A-team: Difference between revisions

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==="Meantone" A-Team tunings===

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.

====~~the~~ 13edo-to-31edo range====

===="19-limit A-Team": The 13edo-to-31edo range====

'''tl;dr:''' [[44edo]] good

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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 get the 23rd harmonic for free.

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. 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.

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.

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.

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.

@@ Line 14: / Line 14: @@
 ==="Meantone" A-Team tunings===
 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.
-====the 13edo-to-31edo range====
+===="19-limit A-Team": The 13edo-to-31edo range====
 '''tl;dr:''' [[44edo]] good
@@ Line 21: / Line 21: @@
 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 get the 23rd harmonic for free.
-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. 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.
+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.
+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.
 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.