source file: mills2.txt
Date: Fri, 7 Feb 1997 21:17:45 -0800

Subject: pitch specification in CET

From: Gary Morrison <MorriSonics@compuserve.com>

Bill Alves raised a very meaningful question about 88CET or other
tunings that don't boil down to some number of steps per so-and-so
interval:  

> if you're using a tuning system that does not replicate any particular 
> interval, how do you specify pitches?  Do you just pick one pitch as the 
> starting point and number them?  That seems rather awkward.

   Well, Bill posed that question in a very general way, but certainly not
an inappropriate way.  I'll suggest though that a better way to approach it
is to assume that you'll devise some sort of organization for the tuning. 
Without some sort of organizational principle, it's chaos.  You have no
framework in which to compose.  

   Some would suggest that establishing a framework or underlying
organization is too restrictive for the freedom needed to compose
innovative new music.  I personally find the alternative too difficult a
nut to crack - there are too many possibilities for creativity to flourish.
 I find that I have to break the problem down to something simpler before I
can get anything done.  

   So, generally speaking, I recommend organizing the notes not into spans
of octaves but into spans of something else.  For example, in the case of
88CET or Carlos' Alpha (78CET), you have good approximations to a 3:2 P5. 
So, you in either of those tunings, you can organize their pitches into
patterns that repeat in fifths.  

   Were I working (so far I haven't yet) with my viola in 88CET, I would
almost certainly build music from the scale pattern 2 1 2 2 1 (i.e.,
whole-step, half-step, whole-step...) repeating in fifths.  That's very
convenient on the viola or violin since their open strings are fifths
apart.  (It's a little trickier on a 'cello since you run out of fingers.)

   But for most of my work, I instead organize 88CET pitches by a "cycle"
of perfect fifths wrapping within a subminor seventh (7:4 approximation). 
That works out to a seven-step-per-cycle scale with the pattern 2 1 2 2 1 2
1, and I give them traditional letter names, in this case A B C D E F G A. 
Any note and the next higher or lower tone of the same note name is a
subminor seventh apart rather than an octave apart.  There's an
inconvenience that G-A is a half-step, but I learned my way around it
pretty quickly.  I simply omit the G#/Ab key from the system.  

   After you do that, you have a complete system of pseudokeys and
pseudokey signatures.  Fifths look like what we normally think of as
sixths, so sharps add in the order of C# A# F# D# B#..., and flats add in
the order of Eb, Gb, Bb, Db, Fb...  So some of the pseudokeys work out to:

Gb: Gb A  Bb C  D  Eb F  Gb
Eb: Eb F  Gb A  B  C  D  Eb
C:  C  D  Eb F  G  A  B  C
A:  A  B  C  D  E  F  G  A
F:  F  G  A  B  C# D  E  F
D:  D  E  F  G  A# B  C# D
B:  B  C# D  E  F# G  A# B

   From that point you learn things analogously with how you'd learn things
in an octave-based tuning (pretty much rote but with auditory reinforcement
and memory), for example:
Intervals:  
   - Subminor third (~7:6) above the root is the third step in the scale,
   - Supramajor third (~9:7) is the fourth step in the scale,
   - The perfect fifth is the sixth step in the scale,
   - The major tenth is the fifth step in the second cycle of the scale,
etc.
Chords:
   - A supramajor third-stack triad is scale steps 1, 4, and 7.
   - A harmonic-series fragment (4:6:7:9:10:11:15) approximation can be
found 
     at scale steps 
     First cycle:  1, 6, 
     Second cycle: 1, 4, 5, #6, 
     Third cycle:  #3.  

   It becomes its own new world of its own with all new nomenclature,
possibly new notation, new harmony rules, and such.  But once you get to
know it, it makes as much sense as the traditional system.  In some cases
it's a similar kind of sense, and others it's a very different kind of
logic.  

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Date: Fri, 7 Feb 1997 21:22:03 -0800
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