Meantone intervals

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This table shows all the simple intervals of POTE septimal meantone, which includes the entire 7-limit tonality diamond. Other relevant tables of meantone intervals are the table of quarter-comma meantone intervals and the table of 31 edo intervals.

In 12edo the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.

In 19edo, in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.

More complex meantone tunings such as 31edo distinguish all intervals listed on this table.

See also Wikipedia's list of meantone intervals

Name Size Ratios
Unisons
Perfect unison (P1) 0 1/1
Augmented unison (A1) 75.5 28/27~25/24~21/20
Seconds
Diminished second (d2) 42.0 128/125~64/63~50/49~36/35
Minor second (m2) 117.5 16/15~15/14
Major second (M2) 193.0 10/9~9/8~28/25
Augmented second (A2) 268.5 7/6
Thirds
Diminished third (d3) 235.0 8/7
Minor third (m3) 310.5 6/5
Major third (M3) 386.0 5/4
Augmented third (A3) 461.5 21/16
Fourths
Double diminished fourth (dd4) 343.5 49/40~60/49
Diminished fourth (d4) 428.0 9/7 (a bit 14/11-ish)
Perfect fourth (P4) 503.5 4/3
Augmented fourth (A4) 579.0 7/5
Double augmented fourth (AA4) 654.5 Close to 16/11
Fifths
Double diminished fifth (dd5) 545.5 Close to 11/8
Diminished fifth (d5) 621.0 10/7
Perfect fifth (P5) 696.5 3/2
Augmented fifth (A5) 772.0 14/9 (a bit 11/7-ish)
Double augmented fifth (AA5) 842.5 49/30~80/49
Sixths
Diminished sixth (d6) 738.5 32/21
Minor sixth (m6) 814.0 8/5
Major sixth (M6) 889.5 5/3
Augmented sixth (A6) 965.0 7/4
Sevenths
Diminished seventh (d7) 931.5 12/7
Minor seventh (m7) 1007.0 16/9~9/5~25/14
Major seventh (M7) 1082.5 15/8~28/15
Augmented seventh (A7) 1158.0 35/18~49/25~63/32
Octaves
Diminished octave (d8) 1124.5 27/14~48/25~40/21
Perfect octave (P8) 1200 2/1
Augmented octave (A8) 1275.5 25/12~21/10