RE: pure? / synth tuning

SPurman writes:

> Anyway, how can I most accurately tune my synth to
> this pure JI scale, in one cent increments?

Gary's intuitive analysis of the tuning table you found in a book was right
on the math except for C# 135:128, and A# which I was unable to decipher - A#
14 cents above 12-tET - anyone? The 16s were not exaggerated. (135:128?
What the heck kinda deal is that?, you may ask. It was reached by taking the
perfect 5th, 3:2, of the F#. See the Danielou site at
http://www.imaginet.fr/~jcloarec/danielou/ANGLAIS/index.html for an
enlightening explanation of a similar kind of approach.)

Someone out there may be able to tell us of a simpler method but here's how I
find the number of cents above the tonic for any JI ratio:

x log-base2(d/n), where n is the numerator of your JI ratio, d is the
denominator of same, x is the answer you seek.

In English - find the log-base2 of the inverse of the JI ratio. Sure, just
get out my trusty slide-rule here, which everyone keeps by their keyboard...

I use a table in a Word Perfect for Windows doc to do the calculations for me
on each scale. WP doesn't do log-base2, only base ten or natural logs, but I
found that multiplying any base10 log by 3986.313713865... will produce the
corresponding base 2 log. Oh, goody. So here is the line from my Formula
Bar for line 2 of the table, which calculates C# (or as I like to call it,
2b):

-3986.313713865*LOG(C2/B2)-100

B2 and C2 are the cells where the numerator and denominator, respectively,
are entered; on line 3 which calculates the D (I call it 2) the formula reads
B3 and C3, instead of B2 and C2, and subtracts 200 instead of 100, and so on
until the last line where you are calculating the B (I call it 7) by
substituting B12 and C12 and subtracting 1100. (Line 1 contains the column
headings.) Make sense?

Make an empty template like that in your word processor or spreadsheet and
you can use it to calculate the synth tuning tables for any JI scale you can
think of. They are legion. Put several sets of these columns side-by-side
in landscape mode to analyze a bunch of scales at once, and to hold several
related scales so you can print it out and keep it near your keyboards. Toss
your slide-rule.

By the way, I've learned that using numbers instead of letters for
scale-degrees allows thinking in terms of various key-centers without having
to transpose your system in your head. 1 2b 2 3b 3 4 4# 5 6b 6 7b
7 is the series I'm comfortable with for use with a 12-key-to-the-octave
keyboard using an octave-repeating scale.

A Q for those who work with non-repeating and/or non-octave scales - what
sort of nomenclature works there for identifying scale degrees?

> is there a JI tuning that is more
> theoretically *pure* or natural than others,...?

Gary M. answers:

> I don't think that any one just intonation
> tuning is necessarily more purely JI than any other. But then again, that
> depends on what you mean by "pure". As far as I can tell, "pure" has no
> specific definition in tuning theory

The way I look at it, there are many possible ways to arrange or develope JI
systems of tunings, so the question belongs the other way around - in what
system would any given scale be the purist JI tuning? E.g., the Danielou
site I mentioned advocates a system based on 5-limit, limiting the
possibilities to ratios involving multiples of 2, 3, and 5. Your original
question stemmed from using a system of dividing successive harmonic
overtones by powers of 2. That's only a small taste of the possible systems.

Have fun.

David J. Finnamore
Just tune it!

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