Thanks, pentagram

I'd like to thank everyone who sent me ordering information for microtonal
recordings. I've collated it and sent it off to the requester.

"Pentagrama" is used in Spanish to refer to the 5-line staff and staves
with more lines may be called "octograms" (8-lines), etc.(
Sabat-Garibaldi).

--John



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From: Manuel.Op.de.Coul@ezh.nl
Subject: Announcement of radio programmes
PostedDate: 19-08-97 12:08:38
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The description in terms of 22 srutis dates to something like the 11th
century. Certainly the evidence suggests that India achieved the 5-limit
before the West.

Graham, I really don't understand what you're trying to do here,
although it seems that my bringing in a JI interpretation of the
historical description of the sruti system was somehow helpful to you. 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
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"? Feel free to review you matrix/determinant
stuff. It may actually be valuable for understanding ancient Hindu
music. Then let's talk 7-limit.



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From: alves@orion.ac.hmc.edu
Subject: Srutis
PostedDate: 25-08-97 22:11:43
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> No Indian theorist I've read ever claimed 22 equal has anything
> to do with Indian music. Some mention a 22 note scale, but
> most emphasise that it is _not_ equally tempered.

You may wish to consult the most famous of all Indian musical theorists,
Sarngadeva, who says (I quote from Levy, Intonation in North Indian
Music, p. 12:)

"Of them, the first [note] must be made as having the lowest possible;
the second must be made to have a slightly higher sound, without any
intermediate sound possible, since no sound could be heard in between
these two srutis."

He goes on to instruct that the same procedure should be used to form
the entire scale of music. To me, who certainly have no bias in favor of
22-tone equal, this sounds like 22-tone equal.

Innumerable other authorities, both traditional and modern, say much the
same thing. It is true that other authorities give - very slightly -
unequal scales (Ellis, for example), and one, the quite untrustworthy
Westerner Danielou, constructs a highly unequal 22-tone
pseudo-pythagorean system, apparently out of thin air. Ravi Shankar, as
I recall, says he uses 22 pitches in the octave - I don't remember
whether equal, nearly equal, or unequal. Normally there would be no
melodic difference between using a slightly unequal and an equal
temperament, but since the tuning degree of 22-tone equal is at the edge
of the critical zone of 55-60 cents, it could possibly make some
difference in this case.

My own opinion, however, is that while the 22-tone equal does seem to
enjoy some instrumental use in India, the evidence - very scanty,
admittedly - supports 19-tone equal for Indian vocal music. So far as I
can determine - but I am no expert - Indian instrumentalists use a wide
variety of tunings, so if you are confused, join the club. The experts
seem to think that the Indians tried to use something very like 22-tone
equal in antiquity, but under Arab influence (ultimately Greek
influence?) gave it up in favor of pythagoreanism. The attitude of the
typical modern Westerner (at least until very recently) seems to be: why
don't those crazy Indians use God's gift to humanity, 12-tone equal
temperament... uhhh... they probably do... they just don't know it...


SMTPOriginator: tuning@eartha.mills.edu
From: Gregg Gibson
Subject: Reply 2 to Graham Breed
PostedDate: 14-12-97 02:42:55
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Unequal tunings waste the melodic substance no less surely than equal
temperaments whose tuning degrees are either much narrower or much wider
than 60 cents or so. A 19-tone unequal tuning would result in
appreciably more intervallic confusion, given the fact that the 19-tone
equal tuning degree is close to the limen. Modulation would also be made
more difficult, and certain consonances would be too deteriorated for
use.

Gregg Gibson said:
> > To give some idea of just how immense this universe is - and remember
> > these are _real_, usable resources, not purely theoretical, aurally
> > imperceptible variations - compare the number of modes (I here use the
> > term loosely merely to indicate a collection of seven notes used in a
> > melody) consisting of 12 notes taken 7 at a time:
> >
> > nCr where n = 12 & r = 7
> > = 792
> > with the number of modes consisting of 19 notes taken 7 at a time:
> > nCr where n = 19 & r = 7
> > = 50,388

Graham Breed replied:
>
> Really, you should define modes relative to a tonic, so the numbers
> get smaller. With 12, you're choosing 6 from 11, which gives 462
> modes. 19C6 is 12376. The contrast is still striking. Including
> all the transpositions, you get 462*12=5544 and 12376*19=235144.

Thanks very much for your correction regarding the method of finding
numbers of 12-tone versus 19-tone modes. You are quite right that the
tonic must be omitted from the calculation. The corrected numbers would
be nCr (n=11, r=6) = 462 and nCr (n=18, r=6) = 18564 (not 12376, you
must have used n=17), or a factor of about 40 to 1. Since I am concerned
here with modes, not keys, the transpositions I neglect.

Gregg Gibson said:
> > ... but the resulting modes conflate a number of
> > aurally distinct modes into single, neutral modes, and lack much of the
> > distinct melodic character of the 19-tone versions.

Graham Breed replied:
>
> Which modes cease to be distinct? Of course, the melodic character
> will be different in different tunings, but that is not to say one is
> superior to the others. In good old 31 equal, you get the added bonus
> of septimal chords, consonant or otherwise.

I have enumerated all the possible diatonic and chromatic modes in
12-tone equal and 19-tone equal, using tree diagrams. I can assure you
that many chromatic modes distinct in 19-tone equal are merged into a
much smaller number of such modes in the 12-tone equal. The exact number
of these modes depends on what criteria one uses, e.g. numbers and
character of consonances, number of consonant chords, inclusion of
septimals, etc. I will post these modes in due course - meaning when I
have a solid day or two free for the purpose.


SMTPOriginator: tuning@eartha.mills.edu
From: Gregg Gibson
Subject: Reply to John Starrett
PostedDate: 16-12-97 08:47:12
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>Unequal tunings waste the melodic substance no less surely than equal
>temperaments whose tuning degrees are either much narrower or much wider
>than 60 cents or so.

I think you need to pick up copies of CDs such as Easley Blackwood's, or
Neil Haverstick's two CDs. They have numerous examples of very effective
melodies that require 60-cent or greater resolution.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: stretched-octave eq temps
PostedDate: 16-12-97 18:44:10
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>I mentioned this business with conical tubes to show that there is
>a quantifiable physical origin for inharmonicity.

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.

>The periodicity
>of the sounds presumably defaults to the player's ability to
>blow a raspberry.

Right.

>Does this mean the resulting overtone series
>is constructed from undertones of the original vibration?

Can you rephrase/clarify this question?

>>> 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.

Sure. I'm glad you think the prime intervals don't have to be prime
numbers. It bears pointing out that depending on which "prime intervals"
you choose, you will end up with different approximations to the rest.
Different tunings will have different "prime intervals" which lead to an
optimal set of compatible approximations -- for the 5-limit, for
example, some ETs will yield greater accuracy if 5/4 and 3/2 are chosen
as the prime intervals, some if 5/3 and 3/2 are chosen, and some if 5/4
and 5/3 are chosen. Consistency simply means that all choices lead to
the same result. That's how I interpreted your statement "For exactness
. ." above.

>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

Of course that's only correct if the worst error is 1/3 of the step size
or less (Paul Hahn originally pointed this out).