio and three of them approximating both 9/8 and 10/9, thus approximating their [[mediant]] 19/17. The 11th harmonic makes an appearance too, because it is approximated by both 13edo and 31edo. Thus A-Team can be viewed as representing the no-3, no-7 19-odd limit. If you optimize for this 19 limit harmony you also pretty much get the 23rd harmonic for free.
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| The most plentiful consonant triad in A-Team scales is 4:9:21 or 8:18:21 (the voicing of the 21th harmonic is important for making it sound smooth), followed by 13:17:19 and 4:5:9.
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| A-Team[13] in the following tuning has five copies of 4:5:9:13:17:21, five copies of 5:9:13:17:19:21, three copies of 4:5:9:11, two copies of 4:5:9:11:13, two copies of 4:5:9:13:17:19:21, and one copy of 4:9:11:(15):21:23. The 13-note [[MODMOS]] given by flattening just the seventh and eighth degrees of the LsLssLsLssLss mode of A-Team[13] by the 13-generator interval (~25.5 cents) gives the entire 4:5:9:11:13:17:19:21:23 otonal chord over a single root.
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| Extending the chain beyond 13 notes gives good, though irregular, mappings of 3/2 (with -17 generators) and 7/4 (with -15 generators) in the "better" tunings.
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| =====Interval chain=====
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| {| class="wikitable"
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| |-
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| ! | Generators
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| ! | Cents (*)
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| ! | Ratios (**)
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| ! | Octatonic notation
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| ! | Generators
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| ! | 2/1 inverse (*)
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| ! | Ratios (**)
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| ! | Octatonic notation
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| |-
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| |colspan=8|The "diatonic" 8-note scale has the following intervals:
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| | style="text-align:center;" | 0
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| | style="text-align:right;" | 0
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| | style="text-align:center;" | 1/1
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| | style="text-align:center;" | P1
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| | style="text-align:center;" | 0
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| | style="text-align:right;" | 1200
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| | style="text-align:center;" | 2/1
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| | style="text-align:center;" | P9
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| |-
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| | style="text-align:center;" | 1
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| | style="text-align:right;" | 463.17
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| | style="text-align:center;" | '''21/16''', 13/10, 17/13
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| | style="text-align:center;" | P4
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| | style="text-align:center;" | -1
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| | style="text-align:right;" | 736.83
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| | style="text-align:center;" | 32/21, 20/13
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| | style="text-align:center;" | P6
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| |-
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| | style="text-align:center;" | 2
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| | style="text-align:right;" | 926.35
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| | style="text-align:center;" | 12/7, 17/10
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| | style="text-align:center;" | M7
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| | style="text-align:center;" | -2
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| | style="text-align:right;" | 273.66
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| | style="text-align:center;" | 7/6, 20/17
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| | style="text-align:center;" | m3
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| |-
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| | style="text-align:center;" | 3
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| | style="text-align:right;" | 189.52
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| | style="text-align:center;" | '''9/8''', 10/9, 19/17
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| | style="text-align:center;" | M2
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| | style="text-align:center;" | -3
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| | style="text-align:right;" | 1010.48
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| | style="text-align:center;" | 16/9, 9/5
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| | style="text-align:center;" | m8
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| |-
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| | style="text-align:center;" | 4
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| | style="text-align:right;" | 652.69
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| | style="text-align:center;" | 16/11, 13/9, 19/13
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| | style="text-align:center;" | M5
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| | style="text-align:center;" | -4
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| | style="text-align:right;" | 547.31
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| | style="text-align:center;" | '''11/8''', 18/13
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| | style="text-align:center;" | m5
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| |-
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| | style="text-align:center;" | 5
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| | style="text-align:right;" | 1115.86
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| | style="text-align:center;" | 40/21, 21/11, 19/10
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| | style="text-align:center;" | M8
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| | style="text-align:center;" | -5
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| | style="text-align:right;" | 84.14
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| | style="text-align:center;" | 20/19, 21/20, 22/21, 23/22
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| | style="text-align:center;" | m2
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| |-
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| | style="text-align:center;" | 6
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| | style="text-align:right;" | 379.04
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| | style="text-align:center;" | '''5/4'''
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| | style="text-align:center;" | M3
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| | style="text-align:center;" | -6
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| | style="text-align:right;" | 820.97
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| | style="text-align:center;" | 8/5, 21/13
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| | style="text-align:center;" | m7
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| |-
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| | style="text-align:center;" | 7
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| | style="text-align:right;" | 842.21
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| | style="text-align:center;" | 18/11, '''13/8'''
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| | style="text-align:center;" | A6
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| | style="text-align:center;" | -7
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| | style="text-align:right;" | 357.79
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| | style="text-align:center;" | 11/9, 16/13
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| | style="text-align:center;" | d4
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| |-
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| |colspan=8|The "chromatic" 13-note scale also has the following intervals:
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| |-
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| | style="text-align:center;" | 8
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| | style="text-align:right;" | 105.38
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| | style="text-align:center;" | '''17/16'''
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| | style="text-align:center;" | A1 (the chroma for oneirotonic)
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| | style="text-align:center;" | -8
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| | style="text-align:right;" | 1094.62
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| | style="text-align:center;" | close to '''15/8'''
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| | style="text-align:center;" | d9
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| |-
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| | style="text-align:center;" | 9
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| | style="text-align:right;" | 568.55
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| | style="text-align:center;" | close to 32/23
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| | style="text-align:center;" | A4
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| | style="text-align:center;" | -9
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| | style="text-align:right;" | 631.45
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| | style="text-align:center;" | close to '''23/16'''
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| | style="text-align:center;" | d6
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| |-
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| | style="text-align:center;" | 10
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| | style="text-align:right;" | 1031.73
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| | style="text-align:center;" | 20/11
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| | style="text-align:center;" | A7
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| | style="text-align:center;" | -10
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| | style="text-align:right;" | 168.27
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| | style="text-align:center;" | 11/10
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| | style="text-align:center;" | d3
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| |-
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| | style="text-align:center;" | 11
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| | style="text-align:right;" | 294.90
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| | style="text-align:center;" | 13/11, '''19/16'''
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| | style="text-align:center;" | A2
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| | style="text-align:center;" | -11
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| | style="text-align:right;" | 905.10
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| | style="text-align:center;" | 22/13
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| | style="text-align:center;" | d8
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| |-
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| | style="text-align:center;" | 12
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| | style="text-align:right;" | 758.07
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| | style="text-align:center;" | close to 14/9
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| | style="text-align:center;" | A5
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| | style="text-align:center;" | -12
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| | style="text-align:right;" | 441.93
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| | style="text-align:center;" | close to 9/7
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| | style="text-align:center;" | d5
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| |}
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| (*) using the 2.9.21.5.11.13.17.19 POTE generator; cf. the 44edo generator of 463.64¢ and the 2.9.21.5.11.13 POTE generator of 463.50¢
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| (**) 2.9.21.5.11.13.17.19 interpretations; harmonics are in bold
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| ==="Superpythagorean" tunings===
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| In general using a sharper 21/16 is better if you don't care about approximating 5/4 and only care about optimizing the 4:9:21 triad. Apart from that, there's little common JI interpretation shared by these sharper tunings. One possible tradeoff is that small steps in the oneirotonic scale get smaller than 1/3-tones (as in 18edo) and become quarter-tones (as in 23edo) and thus become less melodically distinct, much like 22edo's superpyth[7].
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| Using a pure 21/16 of 470.78¢ results in an extremely lopsided oneirotonic scale with L/s = 4.60. Harmonically this results in a 9/8 of 212.342 cents which is very much in the superpyth range (for comparison, [[17edo]]'s 9/8 is 211.765 cents). Instead of approximating 16/11, the "major tritone" in oneirotonic (J-N in Kentaku notation) will be a very flat fifth of 683.123 cents, constrasting with the very sharp 40/21 fifth (729.2 cents). The flat fifths give shimmery detuned versions of zo (subminor) triads 6:7:9 and sus2 triads 8:9:12. All these intervals contribute to the scale's overall gently shimmery quality which the 23edo version shares too.
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| [[Category:Temperament]] | |