RE: theories

Neil,

Like you, I am a guitarist who plays best when feeling, not thinking.
But due to my physics background I tend to often think in mathematical
terms. The math involved in tuning theory is quite simple, really. Some
of us like to make it kind of complicated just to stretch our
intellectual muscles. But you don't need anything beyond a basic
understanding of logarithms to grasp any of the really good ideas in
tuning. Many of us have read Harry Partch's "Genesis of a Music" and
throw around terms like "tonality diamond" and assume that others will
understand us. This book does not assume any mathematical ability and is
written by a musician for musicians. A tonality diamond is just a scale
formed by picking a central pitch and including all notes that have
simple frequency ratios with that pitch. Partch dealt mainly with odd
numbers because dividing by 2 is just transposing down an octave, so if
you're talking about non-octave-specific pitch classes, you can dispence
with all factors of 2. So by "simple frequency ratios" we mean that both
the numerator and denominator of the ratios are odd numbers not
exceeding a pre-specified limit. It is easy to find all possible such
ratios by allowing the numerator to increase along one axis and allowing
the denominator to increase along the other. So the tonality diamond is
just a two-dimensional grid, which Partch liked to rotate by 45 degrees
to make it look like a diamond. Let's work out the first few tonality
diamonds. I will call the central pitch 1/1 and G and will give the
scale tones best approximating the tones of the diamond in 72-tone equal
temperament (all errors are 4 cents or less) notated using 1/6ths of the
conventional 12-tone sharps and flats (e.g., 1/2#, 5/6b).


Tonality Diamonds
-----------------




1-limit


1
-
1

or

G



3-limit


1 3
- -
1 1

1 3
- -
3 3

or

G D

C G



5-limit

1 3 5
- - -
1 1 1

1 3 5
- - -
3 3 3

1 3 5
- - -
5 5 5

or

G D B1/6b

C G E1/6b

E5/6b B5/6b G



7-limit

1 3 5 7
- - - -
1 1 1 1

1 3 5 7
- - - -
3 3 3 3

1 3 5 7
- - - -
5 5 5 5

1 3 5 7
- - - -
7 7 7 7

or

G D B1/6b F1/3b

C G E1/6b A2/3#

E5/6b B5/6b G C5/6#

A1/3# E1/3# D5/6b G




9-limit

1 3 5 7 9
- - - - -
1 1 1 1 1

1 3 5 7 9
- - - - -
3 3 3 3 3

1 3 5 7 9
- - - - -
5 5 5 5 5

1 3 5 7 9
- - - - -
7 7 7 7 7

1 3 5 7 9
- - - - -
9 9 9 9 9

or

G D B1/6b F1/3b A

C G E1/6b A2/3# D

E5/6b B5/6b G C5/6# F1/6#

A1/3# E1/3# D5/6b G B1/3#

F C A1/6b D2/3# G




11-limit

1 3 5 7 9 11
- - - - - --
1 1 1 1 1 1

1 3 5 7 9 11
- - - - - --
3 3 3 3 3 3

1 3 5 7 9 11
- - - - - --
5 5 5 5 5 5

1 3 5 7 9 11
- - - - - --
7 7 7 7 7 7

1 3 5 7 9 11
- - - - - --
9 9 9 9 9 9

1 3 5 7 9 11
-- -- -- -- -- --
11 11 11 11 11 11

or

G D B1/6b F1/3b A C1/2#

C G E1/6b A2/3# D F1/2#

E5/6b B5/6b G C5/6# F1/6# A1/3b

A1/3# E1/3# D5/6b G B1/3# D5/6#

F C A1/6b D2/3# G B1/2b

D1/2b A1/2b F1/3# B1/6# E1/2b G



Look at the 5-limit diamond for a familiar case. Each row is a major
chord, and each column is a minor chord. So tradiationally speaking, the
diamond is a compact way of seeing all the consonant chords that contain
the note G. (Of ourse, the 1/1 could mean anything, not just G.) So if
you were writing a song and you knew you wanted G in the melody, you
could just try out the different chords in the tonality diamond of G and
see which one you liked the best. For higher limits, we are talking
about more complicated chords, but the concept is exactly the same.

How's this so far for basic concepts?



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Subject: Re: TUNING digest 1139 (?'s for Gary Morrison)
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