Yahoo Tuning Groups Ultimate Backup tuning Unison vectors (Peppermint, 72-EDO, etc.)

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Hello, there, everyone, and some musical experimentation has led me to
share a few impressions about unison vectors, and also the kinds of
diesic, commatic, or kleismic vectors that might be observed in a
tempered system.

Here I'll take as an example Peppermint 24, because this is one of my
favorite systems and observes an interesting range of vectors (some of
which might be yet to be discovered, of course, as people try new
intervals and come upon new relationships). There will also be some
comparisons with 72-EDO and other systems such as paultone tunings.

Peppermint 24 consists of two 12-note chains in the Wilson/Pepper
temperament based on a ratio of the Golden Section, or Phi (~1.618),
between the larger chromatic semitone and the smaller diatonic
semitone, with fifths at around 704.096 cents, or about 2.141 cents
wide of a pure 2:3. The two chains are placed at a distance equal to
the difference between a regular major second (~208.191 cents) and a
pure 6:7 major third (~266.871 cents), or about 58.680 cents.

For convenience in the discussion that follows, I'll give a keyboard
diagram, with an asterisk (*) showing a note on the upper keyboard
raised by the "quasi-diesis" of about 58.680 cents:

187.349 346.393 683.253 891.445 1050.488
C#* Eb* F#* G#* Bb*
C* D* E* F* G* A* B* C*
58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680
7/6
-------------------------------------------------------------------------
128.669 287.713 624.574 832.765 991.809
C# Eb F# G# Bb
C D E F G A B C
0 208.191 416.382 495.904 704.096 912.287 1120.478 1200

-----------------
1. Unison vectors
-----------------

The most notable unison vectors of this tuning system are 896:891,
352:351, and 364:363. More generally, these unison vectors are
characteristic of tempered systems with fifths around 704 cents and
regular thirds near 11:14 and 11:13 or 28:33.

The 896:891 (~9.688 cents), or "3-11 kleisma" as I call it, marks the
difference between a pure 11:14 major third (~417.508 cents) and a
Pythagorean 64:81 major third (~407.820 cents) formed from four pure
fifths. In Peppermint 24, each fifth is tempered wide by about 1/4 of
this kleisma -- actually a bit less, to produce major thirds at about
416.382 cents, or around 1.126 cents narrow of 11:14.

The 352:351 (~4.925 cents), or "3-13 kleisma," is the difference
between a pure 11:13 minor third (~289.210 cents) and a Pythagorean
minor third at 27:32 (~294.135) formed by three pure fifths. In
Peppermint 24, fifths are tempered by slightly more than 1/3 of this
kleisma, producing minor thirds at about 287.713 cents, around 1.497
cents narrow of 11:13.

The 364:363 (~4.763 cents), or "11-13 schisma," could be defined as
the amount by which 11:14 plus 11:13 would exceed a pure 2:3 fifth --
or the difference between 11:13 and 28:33 (~284.447 cents), the
fifth's complement of 11:14. Likewise, 364:363 is the difference
between major thirds at 11:14 and 26:33 (~412.745 cents), the latter
ratio constituting the fifth's complement of 11:13.

In musical terms, having 896:891 and 352:351 as unison vectors in
Peppermint 24 means that four fifths up are equivalent to 11:14, and
three fifths down to 11:13.

Further, having 364:363 as a unison vector means that the same
interval of 416.382 cents represents either 11:14 (~1.126 cents
narrow) or 26:33 (~3.637 cents wide). Likewise, the interval of
287.713 cents represents either 11:13 (~1.497 cents narroww) or 28:33
(~3.266 cents wide).

Interestingly, 72-EDO provides an example of a system where some of
these rather small ratios, but not all, are treated as unison
vectors. Thus 896:891 is observed, and indeed magnified from 9.688
cents to 16-2/3 cents (~64:81 at 400 cents, ~11:14 at 416-2/3 cents),
while 352:351 is similarly magnified from 4.925 cents to 16-2/3 cents
(~27:32 at 300 cents, ~11:13 at 283-1/3 cents).

However, in 72-EDO, 364:363 is treated as a unison vector: the same
interval of 283-1/3 cents represents either 28:33 or 11:13, while
416-2/3 cents represents either 11:14 or 26:33.

Returning to Peppermint 24, we can find some other unison vectors
involving a range of intervals. Here I give a few examples I've
noticed.

The diminished fourth (e.g. F#4-Bb4, with C4 as middle C) at about
367.235 cents can represent either 17:21 (~365.825 cents), in respect
to which it is about 1.410 cents wide, or 21:26 (~369.747 cents), to
which it is about 2.512 cents narrow. Here the unison vector is
442:441 (~3.921 cents), the difference between these just ratios.
Either ratio, or the Peppermint 24 interval used for both, could be
described as a submajor third or large neutral or semi-neutral third.

Moving to the central region of neutral thirds, we find that the
single interval of about 346.393 cents, a minor third plus
quasi-diesis (e.g. C4-Eb*4), serves either as a very accurate
representation of 9:11 (~347.408 cents), narrow by around 1.015 cents;
or a rather less accurate representation of 32:39 (~342.483 cents),
wide by around 3.910 cents. Here the unison vector, equal to the
difference between 9:11 and the smaller 32:39, is the 352:351 we have
already encountered with regular minor thirds (11:13 vis-a-vis 27:32).

A more "esoteric" interval in Peppermint 24 is the major third formed
by diminished fourth plus quasi-diesis (e.g. B4-Eb*5), about 425.915
cents. This interval can represent either 18:23 (~424.364 cents), to
which it is about 1.550 cents wide; or 25:32 (~427.373 cents), to
which it is about 1.458 cents narrow. Here the unison vector is
576:575 (~3.008 cents). This Peppermint 24 interval could also be
taken as a tribute to 17-EDO, with its colorful major third of about
423.529 cents.

We encounter the same unison vector of 576:575 in relation to the
tempered fifth's counterpart of this Peppermint 24 interval, the
augmented second plus diesis (e.g. Eb*5-F#5) at around 278.181 cents.
This interval is honored as a Monzian third, since it is very close to
the 279-cent size which Joe "Monz" Monzo chose in an Aristoxenian
fashion through trial by ear for one of his pieces; and could also be
honored as a Secorian third, since it is almost identical to the minor
third of around 278.115 cents found in the nearer keys of George
Secor's superb 17-note well-temperament (17-WT).

The Peppermint 24 third at ~278.181 cents is about 0.590 cents wide of
a pure 23:27 (~277.591 cents), and about 3.598 cents wide of a pure
64:75 (~274.582 cents), the just ratio which Monz selected as the best
equivalent for his empirical choice of 279 cents.

-----------------------
2. Some observed ratios
-----------------------

In Peppermint 24, the quasi-diesis of ~58.680 cents (e.g. F4-F*4)
represents two very important observed ratios: 28:27 and 33:32.

The 28:27 (~62.961 cents), or septimal thirdtone much favored by
Archytas, defines the difference for example between a 7:9 major third
(~435.084 cents) and a 3:4 fourth (~498.045 cents). In Peppermint 24,
the quasi-diesis is about 4.282 cents narrow of this ratio, and often
represents it in cadential progressions where a tempered 14:18:21:24
sonority, for example, expands to an approximate 2:3:4, here with the
thirdtone steps of F4-F*4 and C4-C*4:

F4 F*4
D*4 C*4
C4 C*4
G*3 F*3

The quasi-diesis also represents the undecimal diesis of 33:32
(~53.273 cents), the difference between 3:4 and 8:11 (~551.318
cents). Thus a fourth plus quasi-diesis (e.g. C4-F*4) yields the best
approximation of 8:11 at ~554.584 cents (~3.266 cents wide).

The difference of a regular diatonic semitone (e.g. E4-F4) at ~79.522
cents less this quasi-diesis (E4-E*4) yields a commatic interval of
~20.842 cents (E*4-F4).

From one viewpoint, this interval represents the commatic vector of
64:63, aptly named by George Secor as the comma of Archytas, marking
the difference between the Pythagorean 64:81 and 7:9, or the
Pythagorean 27:32 and 6:7. In Peppermint 24, this would be the
difference between a regular major third (e.g. C*4-E*4) and a near-7:9
major third at around 437.225 cents (e.g. C*4-F4, ~2.141 cents wide);
or a regular minor third (e.g. C4-Eb4) and a just 6:7 minor third
(e.g. C4-D*4).

From another viewpoint, this same commatic vector could be taken to
represent 99:98 (~17.576 cents), the difference between 11:14 and 7:9,
or 28:33 and 6:7.

Interestingly, the difference between the 28:27 thirdtone and the
33:32 undecimal diesis is the unison vector of 896:891 -- as is the
difference between 64:63 and 99:98.

Another and slightly smaller type of diesic vector observed in
Peppermint 24 does not itself occur directly as a step: the 34:33
(~51.682 cents), marking the difference between a 28:33 minor third
and a 14:17 subminor or small neutral or semi-neutral third (~336.130
cents), or between an 11:14 major third and a 17:21 submajor third.
In a larger tuning set such as Peppermint 34, with two 17-note chains,
this vector would be represented by the natural diesis, e.g. Ab4-G#4
(e.g. F4-Ab4 vs F3-G#4, or Ab4-C5 vs. G#4-C5), about 49.147 cents).

We might describe as a "chromatic vector" the usual chromatic semitone
(e.g. C4-C#4) at ~128.669 cents, only about 0.371 cents wider than a
pure 13:14 (~128.298 cents).

A range of smaller distinctions are observed. For example, we have
distinct representations both for 23:27 (a tempered 278.181 cents) and
6:7 (pure), observing the vector of 262:261 (~10.720 cents). Likewise
the distinctions within 11:12:13:14 are all observed, with vectors
such as 169:168 (~10.274 cents), or 12:13 less 13:14.

A curious vector observed in Peppermint 24 is the difference between
12:17 (~603.000 cents) and 17:24 (597.000 cents), or 289:288 (~6.001
cents). These ratios are represented in tempered form by the augmented
third plus quasi-diesis (e.g. Eb4-G#*4) at ~603.731 cents, and the
diminished fifth less quasi-diesis (e.g. G#*4-Eb5) at ~596.269 cents,
respectively wide of 12:17 and narrow of 17:24 by around 0.731 cents.

While the near-12:17 is a rare interval, occurring in only one
location, it is very useful, for example, in an approximation of the
isoharmonic sonority 7:12:17 with differences of 5: the 7:12 is just,
with 12:17 and 7:17 (14:17 plus an octave) wide by only ~0.731 cents.

Notably, 289:288 is one of the unison vectors found in 22-EDO, or
indeed in any paultone tuning, where a demioctave of precisely 600
cents provides the best representation of 12:17 and 17:24 alike -- as
also, most characteristically, of both 5:7 (~582.512 cents) and 7:10
(~617.488 cents), the 50:49 commonly cited as one of the defining
unison vectors for such a tuning.

-------------------------------------
3. Vectors predictable and fortuitous
-------------------------------------

One topic of xenharmonic discussions is how some unison vectors or
observed vectors are by design, while others arise more or less
fortuitously. While George Secor's 17-WT, the topic for two articles
in the forthcoming Xenharmonikon 18 (one by the designer himself),
amply illustrates the first kind of process as well as the second here
and there, Peppermint 24 might be taken largely as an example of happy
serendipity.

To what extent did Ervin Wilson consider the musical characteristics
of the regular tuning with a ratio between chromatic and diatonic
semitones equal to Phi that appears on the Scale Tree? In proposing
the same regular temperament as a counterpart of Kornerup's Golden
Meantone (where the same ratio obtains between the large diatonic and
small chromatic semitone), Keenan Pepper was evidently proceeding by
way of an appealing mathematical intuition rather than a calculated
optimization for a given musical style.

As it happens, however, the basic Wilson/Pepper temperament gets the
ratios of 11:14, 11:13, 14:17, and 17:21 -- and also 13:14 -- all
within 1.5 cents of just. Thus it _could have_ been proposed as a
deliberate optimization of these intervals, with predictable unison
vectors (e.g. 896:891, 352:351, 364:363) and observed vectors
(e.g. 34:33, 14:13).

When I got the idea for Peppermint 24, the main intent was to add
accurate ratios of 2-3-7-9, so that the system would have near-just
versions of both 11:14 and 7:9, or 11:13 and 6:7, naturally implying a
thirdtone vector of 27:28 and a commatic vector of 63:64 (or 98:99,
from the viewpoint of 11:14 and 7:9, or 28:33 and 6:7).

As it happens, choosing a quasi-diesis between the two 12-note chains
so as to obtain pure 6:7 thirds also produces a range of other
intervals within about 3.266 cents of just -- 11:12, 9:11, 8:11, for
example, and also 12:17, 18:23 and 23:27, and 23:31.

The most obvious weakness of the system is the tempering of fifths and
fourths by a full 2.14 cents, meaning that 8:9 is wide by twice this
amount, or about 4.282 cents. A proper "near-just" system, in
comparison, should do somewhat better than 12n-EDO (fifths ~1.955
cents narrow, 8:9 ~3.910 cents narrow).

A final point might not be out of place here: often convenient unison
vectors, rather than accuracy, can serve as a major attraction for a
given tuning in a given stylistic context.

For example, while the representation of 11:14 in Peppermint 24 at
~416.382 cents is quite accurate (~1.126 cents narrow), 72-EDO is
actually closer to just at 416-2/3 cents (~0.841 cents narrow).

However, in Peppermint 24, the unison vector of 896:891 permits
the equivalence of a regular major third (e.g. C4-E4) to 11:14.
In 72-EDO, in comparison, the best representation of 11:14 calls for
an inflection of the regular 400-cent major third by 1/12-tone,
observing and (as noted) actually magnifying the size of this kleismic
vector.

Similarly, for music where 4:5 is the preferred major third, meantone
has the advantage over 72-EDO not so much of accuracy (4:5 is just in
1/4-comma, but 2:3 and 5:6 alike notably less accurate) as of the
convenience of having the 80:81 unison vector, the syntonic comma or
comma of Didymus (~21.586 cents) in just intonation. In 72-EDO the
80:81 (e.g. ~4:5 at 383-1/3 cents vs. ~64:81 at 400 cents), like the
891:896, is represented by a 1/12-tone step of 16-2/3 cents.

At the same time, Peppermint 24 shares with 36-EDO or 72-EDO a
commatic vector of 64:63; the usual major or minor third is distinct
from the best (and quite accurate) representation of 7:9 or 6:7.
Further, we note that 72-EDO is more specific regarding 33:32 and
28:27, with these intervals realized respectively as a quartertone (50
cents) and a thirdtone (66-2/3 cents). In contrast, Peppermint 24
relies on the 58.68-cent quasi-diesis to represent both ratios
somewhat less accurately.

Looking at the unison vectors and observed vectors in a tuning system,
or comparing systems, might at once help to identify musical patterns
in use or to suggest new patterns for stylistic exploration; I share
these reflections from both points of view.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, there, Gene, and your 704.2-cent linear temperament raises some
very interesting theoretical points.

For my own purposes of optimization, if I were using notes taken from
a single chain of fifths in a regular temperament around this region,
I might choose a generator of around 704.160 cents. This is the point
where 32 fifths down or fourths up yields a pure 6:7, and 7:9 and 4:7
are off by the same amount as the fifth.

Taking two 12-note chains of fifths in this temperament at 34 fourths
apart would produce something quite similar to Peppermint 24, with the
near-28:27 between the two keyboards defined precisely by the
"34-diesis" as I call it, equal to precisely twice the 17-comma.

Another way to put this is that the 34-diesis (~58.550 cents) is equal
to the 12-diesis (about 49.923 cents) plus the 46-comma (~8.627 cents).

I picked up on the general relationship when I noticed that in your
temperament, 7/4 was 31 fifths down or -31, while in something like
the e-based temperament at ~704.61 cents, it's 15 fifths up or 15. The
difference is 46, making me realize that the 46-comma is involved in
the 704.2-cent region as the difference between the 12-diesis (12
fifths up) and the 34-diesis (34 fifths down).

This is a bit analogous to the situation with Pythagorean, where the
12-comma or Pythagorean comma (531441:524288, ~23.46 cents) plus the
comma of Mercator (53 pure fifths less 31 octaves, ~3.62 cents) is
almost identical to the comma of Archytas or 64:63 (~27.26 cents)
which George Secor has aptly named. Thus 68 fourths less the requisite
number of octaves is almost exactly equal to 6:7, or 68 fifths to 7:12.

We could break this down as (3 + 12 + 53) fifths forming a usual
Pythagorean major sixth at 16:27 (~905.87 cents) plus a Pythagorean
comma plus a comma of Mercator, together only a nanisma (~0.189 cents)
short of a pure 7:12.

This means that having two chains of Pythagorean fifths at 68 pure
fourths apart is almost identical to two such chains at a pure 6:7
apart; and two such chains at 65 pure fifths apart (a 12-comma plus a
53-comma) would be almost identical to having the chains a pure 64:63
apart.

We might call such arrangements "linear noncontiguous systems": a
single generator can serve as the source of all the notes in the
tuning, but we have two chains of a given size that are not
contiguous. They are separated by a distance of 65 fifths or 68
fourths for the Pythagorean example with pure 2-3-7-9 intervals, or by
34 fourths for the 704.160-cent temperament with pure 6:7 minor
thirds.

Here's the 704.160-cent scheme I describe in Scala format:

! lin76-34.scl
!
Two 12-note chains, ~704.160 cents, 34 4ths apart (32 4ths = 7:6)
24
!
58.55034
129.12199
187.67233
208.32057
266.87091
287.51915
346.06949
416.64114
475.19148
495.83972
554.39006
624.96171
683.51205
704.16029
762.71062
833.28228
891.83261
912.48086
971.03119
991.67943
1050.22977
1120.80142
1179.35176
2/1

In this scheme, the representations of 2-3-7-9-11-13 and 14:17:21 are
much as in Peppermint 24, where an arbitrary "quasi-diesis" of ~58.680
cents is used to achieve a pure 7:6.

Note, however, that from your 17-limit perspective, there's an
important disadvantage in this approach: the noncontiguous chains
don't include the excellent approximation of 4:5 as 21 fifths up
(augmented second plus natural 12-diesis).

From my own point of view, I might prefer Peppermint 24, because it's
minutely more accurate for fifths and fourths (~2.141 cents vs. ~2.205
cents wide for fifths and narrow for fourths), with the temperament of
these intervals, and of 8:9 and 9:16 by twice as much, as the main
compromise from a "near-just" perspective. While Peppermint is also
more accurate for the almost pure 12:13 and 13:14, the 704.160-cent
system is more accurate for 11:12 and 8:11.

From my practical perspective as a keyboardist, either scheme would
have much the same "look and feel," producing an arrangement I call
"metachromatic," with the distance between the keyboards representing
a step of 28:27 (and also 33:32). Thus 6:7 is "major second plus
diesis," and 7:9 "fourth less diesis," etc. -- here actually the
"quasi-diesis" of Peppermint, or the 34-diesis of the 704.160-cent
scheme.

The main advantage of a regular and contiguous temperament such as the
e-based tuning (Blackwood's R=e) is that we have free transposibility
over the range of the gamut; in other words, G#-Eb*=G#-D#, a usual
fifth. In Peppermint 24, that same interval is about 11.673 cents wide
of 2:3, or in the 704.160-cent temperament scheme, about 10.832 cents
wide.

However, in these noncontiguous schemes, the fifths are more
moderately tempered, with 6:7 pure, to keep all intervals of
12:14:18:21 or 14:18:21:24 within 2.141 or 2.205 cents of pure. Also,
ratios of 11:12:13:14 are overall more accurately represented. Each
scheme has its own charms.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Gene W Smith <genewardsmith@juno.com>

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, there, Gene, and thank you for an interesting point about
Wilson/Pepper and the e-based temperament (and your optimized RMS
values) on which you are exactly right.

For Wilson/Pepper at ~704.096 cents, 21 fifths up make almost
precisely a pure 4:5 (giving something like 386.00 cents) -- another
odd 0.01 cent of temperament on the fifth would do it. In your linear
temperament at ~704.16, this same relationship gives a near-pure
rendition for the 5 limit.

For the e-based temperament at ~704.61 cents, or your RMS optimization
at around 704.56 or whatever, the 46-comma is a similar size, but in
the opposite direction (46-EDO is something like 704.35 cents). As you
have pointed it, here -25 fifths or 25 fourths give a better 5-limit
approximation that 21 fifths up -- for the e-based temperament,
something like 384.75 cents (this was using 704.61 as a rough
approximation of the generator). That's a bit closer to just than a
Pythagorean schisma third at 8192:6561 (~384.36 cents).

While I noticed the near-pure 4:5 in the 24-note linear version of
Wilson/Pepper when I looked at in October of 2000, the -25
relationship in the e-based temperament is something that hadn't
occurred to me, since it doesn't arise in a 24-note tuning. However,
in applications where one has a larger tuning set, and especially
where an accurate 5-limit representation is a priority, that -25 could
be a defining parameter of the tuning.

From my own viewpoint, the defining feature of e-based is the use of
-13 14 15 for 7:9, 6:7, and 4:7 -- but from a 5-limit viewpoint, the
availability of -25 for 4:5 could be at least as important.

This dialogue might also illustrate the role that assumptions can play
regarding the likely size of a tuning for a given purpose.

Thus Dave Keenan, in his germinal articles on chain-of-fifths tunings,
assumes that the longest chain to be considered for his purposes is
generally 14 fifths, with an exception for tunings with multiple
chains of fifths separated by distances like 600 cents.

Often I tend to take 24 as a likely tuning size, so that anything up
to about 16 fifths gets viewed as "usual." For example, in a 24-note
Pythagorean tuning, we have intervals of 2-3-7-9 realized as chains of
-14 -15 16 for 4:7, 6:7, and 7:9. This means that there are as many
7-based minor sevenths (10), minor thirds (9), and major thirds (8) as
their regular Pythagorean counterparts, a satisfactory situation from
my point of view.

However, as your linear temperaments show, being ready to consider
longer chains and larger tuning sets opens up room for all kinds of
optimizations that might not occur applying constraints like those
Dave and I might assume (often implicitly) for smaller tuning sets.

Humorously, some of us might recall that Keenan Pepper himself was rather
dissatisfied with his 704.096-cent temperament because it didn't have any
close approximations of 4:5 or 4:7, two intervals he regarded as
indispensable for his generally preferred harmonic style. However, it
turns it that 21 fifths up is a virtually pure 4:5, while 31 fifths down
(as you point out for your optimization with slightly wider fifths) is
quite close to 4:7, about 973.024 cents, or ~4.20 cents wide.

Of course, my immediate enthusiasm for Wilson/Pepper as a 12-note
tuning _does_ reflect a rather different musical agenda -- but your
remarks are a good reminder that in addition to timbre and style,
tuning size can be a significant variable in judging the possible
qualities of a given temperament.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, there, Gene, and your 17-limit optimizations are very
interesting. What I'd like to do is to clarify how I use these
tunings, while at the same time noting an interesting point raised by
your examples taken from the spectrum between 29-EDO and 17-EDO.

First of all, from your 5-limit point of view (as part of your
17-limit point of view), you note that some of these tunings have a
best value for 4:5 (~386.31 cents) of either 21 or -25 (21 fifths up
or 25 fifths down).

The 21 fifths up, as in Wilson/Pepper (~704.096 cents) or 121-EDO
(very close to it, and one of the choices you list), could be analyzed
as 4 + 17 -- that is, is a regular major third (near 11:14, ~417.51
cents, in this region) less a 17-comma, which is here in effect
equivalent to an 81:80 -- or a 56:55, if we take the comparison as
11:14 vis-a-vis 4:5. Note that in Wilson/Pepper, for example, we have
a regular major third at about 416.38 cents, and a 21-fifths major
third around 386.01 cents -- a difference of ~30.37 cents, quite close
to 56:55 (~31.19 cents).

The 25 fifths down or fourths up which we find to be optimal for
5-limit with slightly larger fifths could be taken as -8 + -17, or a
diminished fourth (rather close to 17:21, ~365.83 cents, in this
region) plus a 17-comma. Here we might say that the comma represents
85:84, the difference between 17:21 and 4:5. This approximation would
hold on the other side of 46-EDO, as with your 704.56-cent
optimization, which yields a -25 major third almost identical in size
to the 21 major third of Pepper/Wilson.

An interesting feature of 29-EDO (~703.45 cents), by the way, is that
we might say that 85:84 is a unison vector. Thus people might regard
the interval of 8/29 octave (~331.03 cents) as either a small
supraminor third not too far from 14:17 (~336.13 cents), or as a large
and not-so-accurate representation of 5:6 (~315.64 cents). While I
take the former approach, you might well take the latter. The point is
that one interval in 29-EDO can represent both ratios. In contrast, in
the around around Pepper/Wilson or the e-based temperament (~704.61
cents), the two ratios have separate representations, so that 85:84 is
observed.

Now for the question of how I actually use tunings in this general
region between 29-EDO and 17-EDO and go about optimizing, an area
where our approaches might be quite distinct.

Generally I'm interested in the following considerations:

(1) Regular thirds (4, -3) near 11:14 and 11:13 or 28:33;

(2) Supraminor/submajor thirds (9, -8) near 14:17 and 17:21;

(3) 2-3-7-9 intervals in 14:18:21:24 or 12:14:18:21; and

(4) Approximate steps of 11:12:13:14.

Note that in this region, the usual mapping and keyboard notation of
the 7-based intervals could be written like this:

6:7 = 2* 7:9 = -1d 4:7 = 3*
7:12 = 3d 9:14 = 1* 7:8 = -3d

Here "*" shows the indicated number of fifths up or down plus a diesis
or quasi-diesis separating the two keyboards; "d" likely shows the
indicated chain less a diesis or quasi-diesis. For example, 6:7 is
represented as a major second (2) plus a diesis or quasi-diesis.

In the e-based tuning or a tuning in that region, the natural diesis
is the interval between the keyboards, so that we might write *=12.
This means 6:7=14, 7:9=-13, and 4:7=15.

For Peppermint and other 24-note systems optimized for 2-3-7-9 which,
in contrast to the e-based tuning, do not use a single 24-note chain,
my approach is to define 2*=6:7 so that 6:7 is just; thus the
quasi-diesis is equal to (6:7 - 2), or a pure 6:7 less a major
second. This is about 58.680 cents for Peppermint 24.

Another way of defining this quasi-diesis is to say that it is equal
to a just 27:28 (~62.96 cents) less twice the tempering of the fifth
in the wide direction.

Thus my approach focuses on defining a distance between the chains so
as to optimize 12:14:18:21 or 14:18:21:24; each chain provides
optimized versions of the regular thirds (~22:28:33 or ~22:26:33, the
former for most of the region and the latter near 29-EDO at the lower
end), and of supraminor/subminor thirds. The latter might be said to
range from something like 52:63:78 near 29-EDO to 14:17:21 in the
Peppermint region to something like 46:56:69 (23:28 and 56:59) in the
e-based region.

Note that 5-based thirds are tangential in this kind of musical style,
with the optimization determined by other factors. In either
Peppermint 24 or the e-based tuning, we get some 9* thirds (augmented
second plus diesis or quasi-diesis) at around 395.540 or 396.745
cents, maybe something like 1/8-comma meantone, and a bit smaller than
in 12-EDO. This is fine -- they add another somewhat exotic "flavor"
at a few positions, thus increasing the diversity of the tuning, but
the optimization focuses on the regular intervals in each chain plus
2-3-7-9.

What I would emphasize is that these are only my musical priorities;
your 17-limit method is a very logical approach for the kind of
optimization you are seeking, and your 704.16-cent optimization with a
long enough chain of fifths, or one of your other temperaments, looks
like an ideal solution.

Of course, your conclusions are of interest from any viewpoint,
because they show a bit more about the structure of these tunings,
including the vectors that we're discussing in this thread.

One conclusion is that in using the larger tuning sets your 17-limit
optimizations might imply, one should take note that in the area right
around 704 cents, the 17-comma is just the right size to serve as a
kind of 81:80 or 56:55 (11:14 vis-a-vis 4:5), but a bit large to serve
as an ideal 99:98 (11:14 vis-a-vis 7:9, or 28:33 vis-a-vis 6:7). To
put the point another way, the 12-diesis (close to 50 cents) is quite
small for a 27:28, and we want a larger quasi-diesis.

If we go to the area close to but a bit past 46-EDO, however, then the
17-comma makes a good 56:55 (e.g. 4 vis-a-vis -13 for 11:14 and 7:9 in
the e-based tuning or your 704.56-cent optimization, or 109-EDO), but
is rather large to add to a supraminor third (9* or 21) if the idea is to
get a really accurate 4:5. Here a submajor third plus a 17-comma would
be more accurate -- as in your 704.56-cent temperament, where -25
yields a virtually just 4:5.

The implication is that to get optimal results for something like
4:5:6:7, you'd want a tuning set large enough to observe these
differences and choose the best approximation for each interval.

Most appreciatively,

Margo Schulter
mschulter@value.net