5/4 vs 81/64 etc

The subject of 81/64 tuning vs 5/4 has appeared in several posts in the
short time that I have been here. There are several examples of very
near matches in harmonic ratios. They centre around the near equality
of the numbers 81 and 80, 64 and 63 and some other lesser ones. These
numbers are all important in my view because they are all numbers with
many ways to be factorised and so have many harmonic relationships.
Changes in the relative importance of notes come into play when we shift
around in the multidimensional world of tonality.

The 81/80 shift is the easiest one to get to happen and is no doubt
familiar to all, but I repeat it just to set the scene. If we have a
Just Intonation scale of 24 27 30 32 36 40 45 48 and move by a fifth
then we multiply all these by 3/2 and get 36 40.5 45 48 54 67.5 72 which
when brought back into the original range is 24 27 33.75 36 40.5 45 48.
So we dropped 30 and added 33.75 which is our sharp coming in, and
replaced 40 by 40.5 or an 81/80 shift. Of course if we repeat this 12
times they will all shift and we get Pythagoras comma problem.

If we accept the ratios 28 and 42 as being the minor 3rd and minor 6th
(please excuse my lack of proper musical jargon) then we can see that
these will become 42 and 63 (or 31.5) when we transpose by a fifth.

This 63 introduces another near miss with a difference of 63/64.
There are other more subtle near misses.

It is also possible to shift key by a major third and so we can multiply
the JI scale by 5/4 to get 30 33.75 37.5 40 45 50 56.25 60 which in the
original range is 25 28.125 30 33.75 37.5 40 45 and so lots of notes
have gone wandering. The 28.125 value is however very close to 28 and
the ratio of 225/224 is one of the many lesser near misses.

When I used my little BASIC program to calculate all the strong
harmonics according to the factorisations of each number (which is a
measure of their total number of musical relationships) then I found a
couple of interesting things happening as I went to higher numbers.
Sometimes as I compared the pattern as I went up octaves one harmonic
would gradually fade and a nearby one would come in. This happens with
both 81/80 and 64/63 ratios. In fact, if we consider a tonic as 1 then
although 6 octaves above is 64 alright, after a few more octaves the
most dominant note is 63*2^n not 64*2^n. There is evidence of this type
of thing in the Indian musical scale although I don't know enough about
the music to say whether it happens in the music itself.

Does anyone else have thoughts on whether there might reasonably be such
a funny scale used which had the tonic tuned at successive octaves to
say 16 32 64 126 252? This would be most likely to occur when the
.. oh bugger, my musical knowledge doesn't allow me to express myself
(HELP) let me go back a bit. Say we are in C major and we have Cs at
frequency 16 32 64 126 252 with this funny discontinuity between 64 and
126. In parctice 63 wants to exist also and 128, but we don't have
enough keys. ... anyway, to resume, this is likely to happen when we
are also using Eb and Bb in our music because these have the ratios 28
and 42 already, which want the 63 and 126 rather than 64 and 128. Does
this make sense? I am suggesting that the emphasis might change between
the low and the high octaves.

Here is a grand chord that demonstrates the simultaneous use of such
"out of tune" octave notes: 8 12 16 20 24 28 32 42 56 84 126

Similar examples can be found that want both a 5/4 and 81/80 tuning in
different octaves. Is this something that has ever been suggested
before?

Of course you realise that there is a secret plot. I have come here to
drive you all mad HA HA HEE HEE ...

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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Jonathan Wild wrote:

> "Repugnance" towards irrationals was strictly limited to disciplines
> regarded as subordinate to mathematics. Look at the Greek quadrivium
> to see how music fits in:
>
> MATHEMATICS GEOMETRY
> (number) (magnitude)
>
> MUSIC ASTRONOMY
> (number as embodied (magnitude as embodied
> in sound) in celestial motion)
>
> The schism between the sciences dealing with the discrete vs those
> dealing with the continuous (number vs magnitude) was a huge factor
> in the development of mathematical methods.[etc.]

Many thanks to Jonathan Wild for such a fine posting, which has brought
the thread, I would imagine, near to a consensual close. I'll confess I
was feeling a little guilty at having addressed only Paul Erlich's
general statement about Greeks and irrationals, without moving on to
their treatment of music, where, for the reasons Jonathan has given,
irrationals were indeed shunned. I would still maintain, of course, that
if the Greeks had broken their quadrivial constraints, and tried to
reinterpret Aristoxenus in terms of irrational equal divisions, the 30th
root of 4/3 would have been by far the most likely choice of a basic
unit. The fact that when this break was eventually made, in the 16th
century, Aristoxenus was taken to be the father of 12TET, reflects the
needs of Renaissance theorists, and not those of Greek antiquity.
Aristoxenus was needed as a precedent, to justify the abandonment of
numerus sonorus dogma; without citing a precedent, these ideas could, in
the intellectual environment of the times, have been summarily
dismissed. Renaissance theorists were therefore primarily interested in
the use that could be made of Aristoxenus to serve their purposes; the
truth of their interpretation of Aristoxenus was only a secondary
consideration.

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/



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