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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).