Peppermint 575.4 cents as 39/28

Hello, all.

This message is a response to a post about a "pepperoni"
temperament which I'm guessing is a variant on either
the original Kennan Pepper temperament with fifths at
704.096 cents, or Peppermint, my "rank 3" version as
it would now be called here with two 12-note chains
of these fifths at 58.680 cents apart for some pure
septimal minor thirds at 7/6.

In either the original Pepper temperament or Peppermint,
there is a diminished fifth (e.g. B-F) at 575.4 cents,
and someone raised the question of how I would describe
this interval. Of course, I agree that it's not too
far from 7/5, but would typically describe it as a
tempered version of 39/28 at 573.657 cents.

The 39/28 is the inversion of 56/39, a ratio occurring
in the medieval Near Eastern mode called Buzurg described
around 1300. While Safi al-Din al-Urmawi gives the general
form of Buzurg, one of his commentators (Anonymous LXII if
I'm correct) gives the form of a Buzurg pentachord as
1/1-14/13-16/13-4/3-56/39-3/2 (0-128-359-498-626-702 cents).

In a maqam context, 56/39 is equal to 4/3 plus 14/13 -- and
its inversion, 39/28, to 3/2 less 14/13, a small neutral
second favored by Ibn Sina and Safi al-Din, the latter of
whom tells us in one work that it is common to find 14/13
as the most favored neutral second step in frettings.

Another interval in this general range even closer to 7/5
is 88/63 or 578.582 cents, equal to 4/3 plus 22/21. It seems
the best JI equivalent for an interval of 577.7 cents I have
in MET-24, a system which, as a strict rank 3 temperament
in 2048-EDO, would have generators of 703.711 cents, 2/1,
and 57.422 cents.

This interval plays a very prominent role in a possible
historical Ottoman shading of a maqam called Penchgah,
open to at least two interpretations. One has the first
three notes as in Rast, but with T-J-T-J instead of
T-J-J-T, where T is a regular tone and J a neutral
second. Thus Rast would here be around 1/1-9/8-26/21-4/3-3/2,
and Penchgah 1/1-9/8-26/21-88/63-3/2. In MET-24, this comes
to 0-207.4-370.3-577.7-703.7 (in the 2048-EDO version). In a
1024-EDO version, not precisely regular because two different
fifth sizes are required (703.125 or 704.297 cents), the intervals
would be identical aside from the 3/2 step which would have one
of these two sizes.

A recommended 5-limit tuning of Ozan Yarman -- who seems open
to historical tunings also with wide neutral rather than 5-limit
thirds and sevenths -- is 1/1-9/8-5/4-7/5-3/2-27/16-15/8-2/1, if
I recall correctly. Here, interestingly, the tritone step of
1/1-9/8-26/21-88/63-3/2-22/13-13/7-2/1, as a JI interpretation
of 0-207.4-370.3-577.7-703.1-910.5-1073.4-1200 cents, is within
five cents of Yarman's 7/5.

In temperaments of this range around 704 cents, the realm of
"gentle temperaments" or "the 704-cent neighborhood," a regular
tone somewhere around 207-209 cents may be taken as a representation
of either 9/8 or 44/39. In the latter interpretation, the tone
is divided into a diatonic semitone close to 22/21 plus a chromatic
semitone close to 14/13. Thus we're likely to find a diminished
fifth (six fifths down) somewhere around 39/28 (3/2 less 14/13)
or 88/63 (4/3 plus 22/21).

Best,

Margo Schulter
mschulter@...

Hi Margo! Good to see one of your posts again. Things have been slow
here lately, but hopefully they're starting back up again.

So if I understand your thesis outlined here correctly, it's that you
think a medieval Eastern musician would perceive this interval as
being a compound interval, being attained by going down "a fourth"
from the octave, and then also going down an additional "neutral
second" of a specific size that they're conditioned to recognize? Kind
of like how I can hear a 12-EDO tritone as a combination of a fourth
and a diatonic semitone, or something (among other things).

-Mike

On Mon, Jul 23, 2012 at 11:48 PM, mschulter <mschulter@...> wrote:
>
> In either the original Pepper temperament or Peppermint,
> there is a diminished fifth (e.g. B-F) at 575.4 cents,
> and someone raised the question of how I would describe
> this interval. Of course, I agree that it's not too
> far from 7/5, but would typically describe it as a
> tempered version of 39/28 at 573.657 cents.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi Margo! Good to see one of your posts again. Things have been slow
> here lately, but hopefully they're starting back up again.

Please let me wish you the best with more activity!

> So if I understand your thesis outlined here correctly,
> it's that you
> think a medieval Eastern musician would perceive this interval as
> being a compound interval, being attained by going down "a fourth"
> from the octave, and then also going down an additional "neutral
> second" of a specific size that they're conditioned to recognize?

At least that's the kind of analysis suggested by the steps of a
Buzurg pentachord:

14:13-8:7-13:12-14:13-117:112

Here the 4/3 of the lower tetrachord (14:13-8:7-13:12)
has another 14:13 above it, producing 56/39. And, by
analogy, 39/28 could be viewed as 3/2 less 22:21, although
I'm not sure if this variant appears in any medieval Near
Eastern source.

> Kind
> of like how I can hear a 12-EDO tritone as a combination of a fourth
> and a diatonic semitone, or something (among other things).

The analysis seems analogous. It's another question, of course,
how much a given listener (medieval Near Eastern or other) could
distinguish by ear 39/28, 88/63, and 7/5, for example.

Incidentally, either 14/29 octave (around 579.3 cents) or 1/4-comma
meantone (around 579.5 cents) has an interval close to 7/5, and
even closer to 88/63. The enharmonic dieses are almost identical
(41.38 cents vs. 41.06 cents), of course respectively in the
positive and negative directions.

> -Mike

Best,

Margo