alternate tuning for the Korg O5R/W

I've got a very specific question that only someone familiar with
microtonality would understand (and hopefully appreciate). I have a Korg
O5R/W syth module that hooks into an old DW-6000 synth. The O5R/W allows
for microtonality, but it only allows you to alter the 12 chromatic steps
of Western harmony (and automatically tunes every octave of the keyboard to
the specs that one specifies). If you want to tune a synth keyboard for
more than 12 steps you are out of luck. I read a book by Scott R. Wilkinson
called TUNING IN. He made reference to an article by Carter Scholz that
appeared in 2/86 copy of KEYBOARD magazine; page 49. He showed how one get
any keyboard to play in microtones by writing a program in Pascal and
having it interface with a Roland MPU@1 and a Yamaha DX7. The computer
waits in an infinite loop in a standby mode and whenever one presses a key
on the keyboard, it triggers the correct pitch bend midi data that is
written in the program. Is this possible with my Korg O5R/W? Do you know
of anyone who may be of help.



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Here is one approach to Partch:

(1) Assume for the moment that he used stable and accurate instruments and
that all intervals in the book expressed as ratios are tuned precisely. (In
fact, he did a pretty good job under very difficult material
circumstances).

(2) Partch's _limit_ idea is based upon the series of odd whole numbers,
and his particular expansion of tonal resources was based upon adding
higher odd integers; some in the tuning community prefer to speak of
_limit_ in terms of highest prime numbers, some forego the term limit
altogether in favor of _genus_ (or set of generating factors). In Partch's
case, it is sufficient to think of a limit as the defined by the highest
odd number (x 2^n) used in either a numerator or denominator. The ratio
16/15, then, has to wait for a 15 limit to appear in a _diamond_ (Partch
includes it as a _secondary ratio_, to both fill in the gaps in his scale
and to allow some modulations, as well as an implied 15 limit tone
(although he called it _inharmonic_).

(3) As a music theory teacher, you may have probably noticed already that
Partch's harmonic scheme has three principals: expansion through a kind of
stacked third harmony, common tone chord modulations, and intervallic
inversion. (To my ears, this puts him smack in the mainstream of harmonic
practice and theory in the first half of this century.) In particular, the
use of inversion will become very important to building his diamond, and
it is critical that he identifies inverted harmonic structures by their
_highest_ tone.

(4) There are a couple of ways of looking at the diamond. Here's one
without ratios: build a Major triad on c: c e g. Now invert these intervals
and build a minor triad _down_ from c: c ab f. Next build similar minor
triads down from e (e c a) and g (g eb c)and Major triads up from ab (ab c
eb) and f (f a c). Note the shared tones between the triads, and the fact
that all contain c, but in each chord in a different function. This
arrangement is the diamond:

G
E Eb
C C C
Ab A
F

And transposed to G is the same as Partch's 5-limit diamond.

A more generalized way of talking about the diamond is to think in terms of
a set of intervals multiplied by its inversion (a technique familiar in the
- albeit 12tet - music of Pierre Boulez), so we could map it onto twelve
pitch classes as:

0 4 7

8 0 3

5 9 0

which starts to look like the corner of a twelve-tone row box (who says
that Schoenberg and Partch didn't have anything in common?).

In Partch's case, we are talking about Just intonation, however, so these
pitches can be expressed as the ratios of small whole numbers in relation
to a given 1/1, or tonic.
In the five limit diamond, there are six triads, and each has a distinctive
odd whole number (again times 2^n, to make it all fit in an octave) common
to either the numerators of the minor triads (_utonalities_) or the
denominators of the Major triads (_Otonalities_). For example, locate the
minor triad with 5's in the numerator (5/4, 5/51, 5/3). Ignoring this
common tone, now, the denominators will show the _function_ of each tone in
the triad, which Partch called the _identity_: 5/4: 4 is 2^2, thus a 1
_identity; 5/5: 5 is the 5 _identity_; and 5/3: 3 is the 3 identity.

Let me know if this was of any help...

Daniel Wolf, Frankfurt

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> What is a 5 limit (or any "n" limit for that
> matter)? ... Is the highest prime 5 that any number in numerator or
> denominator can be divided into?

Yes, either numerator or denominator.



> What about 16/15. This ratio's numerator and denominator are
> divisible by primes that are 5 or less.

16:15 is a 5-limit harmony (a half step in particular.



> Does he choose his scale based on
> what sounded good, on how close to just he could get?

Partch's tuning system, called "Monophony", IS just, rather than close
to it. He chose it based on many goals, including what sounded good I'm
sure, but also on provisions for modulation, and other goals.


I'd probably better not say much more, because I don't claim to be an
authority on Monophony, and because there are many others on the list who
know more.

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> From: malkin@iwaynet.net (David Malkin)
> Subject: Genesis of a Music question
>
> I am confused with Harry Partch's description of the 5 limit in Chapter 7
> of "Genesis of a Music." What is a 5 limit (or any "n" limit for that
> matter)? Is it a scale that only includes ratios with a 5 in numerator or
> denominator? Is the highest prime 5 that any number in numerator or
> denominator can be divided into?
> He uses a scale as follows: 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1
> on page 109.
> What about 16/15. This ratio's numerator and denominator are
> divisible by primes that are 5 or less. Does he choose his scale based on
> what sounded good, on how close to just he could get?

I'm responding only on memory, not having Partch's "Genesis of a Music" here to refer to. I'm sure others will
pipe in with refinements if needed.

Partch used the idea of "identities" which are odd numbers in either the overnumber or undernumber position of
the ratios which describe pitches ignoring octave transposition. A limit sets the highest identity he might
use to generate a pitch set (ignoring octave transpositions). Identities are not factored together as in 3*5,
so 16/15 is not in 5-limit, but requires 15-limit or higher.

In my idiolect, a limit is different. A limit is the highest prime factor needed to analyse a given just pitch
set. In this meaning, the interval 16/15 is a 5-limit interval, as is 243/125, etc.

> On page 110 he speaks of the coexistence of Major and Minor and
> draws the first Tonality diamond in the book. Why do these major and minor
> sounds (ie Otonalites and Utonalities) result. It seems like they just
> appear out of thin air.
> What is a Numerary Nexus? His definition on page 72 is confusing to me.

A Numerary Nexus is the number you keep the same in either the undernumber or over number while iterating the
identities in the other position to generate a tonality.

For instance, to generate the Otonality of 3, iterate the identities in the overnumber position while keeping
3 as the Numerary Nexus like so: 1/3, 3/3, 5/3, 7/3, etc. up to your limit.

An Otonality is so called to match with "o"vernumber, since it is the set of overnumber identities which share
a Numerary Nexus. All Otonalities follow the form 1/n, 3/n, 5/n, etc. (n being the numerary nexus).

An Otonality is roughly analogous to major tonality, but may be closer to major chord quality, since we
usually think of a major tonality as including the various diatonic chords available from a major scale,
whereas Partch's tonalities move around with the individual chords, so to speak. The analogies drift apart
rapidly with higher ratios. The septimal dominant (or whatever you want to call it) 4:5:6:7 would be a purely
otonal chord, but would probably not be thought of by most as "major". Otonality is really an unique term
which must be savored for its own theoretical functionality.

Otonality is distinct from the overtone series, with which it is sometimes confused. The o in Otonality does
not stand for overtone. An overtone series includes all positive integers in the numerator but does not
include all octave transpositions of those pitches, while an Otonality includes only the odd overnumbers, but
implicitely generates all octave transpositions of those pitches.

An Utonality (u for "u"ndernumber) is the set of pitches you get from iterating the identities in the
undernumber position, while keeping the same Numerary Nexus in the overnumber, of form n/1, n/3, n/5, etc.

For example, the Utonality of 5 with 11 limit is: 5/1, 5/3, 5/5, 5/7, 5/9, 5/11.

Utonality is roughly analogous to minor tonality or minor chord quality. Utonality is distinct from the
undertone series.

The pitch set you quoted 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1 is what you get when you permute the
identities in both the overnumber and the undernumber up to a limit of 5, remove the redundant instances, and
express them in smallest ascending form (6/5 instead of 3/5 for instance), then throw in a 2/1 for the sake of
"scaleness". The raw set would be:

1/1 3/1 5/1
1/3 3/3 5/3
1/5 3/5 5/5

The raw set may be what a tonality diamond is; I don't remember.

Matt Nathan

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