"Numbers Separated by Colons" Notation

Hi Bob,

At 08:19 AM 3/25/98 -0800, you wrote:
>Thanks for your excellent thinking about diminished 7th chords on the pedal
>steel.

Thanks in return, for your informative web pages, especially "Just
Intonation of the E9th Pedal Steel"
(http://www.wco.com/~quasar/articles/just_e9.html). I'm totally inspired to
take up playing steel guitar! (It's gotta be kind of like trombone, right?)

>I'm always confused by the terminology of numbers separated by colons. You
>wrote:
>
>> Here's your E triad (4:5:6):
>> E: E (1/1) G# (5/4) B (3/4)
>
>Maybe it's my lack of formal music training, but I don't understand the
>"4:5:6" reference.

I don't think you'd learn this stuff in "formal music training". I think
it's more of a "higher math" kind of thing.

>What do these numbers mean? Later you use them a lot. Lacking the
>foundation knowledge, I'm having a hard time with equations like:
>
>> a D# diminished triad (45:54:64=5:6:4*16/9)
>
>I see that you know what you're talking about, and I really want to
>understand it. Can you give me a short explanation of this shorthand?

I'll give it my best shot:

Basically, the colons are a handy way to show the relation between three or
more notes.

To show the relation between TWO notes, you can use a ratio. For example, a
major third, say, C to E, is 5/4. And a minor third, like E to G, is 6/5.

You could also use the colon for this:
C:E is 4:5 (The C is the 4, and the E is the 5.)
E:G is 5:6 (The E is the 5, and the G is the 6.)

But to show the relation between THREE or more notes, the colons have to be
used. C:E:G is 4:5:6. (C is 4, E is 5, and G is 6.)

It would be misleading to use the slash for a triad, as in "6/5/4", because
it would be interpreted as "6 divided by 5, divided by 4" (which is 3/10).

>From C:E:G = 4:5:6, you have these combinations:
E/C = 5/4
G/E = 6/5
G/C = 6/4 (= 3/2)

As for things like:

C full diminished seventh (192:225:270:320)=(4*16/15:5:6:4*16/9):
Cdim7: C (4/5) D# (15/8) F# (9/8) A (4/3)

There are four notes instead of three, so three colons to separate them.
And the "*" is computerese for "times" (multiply). The chord is C:D#:F#:A.
The values line up as follows:

note name whole number small (possibly fractional) number
--------- ------------ ----------------------------------
C 192 4*16/15 (= 64/15)
D# 225 5
F# 270 6
A 320 4*16/9 (= 64/9)

The "whole numbers" are the smallest numbers I could find that were not
fractions. Unfortunately, the more notes there are in the chord, the bigger
these numbers tend to get. That's why I showed the alternative, "small
(possibly fractional) numbers".

By using the small numbers, you can easily see that D#:F# = 5:6 (instead of
having to take 225:270, dividing each number by 45; " 225/5 : 270/5 " =
5:6). If you take parts of the chord two-notes-at-a-time, the numbers stay
smaller. This table shows all the two-note combinations in the C:D#:F#:A chord:

C D# F# A
+---------------------------------
C | 1/1 75/64 45/32 5/3
D# | 1/1 6/5 64/45
F# | 1/1 32/27
A | 1/1

Another way to look at it (if you really want to go nuts) is to represent
each of the whole numbers as products of prime numbers, as follows:

note name whole number prime number breakdown
--------- ------------ -----------------------------
C 192 2*2*2*2*2*2 * 3
D# 225 3*3 * 5*5
F# 270 2 * 3*3*3 * 5
A 320 2*2*2*2*2*2 * 5

You might ask "Why (on earth) do that?" Because it's easy to cancel common
factors when all the prime factors are known. For example, take F#/D#:

F# 270 2 * 3*3*3 * 5 2 * 3 6
-- = ----- --------------- = ------- = ---
D# 225 3*3 * 5*5 5 5

(Notice the "canceling" two of the 3's and one of the 5's.)

By the way, this is where that term "5 limit just intonation" comes from -
none of the "prime number factors" are bigger than 5. If you considered "7
limit", then 7's would end up in some of the prime number breakdowns.

So that's my short explanation. (I know; with short explanations like that,
who needs long ones.)

Thanks again,
--Mark
+------------------------------------------------------+
| Mark Nowitzky |
| email: nowitzky@pacificnet.net |
| www: http://www.pacificnet.net/~nowitzky |
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