RE: Optomizing synthesizer tuning

In order to optimize tuning you must know the tuning
characteristics of the target instrument. This is a practical
impossibility with wavetable synthesis, since manufacturers's
won't release the specs.

However the specs for FM synthesis as implemented in the Sound
Blaster are readily available. The output frequency in
proportional to a number written into a tuning register, and this
number can be any integer up to 1024.

With this information, it is relatively simple to determine
whether a JI scale can be exactly represented on the Sound
Blaster. You just have to convert the scale to a multi-level
fraction. To do that, just find the least common multiple of the
numerators of the fractions that make up the scale. If that
number is less than or equal to 1/2 the highest integer you can
user to program the hardware, then your scale can be exactly
represented. Otherwise it can't. That procedure will work for
most systems using frequency linear tuning. The notable
exception to this is the Amiga, which uses wavelength linear
tuning. In that case you need to find the LCM of the
denominators of the fractions of your scales.

What Manuel suggested in his post was letting the fundamental of
the scale vary in order to maximize the tuning accuracy for the
whole scale. That is also the basis of the technique discussed
above.

It is also possible to use this technique to find best fits for
scales that cannot be represented exactly. For example, let's
find a couple of them for Agricola's Monochord posted on this
forum some time ago by John Chalmer's.

Agricola's Monochord

LCM: 2^14*3^7*5

12 17496 243/ 128
11 16384 16/ 9 Grave Minor Seventh
10 15552 27/ 16 Pyth Major Sixth
9 14580 405/ 256
8 13824 3/ 2 Perfect Fifth
7 12960 45/ 32 Augmented Fourth
6 12288 4/ 3 Perfect Fourth
5 11664 81/ 64 Pyth Major Third
4 10935 1215/ 1024
3 10368 9/ 8 Major Tone
2 9720 135/ 128
1 9216 1/ 1 Unison

The fundamental, 9216, factors into: 9216 = 2^10*3^2

It's easy to see from this that if we want to have the
fundamental be one of the notes represented exactly, we are
limited to numbers with factors of 2 and 3. 18 and 24 provide
some pretty good approximations, as show below.

18 24

b 17496 972 729
a# 16384 910.2222 682.6666
a 15552 864 648
g# 14580 810 607.5
g 13824 768 576
f# 12960 720 540
f 12288 682.6666 512
e 11664 648 486
d# 10935 607.5 455.625
d 10368 576 432
c# 9720 540 405
c 9216 512 384

Since all scales can be represented as collections of rational
fractions to any desired degree of accuracy, this technique is
theoretically extensible to all scales, although fractions with
numerators having large prime factors make it rapidly more
difficult.

Marion

Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 30 Jan 1996 08:02 +0100
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id XAA16527; Mon, 29 Jan 1996 23:02:31 -0800
Date: Mon, 29 Jan 1996 23:02:31 -0800
Message-Id: <960130064739_71670.2576_HHB45-8@CompuServe.COM>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu