Looking for Rayna Software

I'm looking for a PC version of David Rayna's RTMP program. Please respon=
d
privately.

Daniel Wolf
DJWOLF_MATERIAL@COMPUSERVE.COM

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End of TUNING Digest 1558
*************************

On Wed, 21 Oct 1998, M. Schulter wrote:
> Possibly less well known, the septimal schisma of 33554432:33480783 or
> about 3.80 cents is a bridge to a quasi-7-based world of
> "superefficient" cadences and third-tone steps. It is the difference
> between the Pythagorean comma and the septimal comma (64:63, about
> 27.26 cents) which separates basic Pythagorean intervals from their
> 7-based counterparts.

Oddly enough, I was just poking around in some old tuning list messages
today, and guess what I found:

| Date: Mon, 13 Mar 95 08:52:11 -0800
| From: Manuel Op de Coul
| Reply-To: tuning@eartha.mills.edu
| To: manynote@library.wustl.edu
| Subject: Other harmonic 7th comma
|
| Bosanquet has written that 14 fifths downwards (the Pythagorean double
| diminished octave) is very close to the harmonic seventh. Is the
| comma belonging to it, 33554432/33480783 = 2^25 * 3^-14 * 7^-1 =
| 3.8041 cents ever called Bosanquet's comma, does anyone know?
|
| Manuel Op de Coul coul@ezh.nl

Another interval that might be a candidate for the name "septimal
schisma" is the 2401/2400, about .72 cents. It is the difference
between the 50/49 and the 49/48, both intervals which result fairly
directly from septimal voice-leading.

--pH http://library.wustl.edu/~manynote
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

NOTE: dehyphenate node to remove spamblock. <*>

The interval 33554432/33480783 has also been named by Eduardo Sa'bat,
Beta 2. Septimal schisma seems a good name to me.
The bridges from Margo's post are easily found with Scala. It can take
all the combinations of two intervals and check whether a given
interval (some comma for example) is a sum or difference of them. The
list of interval names intnam.par that is provided can be used for
that. So do:

load intnam.par
show combination 33554432/33480783

In this case, only differences are found:

5120/5103 - 32805/32768
Beta 5 - schisma
64/63 - 531441/524288
septimal comma - Pythagorean comma
15625/15309 - 34171875/33554432
great BP diesis - Ampersand's comma
134217728/129140163 - 28/27
Pythagorean double diminished third - 1/3-tone
8/7 - 4782969/4194304
septimal whole tone - Pythagorean double augmented prime
16777216/14348907 - 7/6
Pythagorean double diminished fourth - septimal minor third
9/7 - 43046721/33554432
septimal major third - Pythagorean double augmented second
2097152/1594323 - 21/16
Pythagorean double diminished fifth - narrow fourth
32/21 - 1594323/1048576
wide fifth - Pythagorean double augmented fourth
67108864/43046721 - 14/9
Pythagorean double diminished seventh - septimal minor sixth
12/7 - 14348907/8388608
septimal major sixth - Pythagorean double augmented fifth
8388608/4782969 - 7/4
Pythagorean double diminished octave - harmonic seventh
27/14 - 129140163/67108864
septimal major seventh - Pythagorean double augmented sixth
1048576/531441 - 63/32
Pythagorean diminished ninth - octave - septimal comma

Manuel Op de Coul coul@ezh.nl

In TD # 1559, 20 Oct 1998, Margo Schulter wrote:

> An open question: might an extended Pythagorean
> tuning with its contrasts of basic 3-based,
> quasi-5-based, and quasi-7-based intervals in
> some way have a kinship to the three genera
> (diatonic, chromatic, enharmonic) or to Guido
> d'Arezzo's three hexachords (soft, natural, and
> hard)?

I wish I had read this while the discussion on it
was hot, because I've been thinking along exactly
these lines.

In my book, I demonstrate how it would have been
possible for the ancient Greeks to formulate both
the chromatic and enharmonic genera in their tuning
of the famous "lyre with 12-strings" for which
Timotheus was ridiculed, by retuning the strings
continuously by 3:2 "5th"s and 4:3 "4th"s until
the proper intervals were reached.

I'm not well-read on the following idea, but it's
been documented that the Babylonians (c. 2000 BC)
tuned by ear by means of consecutive 3:2 "5th"s
and 4:3 "4th"s, and I think it's quite possible
that all four of the great ancient centers of
civilization -- China, Egypt, the Indus valley,
and Mesopotamia -- could have either independently
"discovered" extended 3-Limit tuning or adopted
it thru trade from whichever developed it first.
This in turn would have been incorporated into
Greek practice, as the Greeks imported much of
the scientific knowledge of all but China.

Knowing first-hand how far a curious mind will
go with what he thinks is a great new idea, I
think that when any brilliant ancient theorist
first discovered the principle of generating new
notes from a succession of 3:2s, it would be
highly unlikely that he would stop after the 12th
just because it sounded so close to the starting
pitch. Indeed, the fact that it _was_ so close
and yet not the same certainly would have impelled
him on to see what other kinds of relationships
there were. He probably wouldn't have stopped
until he just plain got tired of doing all that
calculating.

If he had the same kind of inquisitiveness that
Darreg, Partch, and Schoenberg had, and many of
us have, he would have written pieces that
utilized all those unusual notes, to see just
what that would sound like. So maybe there
was some ancient Greek walking around with a
toga that said "quasi-7-Limit Rules!".

- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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