Just Intonation Scales for Modes[Revised version added (mo/dy/yr) 03/29/02, additions, chords- 10/06/02] Just intonation scales for normal ('Greek') 7-note modes (here with A=440 Hz). Sequences symbols and letters for notes don't really tell the whole story (see MODES file), so below is given the information for the just intonation scale. Starting the minor scale at A and using the same notes as the C ionian/major scale to get A aeolian/minor works for an equal tempered scale, but not otherwise. The following table gives the ratio of the frequency of the nth note to the keynote for the 7 normal septatonic 'Greek' modes. The numerical scale is also given for the keynote where no sharps or flats appear in the scale. It looks a bit strange at first with some of the notes a factor of 81/80, or 80/81, times the normal C major/ionian frequencies, but transposing the lydian to ionian by flattening the 4th comes out right for C ionian. Then flattening the 7th gives C mixolydian. Then flattening the 3rd then gives Dorian, and so on (i.e., then flatten 6, then 2, then 5). The first thing we note is that we have two Ds, depending on where we find them. If keys of F and A we get D= 293.33, and in the others D= 297. Chords of C major/ionian are 3 consecutive notes of FACEGBD with D=297, alternating major-minor-major but for A minor/aeolian, it's DFACEGBD with D= 880/3=293.3+, alternating minor-major-minor. Minor DFA in F-lyd and A-ael is 293.3, 352, 440. Somewhat of a surprise is that there isn't one in E-phy, but there is in D-dor, G-mix and B-loc, where it's 297, 356.4, 445.5. [Modes based on letter notes are really rather ridiculous, since, in just intonation, they're determined by modes, not notes or keynote, as noted in a new addition under chords, below.] lydian 1 9/8 5/4 25/18 3/2 5/3 15/8 key = F 352 396 440 488.89 264 293.33 330 ionian 1 9/8 5/4 4/3 3/2 5/3 15/8 key = C 264 297 330 352 396 440 495 mixolydian 1 9/8 5/4 4/3 3/2 5/3 9/5 key = G 396 445.50 495 264 297 330 356.40 dorian 1 9/8 6/5 4/3 3/2 5/3 9/5 key = D- 293.3 330 352 391.1 440 488.9 264 aeolian 1 9/8 6/5 4/3 3/2 8/5 9/5 key = A 440 495 264 293.33 330 352 396 phrygian 1 27/25 6/5 4/3 3/2 8/5 9/5 key = E 330 356.40 396 440 495 264 297 locrian 1 27/25 6/5 4/3 36/25 8/5 9/5 key = B 495 267.30 297 330 356.4 396 445.50 Note that each time we change a ratio we just flatten it by multiplying by 24/25. If we do that to the 1st (=keynote) in the locrian mode we, are back to an offset lydian. For example after that first step we have: 24/25 27/25 6/5 4/3 36/25 8/5 9/5 and renormalizing by multiplying all by 25/24 we get 1 9/8 5/4 25/18 3/2 5/3 15/8 which is the lydian we started with. Sharps are just 25/24 times the note, and for double sharps use another factor of 25/24. For flats we use a factor of 24/25, and use it again for double flats. Let's rearrange the scale to start on C, and use a minus sign for frequencies which are 80/81 times the diatonic note frequency and a + sign when they are times 81/80. We also rearrange the order so that only one note at a time changes between these modes. (* = key) dorian C D- E F G- A B- key=D- 264 293.33* 330 352 391.1 440 488.9 lydian C D- E F G A B- key=F 264 293.33 330 352* 396 440 488.9 aeolian C D- E F G A B key=A 264 293.33 330 352 396 440* 495 ionian C D E F G A B key=C 264* 297 330 352 396 440 495 phrygian C D E F+ G A B key=E 264 297 330* 356.40 396 440 495 mixolyd. C D E F+ G A+ B key=G 264 297 330 356.40 396* 445.50 495 locrian C+ D E F+ G A+ B key=B 267.30 297 330 356.4 396 445.50 495* CHORDS: We can arrange the notes of the normal, - and + scales in a form that is convenient for figuring out chords. The note to the right of any chosen base note a fifth (x3/2) above it (multiply or divide by 2 when necessary to get the right octave). The note to the upper right of a starting base note is a major third (x5/4) above the base note, and the one to the lower right is a minor 3rd (x 6/5) of it. This gives an array as follows. This is only the 'center' of an infinite array. As we to up we add a factor of 25/24 to each pair of lines for an additional sharp, and as we go to the right we add a factor of 81/80. New 06/10/02 With the just intonation scales as above, available chords can be stated concisely. (Keynote is irrelevant.) Starting numbers for perfect chords from key = 1 majors minors diminished 7ths lydian 1, 5 3, 6 3, 4, 6 ionian 1, 4, 5 3, 6 3, 6 mixolydian 1, 4 5, 6 1, 3, 6 dorian 3, 4 1, 5 1, 4, 6 aeolian 3, 6 1, 4, 5 1, 4 phrygian 3, 6 1, 4 1, 3, 4 locrian 5, 6 3, 4 1, 3, 4, 6 Now New 07/24/02 Here's the classical way to do it. lydian F 352, G 396, A 440, B 495, C 528, D 2*297, E 660 1: 9/8: 5/4: 45/32: 3/2: 27/16: 15/8 ionian C 264, D 297, E 330, F 352, G 396, A 440, B 495 Standard 1: 9/8: 5/4: 4/3: 3/2: 5/3: 15/8 mixold G 396, A 440, B 495, C 528, D 594, E 660, F 704/ 1: 10/9: 5/4: 4/3: 3/2: 5/3: 16/9 dorian D 297, E 330, F 352, G 396, A 440, B 495, C 528 1: 10/9: 32/27: 4/3: 40/27: 5/3: 16/9 aeol. A 440, B 495, C 528, D 594, E 660, F 704, G 792 1: 9/8: 6/5: 27/20: 3/2: 8/5: 9/5 phry E 330, F 352, G 396, A 440, B 495, C 528, D 594 1: 16/15: 6/5: 4/3: 3/2: 8/5: 9/5 locr. B 495, C 528, D 594, E 660, F 704, G 792, A 880 1: 16/15: 6/5: 4/3: 64/45: 8/5: 16/9 Unfortunately it doesn't work well. If we sharpen the 4th of ionian we get 4/3 x 25/24 = 25/18, which doesn't match the 45/32 that we got from the FGABCDE series with C major frequencies. And the 6th in lydian, 27/16, and the 6th in ionian, 5/3, should be the same. Flattening the 7th of ionian should give us the ratio of the 7th in mixolydian. It doesn't, and the 2nds should be the same in mixolydian, 10/9, and ionian, 9/8, and they aren't. In A aeolian we're back to the lousy DFA minor chord (1:32/27:40/27), not the good one (10:12:15).