Mode numbers, March 5, 2000, reformated Oct 10, 2002:
With enough 'accidentals' any tune can be scored in any of the 7
7-note 'Greek' modes. These different representations can all be
taken care of for the normal 7, 6, and 5 note modes and a few
others where there are assigned names for modes that are
unambiguous, e.g., lydian, or ionian or mixolydian all with the
4th, 6th and 7th missing are pentatonic pi1 (Bronson's notation).
There are also names for a few other modes, e.g., harmonic minor,
but not enough names for most other tune scales (and there are a
lot of them) to give an umambiguous basis for identifing the
scale of a tune from a mode name. In yet other cases there are a
bewildering variety of names for a given mode. What to do?-Mode
number.
For the tune mode, not necessarily the scoring mode we do the
following.
Consider a tune in C (or D) major/ionian and use 1 for note in
and 0 for note out. Then we have:
(D D#/Eb E F F#/Gb G G3/Ab A A#/Bb B C C#/Db)
C C#/Db D D#/Eb E F F#/Gb G G#/Ab A A#/Bb B
(1 1#/2b 2 2#/3b 3 4 4#/5b 5 5#/6b 6 6#/7b 7)
1 0 1 0 1 1 0 1 0 1 0 1
[Semitn sq 2 2 1 2 2 2 1]
The string of 1's and 0's in the bottom row can be taken as a
unique binary number representation of the ionian mode which
doesn't depend on keynote or even the scoring mode. It just
depends on the key and the notes in the tune. The first 1 on the
left is always there, so we can ignore it (unless someone knows
of a tune that doesn't contain the keynote).
I've now chosen to use a mode identifier which is the decimal
equivalant of taking of the binary number consisting of the
rightmost 11 bits in reverse order, so the 1#/2b is the least
significant bit and th 7th the most significant. This order is
reverse in a mathematical sense, but forward for the scale so the
farther from the keynote is the biggest number. See note below on
computer coding for 'reverse order'. We can do this for the scale
of any tune that can be represented by a 12 semitone scale.
Now no matter what the scoring mode is for any tune we have a
unique identifier for the scale or 'mode' of the tune, and from
fairly simple math we can recover the complete scale from the
mode number identifier.
Reverse order is actually forward order for a DO/FOR loop in a
computer to generate the mode #. Let M(I,j) be the binary code of
the mode of tune # i, and j run from 1 to 12 over the notes of
the scale with bits 1 or 0 in each of the 12 positions, with the
bit for the keynote at j = 1.
FOR/DO i = ifirst to ilast
let Mode# = 0
FOR/DO j = 2 to 12
mode# = mode# + M(i,j)*2^(j-2)
next J
nmod(i)= mode#
next i
For forward order we would have
.....
FOR/DO j= 2 to 12
mode#=mode# + M(i,j)*2^(12-j)
next j
...
All modes I've observed, 181 of them, are listed with mode
number in file MODETABL.TXT