More About the Strange 9-limit Tuning Otonal chords B C C# D Eb E F F# G G# A Bb B C C# D Eb E F F# G G# A Bb B C 4-----------5--------6 5--------6--------7-----------9 4-----------5--------6--------7-----------9 4-----------5--------6--------7 4-----------5--------6 5--------6--------7-----------9 4-----------5--------6--------7-----------9 4-----------5--------6--------7 B C C# D Eb E F F# G G# A Bb B C C# D Eb E F F# G G# A Bb B C Utonal chords (each number stands for its reciprocal here). B C C# D Eb E F F# G G# A Bb B C C# D Eb E F F# G G# A Bb B C 7--------6--------5-----------4 9-----------7--------6--------5-----------4 9-----------7--------6--------5 6--------5-----------4 7--------6--------5--------4 9-----------7--------6--------5-----------4 9-----------7--------6--------5 6--------5-----------4 B C C# D Eb E F F# G G# A Bb B C C# D Eb E F F# G G# A Bb B C 9-limit lattice. B\ ------- F#\ ======= C#\ ======= G#\ | /| `. , '/| `. , '/| | /,| ' `. /,| ' `. / | | A\ ======== E\ ======== B\ ------- F#\ | ,' ^ `| ,' ^ `| ,' ^ | ,' | | ,' |\ | `,' |\ | `,' |\ | ,' | | ,' | \| ,' ` |.\| ,' ` |.\| ,' | F -------|- C =======|= G =======|= D | | | /| `. , |'/| `. , |'/| | | |/,| ' `. |/,| ' `. |/ | | | Eb ======== Bb ======== F --------- C | ,' ^ `| ,' ^ `| ,' ^ | ,' | | ,' |\ | `,' |\ | `,' |\ | ,' | | ,' | \| ,' ` |.\| ,' ` |.\| ,' | B/ ------| F#\ ======| C#\ ======| G#\ | | / `. , |'/ `. , |'/ | |/, ' `. |/, ' `. |/ | A\ ======== E\ ======== B\ ------- F#\ All available 9-limit intervals shown. Note that I use C ========= G ========= D to mean: Not only are C-G and G-D usable 2:3's but C-D is also a usable 4:9. Note that the pentads, whether otonal or utonal, follow the dominant 9th pattern. The otonal 4567 tetrads and utonal 1/9765 tetrads follow the dominant 7th pattern while the 5679 and 1/7654 tetrads appear as half-diminished 7ths. The 567 and 1/765 triads are the diminished pattern as expected. Both 456 and 1/976 triads are the usual major pattern while both 679 and 1/654 are the minor pattern. In addition to the chords shown above, usable: augmenteds occur everywhere (the three inversions have different flavours depending where the 7:9 is), minor 7ths occur on A, E, Eb, Bb as (10/3):4:5:6, and on C, G, F#, C as 6:7:9:(21/2), minor major 7ths occur on B and F, minor 9ths occur on A and Eb, minor flat 9ths occur on A, E, Eb, Bb, half diminished flat 9ths occur on C, G, F#, C#, half augmented 7ths occur on G and C#. The following chords don't work anywhere (are poorly tuned/contain wolves): major 7th, diminished 7th, major 9th, 6/9, dominant flat 9th. How one would ever remember these without strong cues from the keyboard/fretboard layout I dont know. A suitable keyboard layout might have 6 white and six black keys per octave, making F/E# the new black key. | | | | | | | | | | | | | | |C# | | Eb| | F |F# | |G# | | Bb| | black | ----| ----| ----|---- |---- |---- | | | | | | | | | C | D | E | G | A | B | C white --------- --------- --------- --------- --------- --------- ------ On second thoughts, I'm not sure this will help.