"Numbers Separated by Colons" - Sequel

To: steele@bom2.vsnl.net.in
Cc: tuning@eartha.mills.edu, quasar@wco.com

Hi Drew,

Thanks for your questions.

At 11:58 PM 4/18/98 -0700, you wrote (Digest 1389, Topic 7):
>From: Drew Skyfyre
>Subject: "Numbers Separated by Colons" - Sequel
>
>I could use some clarification about the following :
>
>> 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)
>>
>I do not understand how one arrives at those figures.
>Since the chord in question is a stack of minor 2nds,
>(using 6/5s) - 6/5, 36/25, 216/125.
>How does one go about notating these in colon form ?

(you meant a stack of "minor 3rds")

First I gotta correct myself. As I looked at the letter names I used above,
and flashed back to childhood piano lessons and music theory, I realized I
should've called the chord a "D#dim7" (in first inversion). Mea culpa or
whatever...

Okay, so here's the chord, properly named, and rearranged to be in root
position. The same ratios are shown (they are based on E = 1/1):

D#dim7: D# (15/8) F# (9/8) A (4/3) C (4/5)

As stated above, a full diminished seventh chord is a stack of minor 3rds,
but there's more than one "flavor" of minor 3rd in just intonation. I chose
the 6/5 flavor for D#:F# and for A:C, but I used 32/27 for F#:A. Here's a
table the intervals involved:

From To Ratio Interval name*
---- -- ----- -----------------------
D# F# 6/5 Just minor Third
F# A 32/27 Pythagorean minor Third
A C 6/5 Just minor Third
C D# 75/64 augmented Tone

*Interval names shown are from the Table of Intervals in Ellis' additions to
Helmholtz' "On the Sensations of Tone". (Yeah, you gotta buy the book if
you haven't already. Or you have to live in one of those comfy chairs at a
Crown & Noble bookstore.)

Note that the interval C:D# is actually not a minor 3rd, but an augmented
2nd (because of it's "spelling").

>>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".
>How do these single large numbers represent those notes ?

The numbers were used in the "numbers separated by colons" notation; viz.,
"192:225:270:320". The numbers only have meaning relative to each other.
For example:

A/F# = 320/270 = 32/27

A "tuning lattice" may be used to help visualize intervals and chords.
Graham Breed describes "Tuning Lattices" at:
http://www.cix.co.uk/~gbreed/lattice.htm

Here's my tuning of the D#dim7 chord, represented in a tuning lattice:

D#
/ \
A--(E)-(B)--F#
\ /
C

It shows how the "Pythagorean minor Third" from A to F# is a more "remote"
interval than the "Just minor Thirds" (from A to C, and from D# to F#). You
can also see that you can get to the F# by traversing the "cycle of 5ths"
from A, to E, to B, to F#.

That's probably more than enough for now. Thanks again,
--Mark

P.S.: If you respond to this, you might want to send a copy to both
tuning@eartha.mills.edu and my email address (nowitzky@alum.mit.edu), to
make sure I don't miss it. (It's easy to get left "in the dust" on the
information superhighway.)

P.P.S.: Now I can go check out those "Continued Fractions" web pages you
mentioned in other posts (around 4/17/98). I'll catch up, as long as I
never sleep...
+------------------------------------------------------+
| Mark Nowitzky |
| email: nowitzky@alum.mit.edu |
| www: http://www.pacificnet.net/~nowitzky |
| "If you haven't visited Mark Nowitzky's home |
| page recently, you haven't missed much..." |
+------------------------------------------------------+