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| A-Team is a 2.9.21 temperament generated by a tempered 21/16 with a size ranging from 5\13 (461.54¢) to 7\18 (466.67¢), or to about 470.78¢ (the pure value for 21/16) if you don't care about tempering out 81/80 or having quartertone-sized steps. Three 21/16's are equated to one 9/8, which means that the [[color notation|latrizo]] comma (1029/1024) is tempered out. Hence, any EDO that equates three 8/7's with one 3/2 will support A-Team with its 21/16. | | #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. The generator generates 5-, 8-, and 13-note MOSes, most notably the 8-note "oneirotonic" MOS; see also [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale]].
| | [[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 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 H-C-D-F (in Cryptic Ruse's notation) represents both 8:10:11:13 and 13:16:18:21 in 13edo. 31edo gives the [[optimal patent val]] for the 2.9.21.5 subgroup interpretation. 44edo is similar to 31edo but better approximates primes 11, 13, 17, 19 and 23 with the generator chain.
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| Its name is a pun on the 18 notes in its proper scale, which is a [[13L_5s|13L 5s]] MOS.
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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).
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| #Using the [[pergen]] (P8, M9/3). Though the tuning lacks perfect fifths, three of the 21/16 generator 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 having fifths.
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| ==A-team tuning spectrum==
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| ==="Meantone" tunings: the 13edo-to-31edo range===
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| Occupies the flat end of the spectrum, from 461.54 to 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.
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| The most plentiful consonant triad in A-Team scales is 4:9:21 or 8:18:21, followed by 13:17:19 and 4:5:9. 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:(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 can give good (though irregular) mappings of 3/2 and 7/4 in the "better" tunings.
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| {| class="wikitable"
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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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| | 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;" | 463.5
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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;" | 736.5
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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;" | 927.0
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| | style="text-align:center;" | 12/7
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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;" | 273.0
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| | style="text-align:center;" | 7/6
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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;" | 190.5
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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;" | 1009.5
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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;" | 654.0
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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;" | 546.0
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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;" | 1117.5
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| | style="text-align:center;" | 40/21, 21/11
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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;" | 82.5
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| | style="text-align:center;" | 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;" | 381.0
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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;" | 819.0
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| | style="text-align:center;" | 8/5
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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;" | 844.5
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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;" | 355.5
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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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| | style="text-align:center;" | 8
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| | style="text-align:right;" | 108.0
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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;" | 1092.0
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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;" | 571.5
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| | style="text-align:center;" | 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;" | 628.5
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| | style="text-align:center;" | '''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;" | 1035.0
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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;" | 165.0
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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;" | 298.5
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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;" | 901.5
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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;" | 762.0
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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;" | 438.0
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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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| | style="text-align:center;" | 13
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| | style="text-align:right;" | 25.5
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| | style="text-align:center;" | (***)
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| | style="text-align:center;" | AA8 - octave
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| | style="text-align:center;" | -13
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| | style="text-align:right;" | 1174.5
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| | style="text-align:center;" |
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| | style="text-align:center;" | dd2 + octave
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| |}
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| (*) using the 2.9.21.5.11.13 POTE generator; cf. the 463.64¢ generator in 44edo
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| (**) 2.9.21.5.11.13.17.19.23 interpretations; harmonics are in bold
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| (***) 65/64 and other commas only tempered out by 13edo
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| [[18edo]] (466.67 cents) is an edge case, as it tempers out 81/80 but fails to approximate more diverse intervals with the same identifications used by 13edo, 44edo or 23edo. 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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| '''tl;dr:''' 44edo good
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| ==="Superpythagorean" tunings===
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| In general sharper subfourths are 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 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 superpyth[7].
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| [[23edo]] (469.57 cents) and [[41edo]] (468.29 cents): two tones represent 14/11 rather than 5/4, and J-O# (4 large steps + 1 small step) becomes a 5/3 rather than a 13/8.
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| Using a pure 21/16 of 470.78¢ results in an extremely lopsided oneirotonic scale with L/s = 4.60 close to 5edo. Harmonically this results in a 9/8 of 212.342722 which is very much in the superpyth range (for comparison, 17edo's 9/8 is 211.765 cents). 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).
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