Yahoo Tuning Groups Ultimate Backup mills-tuning-list biblio

Paul wrote:

'' I don't dispute the musical validity of this construct. In fact it
supports
everything I've been saying about odd numbers and about including a factor
of 1 in the CPS. It's just that in graphing this along with
2(1,3,5,7,9,11),
you'd have no idea that 1*9 is the same pitch as 1*3*3, if you have a
separate axis for 9. ''

I find that an attractive part of the discipline of working in a
(non-stellated) CPS is not using compound factors as supplementary tones -
as in your example, using 1*9 as 1*3*3. It may sometimes be handy to have
''extra identities'' in this way, but in fact, placing compound factors on
their own axes makes the whole structure of the set more transparent, and
makes a whole range of operations upon the set considerably more clear. I
enjoy, for example, composing a bit of music and then reassigning the
factors, to generate a variation where the contours and intervals are
totally different, but the set properties maintained.

(By the way, Erv Wilson's elegant graphs of some CPSes - some are abstract,
those with numbers assign compound factors to separate axes - are found on
the covers of Xenharmonikon 1, 4, IX, XV. For the cover of Xenharmonikon
XII, Wilson has assigned tones to the vertices of a Penrose tiling.)

I wrote:

There are some musical contexts (and some temperaments as well -
>the tuning of the TX81Z comes to mind)

You asked:

''What's this?''

A resolution of 1.56 cents. So that a 3/2 is approximated by an offset of 1
unit, but a 9/8 by 3 units. So for the TX81Z the best 9/8 is not equal to
the sum of 3/2s.

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🔗gbreed@cix.compulink.co.uk (Graham Breed)

> [I] get the feeling that you're doing some interesting work, but you're
> speaking a language too different from the majority of the tuning list
> to engage many of us with your ideas. The last message in particular was

This could well be the case. I worked out my ideas, including the
basics of interval matrices, before I joined the tuning list and so
before I realised there were other people interested in the same
stuff as me. Naturally, I worked out my own terminology to explain
things to myself. I'm trying to explain myself using the standard
terms where possible, but it isn't easy because most of them don't
even seem to have precise meanings, and some of my concepts also
appear to be original.

> The last message in particular was
> very obscure. Can you try doing some hand-holding for us and explain
> what you mean by "this works," "that doesn't work," a "0-comma scale,"
> and "this temperament"?

Okay. By "this works" I meant that a 2-D scale could be
constructed such that every 5-limit interval can be defined on
that scale. I originally added the condition that a basis should
be a tempered 2-3 plane.

A 0-comma scale is a tempered scale with a just octave and another
just interval, usually a perfect fifth.

"This temperament" is the doubly positive temperament that Paul
Erlich worked out. I think this can be called doubly positive
temperament, as it covers all the ETs usually described as doubly
positive. It is defined using the following matrix equation:

(1 0)
H' = (0 1)H'
(5.5 -2)

As this matrix involves a fraction, the tempered 2-3 plane is not
a basis. The 0-comma scale involves a just fifth and a tritone
equal to 6 steps in 12 tet. By my original criteria, it "doesn't
work" but non-Pythagorean bases can be chosen. For example:

(s) = (-0.5 1)H'
(r) ( 3 -5)

> Feel free to review you matrix/determinant stuff.

A full description of interval matrices should arrive with this
post. Even people with a non mathematical background should be
able to follow most of it. Please ask if there's anything you
don't understand. This post uses terminology from that one. I've
brushed determinants under the carpet for the time being.

> It may actually be valuable for understanding ancient Hindu
> music. Then let's talk 7-limit.

I'm more interested in making new computer music than describing
old music, but I am nevertheless interested in what scales are
used by different peoples at different times. Particularly when
those scales can be defined economically by mathematics, ie JI and
ETs. FWIW, I've now worked out 3 different 7-limit approximations
to doubly positive temperament.

SMTPOriginator: tuning@eartha.mills.edu
From: gbreed@cix.compulink.co.uk
Subject: Matrices for beginners
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🔗gbreed@cix.compulink.co.uk (Graham Breed)

> I maintain that any systematic inharmonicity in brass instruments, which
> is either nonexistent or extremely small, has little or nothing to do
> with the failure of the resonant modes of the instrument to form an
> exact harmonic series.

Now you're getting suitably equivocal. Before you were saying that no
inharmonicity could possibly occur, which places too much faith in the
simplified models. Ideally, we should develop more complicated models to
give a quantitative estimate of the (upper limit of the) inharmonicity.

Inharmonicity will cause problems with JI whether or not it's systematic.

Any noise in the input will cause peaks at the resonant modes. The result
will be inharmonicity. The level of that inharmonicity can be estimated
from the noise level of the input. I don't know of a good model of the
human embouchure (that's the word, isn't it?) although there may be one
that is too complicated to escape from the academic literature.

If the overtones are not centered at the resonant peaks, the amplitude of
each overtone on output will depend on it's pitch. That means, where
vibrato is present, the corrected waveform will not be perfectly periodic.
This will probably show up as a subharmonic of the heterodyned overtone.
Whatever, it isn't important for tuning as the effect will be swamped by
the vibrato itself.

I've had a look at some books on this today. The pitch from the
embouchure is determined by the mode of resonance, but overtones can still
arise. Also, the nature of the mouthpiece is such that noise may not
propagate to the conical bit. Another thing is that it is not only brass
instruments that have conical tubes. Examples of recorded resonances were
given for a range of wind instruments, and they're all significantly off
from the integer ratios. This means that the flute, where the initial
sound is not periodic, should show the most inharmonicity. Fortunately,
this is one case where experiment agrees with theory, at least on the
basis of the one flute sample analysed and explained here recently. I
couldn't find any discussion in books of the inharmonicity of the sound of
any instruments.

The resonances of flutes were also given relative to 12-equal. They were
up to 20 cents out! It's only the flautist's skill that adjusts them to
the desired scale.

🔗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.

🔗Carl Lumma <clumma@...>

>The 11/8 in 72tET is 1 cent off. If you really think this discrepancy
>makes certain modulation effects impossible, you should (a) perform some
>listening experiments and (b) remember that it was Partch himself who,
>when told that JI made ordinary modulations impossible, allowed the
>22-cent syntonic comma "correction" which would not be noticeable enough
>to impede the sense of modulation.

You might be right. I don't have enough experience with 72 to know. But
I'll keep it in mind. I do know that higher-level consistency is a
beautiful concept that negotiates many factors to create a measure of how
well an equal temperament can represent JI. Obviously, it breaks down
somewhere, when the step size gets very small. But probably more around
612 than 34 or 69.

>Sure, but as far as strictly defined notated pitches, didn't he stick to
>43?

He claimed he never stuck to anything. I think it is most accurate to say
that Partch composed from an infinite set of pitches, but seldom needed
more than 43 at a time.

>No room for transposed tonality diamonds on 11 identities.

Not in 43 pitches, no.

>Third irrelevancy: I consider plain old, simultanoeus-sounding, level-1
>consistency to be important for ETs up to 34, and possibly 69. Beyond
>that, the tones are less than 1% different from one another, so in most
>musical circumstances what you have is tantamount to a continuous
>spectrum of pitch.

I don't buy it.

Carl

biblio

🔗COUL@ezh.nl (Manuel Op de Coul)

🔗Daniel Wolf <106232.3266@...>

🔗gbreed@cix.compulink.co.uk (Graham Breed)

🔗gbreed@cix.compulink.co.uk (Graham Breed)

🔗gbreed@cix.compulink.co.uk (Graham Breed)

🔗Carl Lumma <clumma@...>