fourth based keyboard mapping

I've been working for a while now with a keyboard mapping with 12
schismic intervals to a perfect fourth. I have mentioned it to a
few people off list, and no-one's been fantastically excited,
but then no-one said they'd heard of it before either. It's far
too obvious to be original, but I'll present it to the list
anyway...

If you aren't familiar with schismic temperaments, well, I can't
be bothered to explain them now. The intervals can all be defined
in terms of an octave and a fifth. At the moment I'm using a
just octave and a fifth of exactly 702 cents. This scale is good
for 7-limit harmony (maximum error 5.5 cents for 7/5).

You could define the scale with any basis of 2 notes, though.
Here's an example:

(rho) = ( 8 -5)H'
(pi) (-19 12)

for the tuning above, rho=90 cents and pi=24 cents.

Now, you fit the scale to the keyboard as follows:

C D E F G A B C
pi pi rho-pi pi rho-pi pi rho-pi pi pi rho-pi pi rho-pi

The letters A to G here, and for the whole of this post, refer
to keys and not notes. The scale repeats over this fourth.
The keys are tuned as follows in cents:

C 0000 0498 0996 1494 1992
C# 0024 0522 1020 1518
D 0048 0546 1044 1542
Eb 0114 0612 1110 1608
E 0138 0636 1134 1632
F 0204 0702 1200 1698
F# 0228 0726 1224 1722
G 0294 0792 1290 1788
GA 0318 0816 1314 1812
A 0342 0840 1338 0836
Bb 0408 0906 1404 1902
B 0432 0930 1428 1926

I think these are right anyway: I don't usually use cents. That
gives you a full 4 octave keyboard, which is enough for most of
us. Most synths will allow you to tune up and down in octaves
so that you can choose the right register.


Now, there are 12 black notes to an octave in his mapping. I
think of these as being the base scale, whith the white notes
as alterations. If you worry about absolute pitch, set one
of the Bb keys to 440 Hz. You could retune to a 17 note scale
on the white notes, but I prefer to hit black notes with these
extended mappings.

This has all the same diatonic logic as a 12 note meantone
octave. I find it really liberating to be able to try out
comma shifts so easily. Anyway, see what you think.

A fifth is 17, a major third 9 and a minor third 8 notes.
A triad formed from these intervals will be in tune provided
no more than one note is white.

7/6 covers 6 notes. For this to be in tune, the lower note
must be either the C#, D, E, F#, A or B key. You can work
out 7-limit chords from that.

Seriously folks, if you've got full keyboard tuning, try this
out. It's the next best thing to a generalized keyboard.


SMTPOriginator: tuning@eartha.mills.edu
From: William Sethares
Subject: 13th root of 3
PostedDate: 23-10-97 22:22:38
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