Ratios etc

Thinking about Indian and Arabic tunings (among others) lately has really got me to wondering about the way people around the world have tuned for thousands of years, and why. I know there's probably no real firm answers, but take Indian ragas, for example...why are they not using intervals up to, say, the 13 limit, instead of concentrating on 5 limit? If one is talking about low ratios, 13/8 is certainly lower than many of the 5ths based ratios one commonly finds, yet the 5ths based seem to be very common. And, another thought, between the fundamental and 32nd harmonic, there's a vast world of possible interval relationships, yet it seems like none of the maqam based musical cultures went there. Just lately, this fact has been dawning on me in a deep way...I've been aware of it for years, but it's starting me to wonder about the choices people have made as to how they form and shape their musical expression.
Actually, by not exploring the overtone series in music, we (humans) have really limited ourselves for most of recorded history...and, I'm sitting here wondering why, and I don't quite know. Unless, of course, the studies of numerology and number symbolism figure in somehow, which they most likely do. From my studies, I've found these correspondences are very important to most cultures (except contemporary Europeans, for the most part). Where does music go from here? It would be interesting to see Indian maestros, for example, start composing ragas off the harmonics, and using those untapped upper ratios...and, they're not even that up, the 32nd harmonic is pretty low, in the sceme of things.
No biggie, just sort of thinking aloud...best...HHH
microstick.net

--- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...>
wrote:
>
> Thinking about Indian and Arabic tunings (among others) lately
has really
> got me to wondering about the way people around the world have
tuned for
> thousands of years, and why. I know there's probably no real firm
answers,
> but take Indian ragas, for example...why are they not using
intervals up to,
> say, the 13 limit, instead of concentrating on 5 limit?

It's hard to be sure you're hearing or tuning ratios of 13 unless you
have a big otonal chord with it like 8:9:10:11:12:13:14. Since Indian
music doesn't make use of any chords, let along big multi-voice
chords like this, it's not too surprising that ratios of 13 don't
play a role there.

> If one is talking
> about low ratios, 13/8 is certainly lower than many of the 5ths
based ratios
> one commonly finds, yet the 5ths based seem to be very common.

If you can tune one fifth, you can tune a whole chain of them, and
thus end up with some very complex 3-limit ratios. It's not that each
of the notes so produces will be in a meaningful acoustical
relationship with the fixed drone. But at least each of the fifths in
the chain will be pure, and this seems to be important even in some
of the Indian music I've heard, where two soloists play a figure in
parallel fifths or fourths. It's only when the music is a single
melody against a single drone (at 1/1) that you can ignore the
intervals between the pitch ratios and just look at the pitch ratios
themselves.

> And, another
> thought, between the fundamental and 32nd harmonic, there's a vast
world of
> possible interval relationships, yet it seems like none of the
maqam based
> musical cultures went there.

Really? We've just seen all kinds of ratios of 11 and 13 posted in
connection with Maqam music here!

> Actually, by not exploring the overtone series in music, we
(humans) have
> really limited ourselves for most of recorded history...and, I'm
sitting
> here wondering why, and I don't quite know.

The overtone series is useful for just a few kinds of music -- none
of which feature any harmonic movement -- including some that have
used it for ages. For example, the Tuvan throat singers.

> Unless, of course, the studies
> of numerology and number symbolism figure in somehow, which they
most likely
> do.

It seems to me that the overtone series is often used as a *result*
of numerology and number symbolism, so I don't quite know what you
mean.

> It would be interesting to see Indian maestros, for
> example, start composing ragas off the harmonics, and using those
untapped
> upper ratios...and, they're not even that up, the 32nd harmonic is
pretty
> low, in the sceme of things.

The way most sources give the Indian system, I can easily tune the
whole thing by ear, by tuning each note to a single note I've already
tuned (and starting with a 1/1). S. Ramanathan describes this process
in his 1973 article. But if you had a 31/16 in there, I'd have no way
of tuning it by ear, unless the whole tuning was a big harmonic
series and I could play all the notes at once. A very different
scenario.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...>
> wrote:

> > If one is talking about low ratios, 13/8 is certainly
> > lower than many of the 5ths based ratios one commonly
> > finds, yet the 5ths based seem to be very common.
>
> If you can tune one fifth, you can tune a whole chain
> of them, and thus end up with some very complex 3-limit
> ratios. It's not that each of the notes so produces will
> be in a meaningful acoustical relationship with the fixed
> drone. But at least each of the fifths in the chain will
> be pure, and this seems to be important even in some
> of the Indian music I've heard, where two soloists play
> a figure in parallel fifths or fourths. It's only when
> the music is a single melody against a single drone
> (at 1/1) that you can ignore the intervals between the
> pitch ratios and just look at the pitch ratios themselves.

This got me thinking about the fact that if you tune
a large enough chain of pythagorean ratios, you'll
find lots of xenharmonic-bridges which approximate
higher primes.

It inspired me to make a graph and table of some of
the pythagorean xenharmonic-bridges, to the primes
5, 7, 11, 13 ,17, and 19:

/tuning/files/monz/xenharmonic-bridges_pythagorean.gif

We've been talking in this thread about the 3==5 bridge
which is found at 3^-8, commonly called the skhisma or
schisma -- you can see it on the graph as a blue diamond
labeled "-8" at just under 2 cents.

The table lists the bridges which have the lowest generator
numbers. You can see from the graph that if you go higher
in the series of 3/2s, you can find closer approximations:

3^-67 is a much closer 3==7 bridge,
3^50 is a much closer 3==19 bridge,
3^45 is a slightly closer 3==5 bridge.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> This got me thinking about the fact that if you tune
> a large enough chain of pythagorean ratios, you'll
> find lots of xenharmonic-bridges which approximate
> higher primes.
>
> It inspired me to make a graph and table of some of
> the pythagorean xenharmonic-bridges, to the primes
> 5, 7, 11, 13 ,17, and 19:

What you have here is a 19-limit version of garibaldi. In terms of
rounded (standard) vals, it can be called 19-limit 41&53, and has
comma basis {120/119, 154/153, 171/170, 190/189, 209/208, 225/224}.