source file: m1423.txt
Date: Thu, 21 May 98 20:04 BST-1

Subject: Reply to Paul Erlich

From: gbreed@cix.compulink.co.uk (Graham Breed)

>> There may be ways the ear relates to prime factors.  Different
>> overtones will be reinforced, and the difference tone pattern
>> may change.

> Can you give a concrete example?

Ooh, nothing concrete, no.  Where two notes are related by
a 7-limit interval, this will reinforce the 7th overtone of
one of them.  I think this explains why different timbres
behave differently in the 7-limit: it depends on the strength
of the 7th overtone.  There may be knock on effects in a
chord with two 7-limit intervals producing a factor of 49.
Instinctively, I recognise the possibility that this sort of
thing may lead to prime-limit character, but I haven't worked
out any details.  There are a load of difference and overtones
to take care of.

>> The good intervals in equal temperaments don't usually
>> constitute anything like an odd limit.  It's generally more
>> efficient to say how well different primes are approximated.

> This may not work if consistency doesn't hold.

If the approximation is *good* consistency will hold.  If
you know the intervals you want to use, a consistency test is
more appropriate.  For general rules governing what ratios
*might* work, prime approximations are more flexible.

>> For exactness, state the signed errors, and you can work
>> other intervals out from that.

> Isn't that what Carl Lumma said? But no, you guys are wrong, and that's
> the whole reason for the consistency concept. Wendy Carlos, Yunik and
> Swift, and others also seem to have missed out on the importance of
> consistency.

I think Carl Lumma (mistakes aside) takes the best
approximation for each interval, and tries to describe them in
terms of prime intervals.  I really _mean_ to take the best
approximations to prime intervals (not necessarily prime
numbers) and define the rest from this.  If the scale is
inconsistent, one of the intervals will have an error greater
than half the step size.  If that error is acceptable, so is
the scale.  I'm not wrong.

Consistency makes sense if you happen to be working within an
odd limit.  Expressing the consistency level as a fraction
[(step size) / (2 * worst error)] is more precise, and fairer
on 46-equal, but also harder to remember than an integer.


On the phi-based meantone: some algorithms show it as almost
the optimal 5-limit tuning.  However, it is poor in higher
limits, so won't do as a harmonic standard.  I use 31-equal
for this purpose: find the best approximation to a given
prime interval, take the simplest mapping on the cycle of
fifths, and generalise this to all meantones.


>> Incidentally, having different notes on different strings is
>> a _good_ thing.  It means you get more chords than you would
>> otherwise.

> Huh? It definitely means fewer positions in which to play a given chord.

It means both, of course :-)

With all the chords I've tried so far, it works out fine.
For an example, the 7:9:11 chord G-Cb-D#.  Find G on the
D string.  Cb is two steps of 31-equal nearer the bridge on
the G string.  D# is another step along on the B string.
This chord is easy to play on my guitar with an incomplete
meantone mapping.  It can be built on E, Fb, A, B, C and D,
as well as G, with 19 frets per octave.  As I have an extra
fret, I can also build it on F.  On a 19 note meantone
keyboard, you can only get this chord on G, D and A.  So,
the guitar wins!

With all 31 notes, of course, you can play as many of these
chords as you like.  There are some very common chords that
would be too difficult for me, though, so I'm happy with
what I've got.  JI might work, but is too complicated for me.