Re: Optimizations of 2-3-7 (for Graham Breed)

Hello, there, Graham, and your remarks about 2-3-7 optimizations and
neo-Gothic music in meantone tie in with an article I'm planning on
positive and negative tunings. Certainly it's possible to play
neo-Gothic progressions in meantone, although not necessarily the
ideal way to optimize ratios of 2-3-7.

Really, it's appropriate that we're discussing these questions,
because it was your very patient and helpful experiments with
neo-Gothic progressions in various tunings around the middle of last
year, and your emphasis on really getting 7-based ratios accurate,
that helped start me on my way to a 2-3-7 JI system. In many ways,
it's a simple problem compared to the ratios and limits you juggle in
your tunings.

Thank you also for convincing me that the term "septimal schisma" can
mean different things to different people; your "2-3-7 schisma" nicely
communicates the idea of the schisma equal to the difference of the
2-3 or Pythagorean comma and the 3-7 or septimal comma.

As you noted, the Pythagorean comma is nicely expressed as the
difference between six 9:8 whole-tones and a pure 2:1 octave; this
comma is equal to 531441:524288 (~23.46 cents).

Now for the meantone question. This seems the _ideal_ solution if we
want to mix Renaissance and neo-Gothic styles and intervals in the
same piece using a single tuning, and really have those Renaissance
sonorities shine.

A 24-note archicembalo inspired by the 36-note archicembalo of Nicola
Vicentino (1511-1576) makes possible the wonderful effects for
Renaissance music of the kind he champions, including "enharmonic"
diesis or fifthtone progressions and "proximate minor thirds" (neutral
thirds near 11:9). As a bonus, it provides lots of intervals useful
for neo-Gothic music, such as the wide major third and the narrow
minor third and seventh.

Apart from the special situation of mixing Renaissance and neo-Gothic
styles in a single tuning, playing neo-Gothic intervals in meantone
can be a fascinating intonational "role reversal" from which we can
learn more about the music and the tuning alike. It's quite possible
to demonstrate some sample cadences on a usual 12-note meantone
keyboard, as I explain toward the end of this article.

However, if we're focusing on a 2-3-7 optimization specifically for
neo-Gothic music, then I wouldn't call meantone either the simplest or
the most accurate solution, although it's actually more accurate on
the whole than the one simple solution, 22-tone equal temperament
(22-tET) or something it its vicinity.

If we're focusing on simplicity, I would say that navigating a 24-note
meantone temperament seems to me about as simple, or complicated, as
getting around a more characteristically neo-Gothic scheme such as a
24-note 2-3-7 JI system or e-based system (fifth ~704.61 cents).

Curiously, I consider a 24-note tuning in many ways less intricate
than a 19-note tuning, since the symmetry of the two "generalized"
12-note keyboards can help in finding intervals and cadences.

Here there may be two ways of approaching the 2-3-7 problem. The first
is simply to compare the accuracy of different solutions; the second
is also to look at a system as a whole in a given musical setting,
including the question of how other intervals might fit in with a
given style.

In a 24-note 2-3-7 JI scheme with two Pythagorean manuals a septimal
comma apart, for example, we get all the usual Pythagorean intervals
on either manual, as well as the new 7-based ones. The two types of
intervals can nicely contrast with each other, and both fit a
neo-Gothic style very nicely.

In a 24-note e-based tuning, similarly, we have a very characteristic
12-note neo-Gothic tuning on each manual, as well as excellent
approximations of 7-based intervals and an intriguing new type of
cadential diesis at around 55 cents, large enough to feel like a
semitone step.

Similarly, a 24-note meantone tuning gives us a beautiful set of
regular intervals on each manual -- but this time optimized for
Renaissance music, although we can certainly use it for neo-Gothic as
well.

To borrow the title of a recent thread here, comparing optimizations
is a matter of "cents and sensibility," and the numbers are only part
of the picture.

For example, 22-tET is generally the _least_ accurate 2-3-7 system on
the table that follows, but it's also the one type of solution that I
consider really simple: just tune up a 12-note chain of fifths and
go. Of course, tuning the full set of 22 opens up possibilities for
various special effects, but we only need 12 for a keyboard full of
near-7 ratios as _regular_ intervals.

In fact, the appreciable compromise of the 3:2 and 7:4 in 22-tET
reflects the heroic task this tuning takes on and accomplishes:
dispersing the septimal comma (64:63, ~27.26 cents) in a chain of only
2, 3, or 4 fifths to get regular intervals near 7:4, 7:6, and 9:7.

Let's compare some typical neo-Gothic solutions for 2-3-7 based on
Pythagorean or wider fifths with two forms of Renaissance or
neo-Renaissance meantone, and also a "middle ground" solution notable
for its accuracy. The term "schisma" refers to the 2-3-7 schisma of
around 3.80 cents:

---------------------------------------------------------------------
System Size/5th chains Fifth 7:4 7:6 9:7
(7:4,7:6,9:7)
---------------------------------------------------------------------
Neo-Gothic systems (fifths 3:2 or wider)
---------------------------------------------------------------------
22-tET 12-22 ~709.09 ~981.82 ~272.73 ~436.36
(2-3-4) +~7.14 +~12.99 +~5.86 +~1.28
.....................................................................
2-3-7 JI* 24 ~701.96 ~968.83 ~266.87 ~435.08
(14-15-16) pure pure pure pure
.....................................................................
1/15 schisma 24 ~702.21 ~969.08 ~266.87 ~435.34
(14-15-16) +~0.25 +~0.25 pure +~0.25
.....................................................................
1/14 schisma 24 ~702.23 ~968.83 ~266.60 ~435.63
(~135-tET) (14-15-16) +~0.27 pure -~0.27 +~0.54
.....................................................................
e-based 24 ~704.61 ~969.10 ~264.50 ~440.11
(15-14-13) +~2.65 +~0.28 -~2.37 +~5.03
---------------------------------------------------------------------
Middle ground system (36-tET)
---------------------------------------------------------------------
36-tET 24-36 700.00 ~966.67 ~266.67 ~433.33
(steps: 29-8-13) -~1.95 -~2.16 -~0.20 -~1.75
---------------------------------------------------------------------
Renaissance or Renaissance-like meantone systems
---------------------------------------------------------------------
1/4 comma 24 ~696.58 ~965.78 ~269.20 ~427.37
meantone (10-9-8) -~5.38 -~3.04 +~2.33 -~7.71
.....................................................................
Golden 24 ~696.21 ~962.14 ~265.93 ~430.28
meantone (10-9-8) -~5.74 -~6.68 -~0.94 -~4.80
---------------------------------------------------------------------
* Fifth between manuals (G#-D#) is 2-3-7 schisma wide (~3.80 cents)
---------------------------------------------------------------------

If we were concerned only about 2-3-7 accuracy, then a 2-3-7 JI system
or regular schisma temperament leaving the fifths virtually pure would
be our obvious choice.

Among listed systems, the next most accurate is 36-tET, with two
manuals at 1/6-tone (~33.33 cents) apart. This tempers the fifths by
about 1.95 cents, here in the narrow direction, and gives us, overall,
more accurate 7-based approximations than any of the more heavily
tempered alternatives.

The e-based tuning is next, with fifths about 2.65 cents from pure,
this time in the wide direction; we get a virtually pure 7:4, with all
of our 7-based intervals within just over 5 cents of pure.

With our meantones, the fifths are tempered in the narrow direction
rather appreciably: 5.38 cents for 1/4-comma, likely Vicentino's
tuning, and 5.74 cents for Thorwald Kornerup's Golden Meantone. We get
a 7:6 close to pure in either tuning -- within one cent in Golden
Meantone. However, optimizing both the 7:4 and the 9:7 at the same
time presents a bit of a dilemma; 1/4-comma does nicely for the 7:4,
while Golden Meantone is kinder for the 9:7.

Nevertheless, either of these meantones is more accurate both for the
fifth and for overall 7-based ratios than 22-tET (fifth ~7.14 cents
wide), although 22-tET features a near-pure 9:7, in fact better than
any other system where the fifth is substantially tempered.

In reality, however, accuracy isn't the only issue: other musical
features and values make each of these systems an attractive option.

With the superaccurate JI and schisma systems, we get the regular
Pythagorean intervals on each 12-note keyboard, a delightful mixture
of the old and the new.

With the e-based system, we get a regular minor sixth (~781.57 cents)
only about 0.91 cents narrow of 11:7, and lots of other characteristic
neo-Gothic ratios like "submajor/supraminor" thirds not too far from
17:14 and 21:17, here a bit on the neutralish side, as well as a "Wolf
fourth" (e.g. Eb-G#, 11 fifths up) within one cent of 11:8.

With 22-tET, of course, we get a neo-Gothic equivalent of meantone for
ratios of 2-3-7, with the septimal comma neatly dispersed, unlike any
of the other system where this comma (or some analogous interval) is
in play to complicate as well as sometimes enrich things.

With meantone, we get an optimized tuning for conventional and
experimental Renaissance music alike, both styles described by
Vicentino, as well as the intervals of special interest from a
neo-Gothic perspective. For example, we can use the 427-cent third
both as a meantone diminished fourth (wonderfully illustrated in some
Renaissance pieces) and as a neo-Gothic major third.

With 36-tET, we get a basic 12-tET tuning on either keyboard that in
appropriate timbres might serve as a compromise intonation for mixing
medieval and Renaissance styles, as well as the 7-based intervals and
excellent submajor/supraminor thirds (~333.33 cents, ~366.67 cents)
of the neo-Gothic "17-flavor" (around 21:17 and 17:14).

Having discussed some of the issues of "cents" -- and hopefully also
of "sensibility" -- why don't I briefly give examples of a couple of
neo-Gothic progressions available on a 12-note meantone keyboard in
the usual Eb-G# range. Here I use a MIDI-style octave notation where
C4 is middle C:

Eb4 E4
Bb3 B3
F#3 E3

Let's assume that we're in 1/4-comma meantone with pure 5:4 major
thirds. Here we have a standard Gothic cadence with major third
expanding to fifth, and major sixth to octave, bringing us to a
complete 2:3:4 trine on E3-B3-E4.

In our musical "role reversal," the meantone diminished fourth F#3-Bb3
at 32:25 becomes our 427-cent neo-Gothic major third, and the
diminished seventh F#3-Eb4 becomes our near-12:7 major sixth.

Also, as we see in this cadence, the meantone chromatic semitone, the
_smaller_ semitone of this tuning at around 76.05 cents, becomes our
neo-Gothic diatonic semitone: here Bb3-B3 and Eb4-E4 in the two upper
voices.

How about a meantone version of a cadence with a narrow minor seventh
contracting to a fifth, and an upper minor third contracting to a
unison? Here's another example available on a 12-note keyboard in
Eb-G# tuning:

C#4 B3
Bb3 B3
Eb3 E3

Our neo-Gothic cadence takes the meantone augmented sixth Eb3-C#4 as
our near-7:4 seventh; the augmented second Bb3-C#4 serves as our
near-7:6 minor third. Again the 76-cent meantone chromatic semitone,
here Eb3-E3 or Bb3-B3 (two lower voices), becomes our usual cadential
semitone.

In a 24-note tuning with the manuals a diesis apart, the wide major
thirds and sixths are a diesis larger than the usual meantone
intervals, and the narrow minor thirds and sevenths a diesis smaller.
Here are versions corresponding to our two examples, using an ASCII
asterisk (*) for Vicentino's dot above a note showing that it is
raised by a diesis:

E*4 F4 D4 C4
B*3 C4 B*3 C4
G3 F3 E*3 F3

The "look and feel" of a 24-note meantone is rather similar to that of
a 24-note neo-Gothic scheme, or 24-out-of-36-tET for that matter: we
play fifths and fourths on the same keyboard, with 7-based intervals
combining notes from the two keyboards.

I'd call this "moderately intricate": the chain of fifths doesn't get
disrupted, unlike in some traditional JI systems (e.g. the syntonic
diatonic for 2-3-5), and we can play anything on either keyboard that
we could in the base 12-note system.

Most respectfully,

Margo Schulter
mschulter@value.net