A very curious system

Hello, there, everyone, and I'd like to share some impressions of what
might be a very curious tuning system from an historical as well as
contemporary perspective.

Let's approach this as an enigma of sorts. You are sitting at a
keyboard and find that a tuning system includes some minor sevenths at
about 983 cents; major third sizes of a few cents smaller than a pure
4:5 and very close to a pure 7:9 (the latter at around 434 cents); and
some tritonic intervals of precisely 600 cents.

"Ah," you might say, "this is likely some kind of paultone, with the
shade of temperament a bit milder than Paul Erlich's beloved 22-EDO,
where the minor sevenths at ~981.82 cents are a bit closer to 4:7, and
the large major third a tad larger than a pure 7:9 at ~436.36 cents."
Indeed you find fifths at ~708.38 cents, or ~6.42 cents wide, as
compared to 13/22 octave at ~709.09 cents or ~7.14 cents wide.

However, exploring the keyboard, you soon find other intervals which
don't fit this model, including some minor thirds at precisely 300
cents, and major thirds at around 408 cents and 421 cents. Also, you
find that the near-4:5 major thirds are around 383.24 cents, larger
than would be expected in a paultone model (where nine fifths up at
~708.38 cents each would yield an augmented second at ~375.42 cents).

You quickly confirm that this isn't a paultone -- at least not the
whole tuning system -- but a circulating 12-note tuning with fifths at
around 708.38 cents (as already noted) and 695.81 cents (~6.14 cents
narrow). There are four fifths of the former size, and eight of the
latter at F-C# -- each narrowed by 2/7 of a syntonic comma (81:80,
~21.51 cents).

In other words, this is a temperament extraordinaire based on
Zarlino's 2/7-comma meantone, possibly one of the 12-note circulating
systems with the widest ranges of third sizes, and the best
approximations of 6:7, 7:9, and 4:7 (although not as good, at least
overall, as in either a paultone near 22-EDO or a regular meantone).

As occurs also in a temperament extraordinaire based on 1/4-comma
meantone, all fifths are tempered by almost the same absolute amount:
here -6.14 cents or +6.42 cents; and in 1/4-comma, -5.38 cents or
+4.88 cents. Thus the system is freely circulating as far as 3-limit
harmony is concerned: each of the 12 steps has a reasonable
approximation of 2:3:4, so that Gothic or neo-Gothic cadences on all
steps are possible.

In either system, as in a temperament extraordinaire generally, eight
fifths are tempered in regular meantone, and the others by equal
amounts in the wide direction so as to produce two "even-tones" for
the usual diminished fourth C#-F in a standard meantone with narrow
fifths of the same size. Thus these even-tones are ~213.686 cents in a
1/4-comma temperament extraordinaire (with C#-F at 25:32, ~427.373
cents), and ~216.759 cents in the 2/7-comma version (with C#-F at
~433.517 cents).

Interestingly, in the 2/7-comma scheme this even-tone is only ~0.072
cents larger than a just 15:17 (~216.687 cents), so that Eb-F-G
(~0-216.76-408.38 cents) gives a very close approximation of the
isoharmonic sonority 15:17:19 (~0-216.69-409.24 cents), with all three
intervals within one cent of just.

Here is a keyboard tuning diagram:

25:24
70.672 287.431 574.862 779.052 995.810
C# Eb F# G# Bb
C D E F G A B C
0 191.621 383.241 504.190 695.810 887.431 1079.052 1200

Here is a Scala file:

! zarte84.scl
!
Temperament extraordinaire, Zarlino's 2/7-comma meantone (F-C#)
12
!
70.67243
191.62069
287.43104
383.24139
504.18965
574.86208
695.81035
779.05173
887.43104
995.81035
1079.05173
2/1

Among the set of 12-note circulating temperaments, the Zarlino-based
temperament extraordinaire (or TE for short) might offer some of the
most generous contrasts in semitone sizes, ranging from a just 24:25
or ~70.672 cents (C-C#/Db, F-F#/Gb) to the regular diatonic semitone
of 2/7-comma meantone at ~120.948 cents, a difference of ~50.276
cents.

This 50.28-cent difference, equal to the enharmonic diesis in a
regular 2/7-comma meantone tuning (e.g. G#-Ab), also defines the
amount by which the largest major third or diminished fourth (C#/Db-F)
at ~433.52 cents exceeds the smallest major thirds at their usual
meantone size of ~383.24 cents. These thirds are close respectively to
7:9 (~435.084 cents) and 4:5 (~386.314 cents), so that the difference
is close to 36:35 (~48.770 cents).

One could achieve a 12-note circle with an even greater difference by
choosing a slightly greater amount of temperament for the narrow
meantone fifths, as for example in Ervin Wilson's metameantone tuning,
with fifths at around 695.630 cents (~6.32 cents narrow). A TE based
on this meantone would have four wide fifths producing even-tones of
the virtually pure 7:9 diminished fourth or major third C#/Db-F at
~434.96 cents, or only ~0.12 cents smaller than just -- in other
words, fifths at ~708.74 cents, or ~6.78 cents wide, and even-tones at
~217.48 cents, or very slightly wider than 17:15.

The Zarlino-based circle has the special attraction of being a system
combining nine notes or eight fifths (F-C#) identical to those in the
earliest Western European meantone temperament to be documented in
precise mathematical terms (by Zarlino in 1558), and also such
features as the near-7:9 third and just 24:25 semitones.

In the Zarlino-based TE, interval sizes jump by steps of ~12.57 cents,
or 1/4 of the 50.28-cent diesis in 2/7-comma meantone. Thus we have,
for example (using typical meantone spellings, although enharmonic
equivalence here applies):

-------------------------------------------------------------------
minor2 (can represent diatonic or chromatic semitone)

~70.672 cents (25:24) -- 2 -- C-C#, F-F#
~83.241 cents -- 2 -- Bb-B, G-G#
~95.810 cents -- 2 -- D-Eb, Eb-E
~108.379 cents -- 2 -- G#-A, A-Bb
~120.948 cents -- 4 -- C#-D, E-F, F#-G, B-C
...................................................................
Major2/diminished3

~191.621 cents -- 7 -- C-D, D-E, E-F#, F-G, G-A, A-B, B-C#
~204.190 cents -- 2 -- F#-G#, Bb-C
~216.759 cents -- 3 -- C#-Eb, Eb-F, G#-Bb
...................................................................
minor3/Augmented2

~274.862 cents -- 2 -- Eb-F#, Bb-C#
~287.431 cents -- 2 -- C-Eb, F-G#
300.000 cents -- 2 -- G-Bb, G#-B
~312.569 cents -- 6 -- C#-E, D-F. E-G, F#-A, A-C, B-D
...................................................................
Major3/diminished4

~383.241 cents -- 5 -- C-E, D-F#, F-A, G-B, A-C#
~395.810 cents -- 2 -- E-G#, Bb-D
~408.379 cents -- 2 -- Eb-G, B-Eb
~420.948 cents -- 2 -- F#-Bb, G#-C
~433.517 cents -- 1 -- C#-F
...................................................................
Fourth/Augmented3

~491.621 cents -- 4 -- Bb-Eb, F-Bb, G#-C#, Eb-G#
~695.810 cents -- 8 -- F-C, C-G, G-D, D-A, A-E, E-B, B-F#, F#-C#
...................................................................
Augmented4/diminished 5

~574.862 cents -- 3 -- C-F#, F-B, G-C#
~587.431 cents -- 2 -- D-G#, Bb-E
600.000 cents -- 2 -- A-Eb, Eb-A
~612.569 cents -- 2 -- G#-D, E-Bb
~625.138 cents -- 3 -- F#-C, B-F, C#-G
____________________________________________________________________

From a paultone-like perspective, F-A-C-Eb (~0-383-696-983 cents)
provides the best approximation of 4:5 in a 4:5:6:7-like sonority, and
includes a characteristic 600-cent tritone A-Eb. The upper minor
third at ~287.431 cents, however, is a much better approximation of
11:13 (~289.210 cents) or 28:33 (~284.447 cents) than 6:7 (~266.871
cents).

Another somewhat paultone-style approximation of 4:5:6:7 (albeit with
a diminished fifth equal to ~587.431 cents rather than 600 cents,
actually closer to the ideal 5:7 at ~582.512 cents) is available at
Bb-D-F-Ab (~0-396-708-983 cents). Here the best equivalents for 4:7
and 6:7 (~274.862 cents, only about 0.280 cents larger than 64:75) are
combined with a rather compromised ~4:5 at ~395.810 cents, or about
9.497 cents wide). The wide ~2:3 is like that of a moderate paultone,
as noted above, and a bit closer to pure (~6.415 cents wide) than in
22-EDO (~7.136 cents wide).

From a neo-Gothic perspective, the 287-cent minor thirds are also very
attractive along with the 421-cent and 408-cent major thirds, as well
as the 434-cent major third very close to 7:9.

The 275-cent minor thirds at Bb-C#/Db and Eb-F#/Gb are near the low
end of the range called "Monzian" in honor of Joe Monzo, running from
around 64:75 (as here) to about 280 cents, as with the thirds at
around 279.471 cents in a 1/4-comma meantone TE. The latter value
represents the chosen size for a third at which Monz arrived in a
certain composition through tuning by ear, while the former represents
his choice of 64:75 as an appropriate rational value.

It is thus notable that the spectrum of TE systems from 1/4-comma to
2/7-comma roughly represents the range of Monzian sizes for the
smallest minor thirds in these 12-note circles.

It would go almost without saying that the 2/7-comma TE, like the
1/4-comma TE, is quite different from a conventional 18th-century
well-temperament, where at least marginally "playable" representations
of 5-limit concords such as 4:5:6 or 10:12:15 are expected in all
positions, with Pythagorean thirds at 64:81 (~407.820 cents) and 27:32
(~294.135 cents) as rough limits. In such systems, typically, no fifth
is wider than pure.

In these TE systems based on 16th-century meantones with optimal or
near-optimal 5-limit sonorities, however, major and minor thirds
beyond Pythagorean (in the large and small directions respectively)
are a main attraction, along with impeccable with Renaissance meantone
sonorities in the range of F-C#. The wide fifths are desirable for
neo-medieval progressions, just as the narrow ones are for usual
Renaissance progressions. Consider, for example, this cadence:

Bb4/A#4 C5/B#4
Ab4/G#4 G4/Fx4
F4/E#4 G4/Fx4
Db4/C#4 C4/B#3

(0-434-708-925) (0-696-1200)

Here we have a ~14:18:21:24 sonority (as it might be described) with
descending semitones of 24:25 (Db4-C4) and ~83.241 cents (Ab4-G4). The
ascending whole-tone F4-G4 is the usual 2/7-comma mean-tone of
~191.621 cents, while Bb4-C5 is at ~204.190 cents, only ~0.28 cents
wider than a just 8:9 (~203.910 cents).

It is fascinating to explore such a 12-note circle, identical to
2/7-comma meantone in nine of its steps and eight of its fifths, but
quite different from a typical European historical system in its total
effect if one uses a variety of intervals from all portions of the
circle.

It would also be interesting to investigate the widest major third
specified in an historical temperament ordinaire (TO): the
433.517-cent third at C#/Db-F in this TE might be one of the wider
ones on record for a circulating 12-note system.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@yahoogroups.com, "M. Schulter" <MSCHULTER@V...> wrote:
> Also, you
> find that the near-4:5 major thirds are around 383.24 cents, larger
> than would be expected in a paultone model (where nine fifths up at
> ~708.38 cents each would yield an augmented second at ~375.42
>cents).

just to clarify -- in the "paultone" model, which is known
as "pajara" these days, the near-4:5 is generated not necessarily by
nine fifths up, but always by a half-octave plus two fifths down. if
the fifth is fixed at 708.38 cents, the functional near-4:5 will
necessarily come out as 383.24 cents.

anyway, thanks for sharing your curious system!