Tuning of Phrgian Mode

As for the tuning of the ecclesiastical Dorian (Greek Phrygian) mode,
I think it depends very much on how it is to be used.If one wishes
to harmonize it as if it were what Ellis called a "trichordal", the
best tuning would be 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1 as this scale may
be chorded with a major triad on 4/3 and minor triads on 1/1 and 3/2,
in Ellis's nomenclature, ma.mi.mi. See pages 275 and 460 of Helmholtz
for details. I personally do n�ind the 27/25 wide, epimeric "semitone"
of 133 cents offensive in the upper tetrachord, but some might.

One might also try the "dual" harmonies proposed by David Lewin,
Blainville, etc. and build the triads downwards. In this case, the best
tuning would be 1/1 10/9 6/5 4/3 3/2 5/3 16/9 2/1, with chords 2/1 5/3 4/3,
4/3 10/9 8/9 (16/9 in the lower octave) and 3/2 6/5 1/1.

If the melodic pattern, T S T T T S T, is more important, then as Lydia
stated, one may take it as a mode of the JI C major scale (which is
the C mode of the E mode (Greek Dorian) of Ptolemy's Intense Diatonic genus
16/15 x 9/8 x 10/9). The scale is thus 1/1 10/9 32/27 4/3 40/27 5/3 16/9
2/1, whose tuning may be adjusted to 1/1 9/8 6/5 4/3 3/2 5/3 16/9 2/1.

However, the scale may also be taken as a mode of the un-Greek natural
minor scale, generated by the tetrachord 10/9 x 9/8 x 16/15. In this case
the tuning is 1/1 9/8 6/5 27/20 3/2 27/16 9/5 2/1, the inversion of the
preceeding, and may be retuned as before to 1/1 9/8 6/5 4/3 3/2
5/3 9/5 2/1.

If the Didymos's diatonic 16/15 x 10/9 x 9/8 is employed, the scale
as a mode of the Greek Dorian is 1/1 9/8 6/5 4/3 3/2 27/16 9/5 2/1.
(There is less motivation to invert it as it is less harmonic than
Ptolemy's form.) One could even use the other Greek and Islamic
diatonic genera.

Or one could follow Safiyu-d-Din and the other Islamic theorists and
generate the scale from two identical tetrachords of the form T S T'
(or T' S T) where T and T' are any whole tone-like intervals and S whatever
completes the tetrachord. These Islamic theorists allowed all
permutations of the tetrachord to form scales. Thus from Ptolemy's and
Didymos's tunings we can get Phrygian-like scales with duplicated
tetrachords such as 1/1 10/9 32/27 4/3 3/2 5/3 16/9 2/1 and 1/1 9/8 6/5
4/3 3/2 27/16 9/5 2/1.

The "Pythagorean" tuning 1/1 9/8 32/27 4/3 3/2 27/16 16/9 2/1 has the
advantage of being a "classical" mode generated by the 256/243 x 9/8
x 9/8 tetrachord. Other reduplicated tetrachords could be tried as well.

Of course in ET's such as 12,17, 19, 22, and 31 that do not articulate
the syntonic comma, the melodic pattern T S T T T S T is readily available.


--John

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 17 Jun 1997 16:49 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA31421; Tue, 17 Jun 1997 16:49:15 +0200
Date: Tue, 17 Jun 1997 16:49:15 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA31406
Received: (qmail 9126 invoked from network); 17 Jun 1997 14:49:05 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 17 Jun 1997 14:49:05 -0000
Message-Id: <199706171047_MC2-18A6-4EA9@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

On Fri, 20 Jun 1997, it was written:
> There will be no counterexamples because Paul's guess is correct. In
> general, it is only possible to construct such chords for composite
> limits (such as, in the snipped example, 9). 7 being prime, it is not
> possible.

Once again I correct myself: 7 is unusual in this regard because it is
smaller than 9, the smallest odd composite; actually in the example Paul
gave the chord overall was 15-limit, while each individual interval in
it was 9-limit, which shows that strictly speaking it _is_ possible for
larger primes, such 11 and 13. However, it would not be possible to
construct a chord which included a (say) 11-limit interval, but none
higher, such that the chord overall had a higher limit.

--pH http://library.wustl.edu/~manynote <*>
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 23 Jun 1997 10:38 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01702; Mon, 23 Jun 1997 10:38:21 +0200
Date: Mon, 23 Jun 1997 10:38:21 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA01700
Received: (qmail 11646 invoked from network); 20 Jun 1997 21:20:53 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 20 Jun 1997 21:20:53 -0000
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

On Tue, 17 Jun 1997, Paul Erlich wrote:
> I wonder if there are any 7-limit analogues to these chords (i.e., chords in
> which each interval is within the 7-limit but the chord as a whole is in a
> higher limit). I think the answer is no. Anyone care to come up with a
> counterexample?

There will be no counterexamples because Paul's guess is correct. In
general, it is only possible to construct such chords for composite
limits (such as, in the snipped example, 9). 7 being prime, it is not
possible. I have discovered a truly marvelous proof of this, which this
bandwidth is unfortunately too narrow to contain . . .

(Actually, it's not particularly marvelous; I just haven't figured out
how to translate it from my muttering and making henscratches to myself
to something a sentient being would understand. If anyone wishes to
challenge me on it, though, I'll give it a go.)

--pH http://library.wustl.edu/~manynote <*>
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 23 Jun 1997 11:07 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04343; Mon, 23 Jun 1997 11:07:45 +0200
Date: Mon, 23 Jun 1997 11:07:45 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA04339
Received: (qmail 17199 invoked from network); 22 Jun 1997 23:55:37 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 22 Jun 1997 23:55:37 -0000
Message-Id: <33d7b09b.400615185@kcbbs.gen.nz>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

On Fri, 20 Jun 1997, it was written:
> However, it would not be possible to
> construct a chord which included a (say) 11-limit interval, but none
> higher, such that the chord overall had a higher limit.

Once _again_ I correct myself. (Who is this Hahn guy anyway--he keeps
screwing stuff up!?) The highest-limit interval in the chord
18:22:24:33 is a 11/9, but the chord as a whole is 33-limit.

(But Paul E.'s conjecture is still true for the 7-limit! Honest. Have
I ever been wrong? 8-)> )

--pH http://library.wustl.edu/~manynote <*>
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 23 Jun 1997 22:54 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05066; Mon, 23 Jun 1997 22:54:18 +0200
Date: Mon, 23 Jun 1997 22:54:18 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA05070
Received: (qmail 25441 invoked from network); 23 Jun 1997 20:54:01 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 23 Jun 1997 20:54:02 -0000
Message-Id: <009B63B0C2C11C46.5CEC@vbv40.ezh.nl>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

From: "Paul H. Erlich"

>> (i.e., chords in
>> which each interval is within the 7-limit but the chord as a whole is in a
>> higher limit).

>I know what the first bit means, but I don't know any other definition of
>the limit of a chord. Could someone explain this to me so that I can set
>about (dis)proving it?

I know this whole business can be very confusing, but let me try to
explain this step by step.

First of all, as the title of this message indicated, I am talking about
odd-limit theory, as in Partch's original definition. The limit is
defined as the highest number needed in the representation of the
interval or chord after all factors of two are discarded (octave
equivalence is the justification for discarding them). This also agrees
with Ralph D. Hill's observation that every new odd number adds a new
quality, as well as a higher degree of tension, to a chord. I used to
believe (though I'm not sure in light of my recent obsevation here) that
a given style of music established a certain odd limit as the norm;
higher-limit chords will sound dissonant and need to resolve, while
lower-limit chords will sound incomplete. This is certainly applicable
to Common Practice and the 5-limit; 3-limit pentatonic examples are
easily constructed, and I believe I've taken a first step toward
realizing the 7-limit case (see my upcoming Xenharmonikon paper).

Now let us state what we mean by "representation." For an interval, this
is just a lowest terms ratio p/q; we can define this to be the frequency
ratio, in which case the period (or wavelength, string length, . . . )
ratio in lowest terms is just q/p. Either reprentation implies the same
odd limit for the interval.

Following Partch, we will define integer multiples from 1 to n of a
frequency to be an "n-otonality," and integer multiples from 1 to n of a
period (or wavelength, string length, . . .) to be an "n-utonality." So
we can define the otonality-limit as the lowest n for which a chord
(with all factors of 2 removed) is a subset of an n-otonality, and
similarly for the utonality-limit.

For a chord of 2 notes, the limit of the 1 interval in the chord is the
same as the utonality-limit and the otonality-limit of the chord. For
more than two notes, the otonal or utonal representations may differ in
complexity, in which case the simpler one is often chosen and the chord
is deemed as appropriate. For example, 10:12:15 is a "5-limit utonality"
and 4:5:6:7 is a "7-limit otonality." As Paul Hahn has pointed out, the
3-limit, 5-limit, and 7-limit are particularly easy cases to deal with,
since if all intervals of a chord belong to one of these limits, the
chord itself will belong to that limit either as an otonality or as a
utonality.

However, the 6th chord 12:15:18:20 has an otonality-limit of 15, a
utonality-limit of 15, but no interval is beyond the 9-limit. Kami
Rousseau's 6th chord 14:18:21:24 has an otonality-limit of 21, a
utonality-limit of 21, but again no interval is beyond the 9-limit. Paul
Hahn's example 18:22:24:33 has an otonality-limit of 33, a
utonality-limit of 33, but no interval is beyond the 11-limit.

It appears then that there is a new class of "saturated" chord beyond
the otonal and utonal chords that Partch envisioned. By saturated I mean
that no new notes can be added without increasing the odd-interval-limit
of the chord. I think this finding merits investigation as a natural
extension of Partch's work.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 23 Jun 1997 23:01 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05735; Mon, 23 Jun 1997 23:01:23 +0200
Date: Mon, 23 Jun 1997 23:01:23 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA05728
Received: (qmail 26044 invoked from network); 23 Jun 1997 20:59:26 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 23 Jun 1997 20:59:26 -0000
Message-Id: <009B63B0DDB21546.5CEC@vbv40.ezh.nl>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

From: "Paul H. Erlich"

Thank you, Graham, for the nice proof of my conjecture, that aside from
the otonal and utonal 7-limit tetrads, and subsets thereof, there are no
chords with only 7-limit intervals. Thank you also for explaining why
things go astray in the >7 limit case. Now, can we come up with a
complete enumeration of "saturated" chords; i.e., chords which consist
only of n-limit intervals but cannot retain this property if any note is
added to them? For the 5-limit, the major and minor triads are the only
examples. 7-limit, it's just the two tetrads. For the 9-limit, the
otonal and utonal pentads work, as well the two 6th chords discussed in
this thread, so there are at least 4 saturated 9-limit chords. A
complete enumeration through the 11-limit, at least, should be of great
interest to users of Partch-type JI.

-Paul Erlich

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sun, 29 Jun 1997 01:14 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA31885; Sun, 29 Jun 1997 01:14:32 +0200
Date: Sun, 29 Jun 1997 01:14:32 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA31803
Received: (qmail 12966 invoked from network); 28 Jun 1997 23:14:26 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 28 Jun 1997 23:14:26 -0000
Message-Id: <3.0.2.32.19970628160839.00699f0c@adnc.com>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu