Yahoo Tuning Groups Ultimate Backup tuning Re: Pythagorean 3-speed bikes: for Joseph Pehrson

[Please let me explain that this post was originally written April 30-
May 1, but that problems at my ISP with servers, networks, etc.,
prevented me from accessing e-mail and other Internet services for
several days. I am very happy to be back online, and would clarify
that in this post, "within the last couple of weeks" means "around the
middle of April."]

Hello, there, Joseph Pehrson, and thank you for some very helpful
comments and questions.

Just as you say, actually playing a two-manual keyboard in a 24-note
"two-gear" Pythagorean tuning is a much more effective introduction to
this kind of system than anything I could write here.

By the way, thank you very much for mentioning my _Flugblatt_
distributed at MicroFest, and I'd like to express my warmest
appreciation to Bill Alves for his pivotal role in making this and all
other aspects of the conference possible.

Within the very modest limits of written prose and ASCII keyboard
diagrams, maybe one way that I can share some "Pythagorean biking"
ideas with you is by a "test drive" in which we try to design the
perfect "3-speed" model in 24 notes.

I write this in the hope that you and other readers can reasonably
soon make the "test drive" metaphor more concrete by actually trying
out some of these tunings on a synthesizer, or at least hearing them
on a tape which I may soon by preparing.

We'll start out with a rather offbeat (for neo-Gothic purposes) tuning
providing a 3-speed prototype for our bike design, and then seek to
design a Pythagorean-based model (each keyboard in standard
Pythagorean tuning) including similar 3-speed features.

The result, curiously, is a 3-speed Pythagorean bike with a special
"super-Monzonian" quality: using only powers of 3:2, it almost exactly
emulates an interval which the Monz, in turn, has used to emulate a
ratio of a different prime limit. While I've described this new (for
me) tuning on Usenet's rec.music.theory, this is my first mention of
it on the Tuning List.

-------------------------------------------------
1. An offbeat 3-speed prototype: 24-out-of-36-tET
-------------------------------------------------

Given all the recent discussions about 72-tET, why don't we start with
a prototype of our "Pythagorean bike" design in a very useful subset
of this tuning, 24-out-of-36-tET. Here C4 is middle C, with
Vicentino's dot shown as an ASCII asterisk (*):

133 233 633 833 1033
C#*3 Eb*3 F#*3 G#*3 Bb*3
C*3 D*3 E*3 F*3 G*3 A*3 B*3 C*4
33 233 433 533 733 933 1133 1233
---------------------------------------------------------
100 200 600 800 1000
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 200 400 500 700 900 1100 1200

While 36-tET isn't the most typical neo-Gothic tuning, it features
three of the most popular intonational "flavors" -- a 3-speed bike, to
use our metaphor.

To get our first or standard Pythagorean speed, we use the regular
intervals available on either keyboard -- identical to those of
12-tET. Here we encounter the main compromise in what would otherwise
be very close to the "perfect" neo-Gothic temperament.

In the most characteristic neo-Gothic tunings, fifths range from pure
(Pythagorean-based systems) to not quite four cents wide (17-tET).
Major thirds and sixths are Pythagorean or wider, while minor thirds
and sixths are Pythagorean or narrower, as are diatonic semitones.
These parameters nicely fit a style where active thirds and sixths
often serve as points of directed cadential tension, resolving to
stable intervals by progressions involving incisive melodic
semitones.

In 36-tET, however, the otherwise excellent 700-cent fifths are
tempered by about 2 cents in the narrow or "wrong" direction! Our
regular thirds and sixths are thus a bit "bland" or "subdued," and our
100-cent semitones somewhat less incisive and efficient than their
90-cent Pythagorean counterparts. However, these approximations are
still quite reasonable, as a table may suggest:

--------------------------------------------------------------------
Interval Pythagorean ratio/cents 36-tET variation
--------------------------------------------------------------------
5 3:2 701.96 700 -1.96
4 4:3 498.04 500 +1.96
....................................................................
M2 9:8 203.91 200 -3.91
m7 16:9 996.09 1000 +3.91
....................................................................
m3 32:27 294.13 300 +5.87
M6 27:16 905.87 900 -5.87
....................................................................
M3 81:64 407.82 400 -7.82
m6 128:81 792.18 800 +7.82
....................................................................
m2 256:243 90.22 100 +9.78
M7 243:128 1109.78 1100 -9.78
--------------------------------------------------------------------

We might accordingly describe 36-tET as a "semi-Pythagorean" bike with
regular thirds, sixths, and diatonic semitones a bit "diluted"; but
the other two speeds, as already noted, are absolutely superb!

To shift to our second gear -- or speed -- we may try cadences with
major thirds and sixths 1/6-tone wider than the regular ones, and
minor thirds and sevenths 1/6-tone narrower. These cadences combine
usual fifths and fourths with intervals based on factors of 3 and 7
such as 9:7 major thirds, 12:7 major sixths, 7:6 minor thirds, and 7:4
minor sevenths, and are thus referred to as "7-flavor" cadences. Here
are some examples in 36-tET:

E*4 F4 F4 E*4
D4 C4 D*4 E*4
B*3 C4 Bb3 A*3
G3 F3 G*3 A*3

The unstable sonority in our first cadence of G3-B*3-D4-E*4, with
rounded intervals of 0-433-700-933 cents, is very close to a pure
ratio 14:18:21:24 (around 0-435-702-933 cents). The outer major sixth,
ideally 12:7 (~933.13 cents), is virtually pure (933.33 cents), and
all other intervals within about 2.16 cents of pure.

Likewise, in our second cadence, G*3-Bb3-D*4-F4 at 0-267-700-967 cents
outstandingly approximates 12:14:18:21 (0-267-702-969 cents).

Note that these "7-flavor" cadences in 36-tET involve a melodic
"thirdtone step" such as B*3-C4 or E*4-F4 in the first example, or
Bb3-A*3 or F4-E*4 in the second, with a size of 66-2/3 cents. While
the usual 100-cent semitone may seem a bit wide by neo-Gothic
standards, the contrast between these two sizes of steps can be a very
pleasing one.

These cadences illustrate a pattern common to many neo-Gothic tunings:
vertical fifths and fourths are played on the same keyboard, while
other intervals such as thirds and sixths often combine notes from
different keyboards.

Now let us shift to our third speed, featuring cadences with minor
thirds or sixths 1/6-tone _wider_ than usual, and major thirds or
sixths 1/6-tone _narrower_. Since major and minor thirds of this kind
have ratios around 21:17 and 17:14, this is called the "17-flavor,"
again superbly realized in 36-tET:

B3 C*4
G*3 F*3
E3 F*3

In this cadence, the sonority E3-G*3-B3 (0-333-700 cents) excellently
approximates the 17-flavor ratio of 14:17:21 (0-336-702 cents).

Note that each of the outer parts moves by a "2/3-tone" of E3-F*3 or
B3-C*4, or 133-1/3 cents, a characteristic feature of 17-flavor
cadences in various neo-Gothic tunings with similar steps.

Taken by itself, either the 7-flavor or the 17-flavor of 36-tET would
be very impressive; but _both_ flavors so outstandingly realized make
it a special tuning indeed.

One of the "offbeat" aspects of 36-tET from a neo-Gothic viewpoint,
its precisely symmetrical regular intervals including our 100-cent
semitones, is not without its advantage here: we can build a given
7-flavor or 17-flavor sonority from all 12 positions on the applicable
keyboard. For example, we can build a 14:18:21:24 sonority like
G3-B*3-D4-E*4, or a 14:17:21 sonority like E3-G*3-B3, above any note
on the lower keyboard; or a 12:14:18:21 sonority like G*3-Bb3-D*4-F4
above any note on the upper keyboard.

Now for our game of trying to design the "perfect" 3-speed Pythagorean
bike: our object is to come up with a 24-note tuning featuring pure
Pythagorean intonation on each keyboard and at the same time also
offering the best possible 7-flavor and 17-flavor intervals.

In the process, we'll encounter another neo-Gothic flavor also
associated with a very famous 20th-century tuning -- before arriving
at a "super-Monzonian" solution I find quite amazing.

------------------------------------------------------------
2. A perfect bike for two flavors: the septimal comma tuning
------------------------------------------------------------

Getting two out of our three flavors, or "speeds," perfect, isn't
hard, and let's look at a bike designed for an optimal 7-flavor as
well as the usual Pythagorean intervals.

Our solution is simply to place our two manuals in regular Pythagorean
tunings of Eb-G# a _septimal_ comma apart: 64:63, or ~27.26 cents,
about 3.80 cents larger than the usual Pythagorean comma with its
ratio of 531441:524288 (~23.46 cents).

Using Vicentino's dot, or here our asterisk sign (*), we can give the
notes the same name as in our 36-tET scheme, although the intervals
are different, here shown again in rounded cents:

141 321 629 843 1023
C#*3 Eb*3 F#*3 G#*3 Bb*3
C*3 D*3 E*3 F*3 G*3 A*3 B*3 C*4
27 233 435 525 729 933 1137 1227
---------------------------------------------------------
114 294 612 816 996
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 204 408 498 702 906 1110 1200

Now our standard Pythagorean intervals forming the backbone of the
tuning, as it were, are perfect on either keyboard: the fifths and
fourths are pure, while the thirds and sixths are active and energetic
and the diatonic semitones incisive.

Our 7-flavor sonorities such as 14:18:21:24 and 12:14:18:21 are also
perfect. Here a musician accustomed to 12-tET might note a difference
which someone steeped in neo-Gothic tunings would take for granted:
these four-voice sonorities are now available in 8 positions, rather
than the 12 positions of 24-out-of-36-tET.

For example, we can build G3-B*3-D4-E*3 or G*3-Bb3-D*4-F4 without any
problem because all of the needed spellings are included within our
Eb-G# range (or Eb-G#/Eb*-G#*, to show our full 24-note set). However,
we cannot build something like B3-D#*4-F#4-G#*4 because we do not have
a D#*4 in our set of accidentals; nor F*3-Ab3-C*4-Eb4, which requires
an Ab3. In 36-tET, where Eb*4=D#*4 and G#3=Ab3, there is no problem.

Historical and "neo-historical" tuning advocates such as myself tend
to describe this as a "stimulating asymmetry" rather than a flaw, and
for typical neo-Gothic styles, 8 positions are quite sufficient. A
composer seeking greater transposibility, however, might find this a
motivation to use 24-out-of-36-tET (or 36-out-of-36-tET!).

As in 36-tET, our typical 7-flavor cadences involve "thirdtone"
melodic steps such as B*3-C4 or Bb3-A*3, here tuned at a precise ratio
of 28:27, or around 62.96 cents. This is slightly smaller than the
66.67-cent step of 36-tET, either tuning having an excellent effect.

The Pythagorean-based septimal comma tuning is thus a near-ideal
optimization except for the matter of our 3-speed design: we don't
have anything close to a 17-flavor in any significant number of
positions.

While this is in fact a very attractive neo-Gothic tuning, it
therefore isn't "the tuning to end all tunings" -- and fortunately, we
won't find any one tuning to replace all the others. Let's go "back to
the drawing board" on our 3-speed optimization problem.

-----------------------------------------------------------
3. An ideal model for the 17-flavor -- with something extra
-----------------------------------------------------------

Suppose that instead of seeking perfect Pythagorean and 7-flavor
ratios, we seek a Pythagorean bike with ideal 17-flavor ratios.

Just what an "ideal" or "perfect" 17-flavor ratio might be remains an
open question: is it specifically 14:17:21, for example, or equally
anything within about 5 cents in either direction? Let us here take
14:17:21 as _one_ "ideal" as well as mathematically elegant value,
while wisely leaving this larger question open.

To get a 14:17:21 sonority, we can take an approach similar to that
approximated in 36-tET: place our two Pythagorean manuals at a
distance equal to the difference between a regular minor third and a
17:14 "supraminor" third.

This difference between a usual Pythagorean minor third at 32:27
(~294.13 cents) and 17:14 (~336.13 cents) turns out to be the complex
integer ratio of 459:448, or about 41.99 cents. We thus get a 24-note
Pythagorean scheme like this:

156 336 654 843 1038
C#*3 Eb*3 F#*3 G#*3 Bb*3
C*3 D*3 E*3 F*3 G*3 A*3 B*3 C*4
42 246 450 540 744 948 1152 1242
---------------------------------------------------------
114 294 612 816 996
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 204 408 498 702 906 1110 1200

Let's try out a 17-flavor cadence of the same kind we encountered in
36-tET:

B3 C*4
G*3 F*3
E3 F*3

Here the steps E3-F*3 and B3-C*4 have a size equal to about 132 cents,
or more precisely about 132.22 cents, very close to the 133.33-cent
"2/3-tone" step we used in 36-tET.

From our 3-speed optimization perspective, we find the same kind of
problem as with our septimal comma tuning: we have perfect Pythagorean
tunings on each keyboard, and ideal 17-flavor intervals, but now
cannot get anything very close to a 7-flavor.

Before moving on to our next bike design, however, let's pause a
moment to appreciate a different kind of "third speed" on this model,
illustrated in cadences such as the following with the same spellings
as in our examples for 36-tET or the Pythagorean-based septimal comma
system:

E*3 F4 F4 E*4
D4 C4 D*4 E*4
B*3 C4 Bb3 A*3
G3 F3 G*3 A*3

In the first cadence, G3-B*3-D4-E*4 is a rounded 0-450-702-948 cents,
with a 450-cent major third (close to 13:10) and a 948-cent major
sixth (close to 26:15). The latter interval may be close to what
Marchettus of Padua (1318) described as a cadential major sixth
equally distance from the fifth and the octave.

Similarly, G*3-Bb3-D*4-F4 has a 954-cent minor seventh and a
252-cent minor third, the latter close to 15:13.

In either variety of cadence, we have very narrow semitones or dieses
of about 48 cents: e.g. B*3-C4 or Bb3-A*3. These steps have a size
equal to the difference between a usual 90-cent Pythagorean semitone
and the 42-cent distance between the keyboards, and are very close to
the 36:35 step (~48.77 cents) of some ancient Greek theorists in the
enharmonic genus.

In fact, this "3-speed bicycle" shows one way to combine pure
Pythagorean intervals of the kind described by Marchettus of Padua
(2:1, 3:2, 4:3, 9:8) with the kinds of very wide cadential major
thirds and sixths he may well be describing, and very narrow cadential
"dieses." As I caution in my _Flugblatt_, of course, any keyboard
model can only approximate the fluidity of voices or other
flexible-pitch instruments.

We can get similarly striking "Marchettan-like(?)" cadences in 24-tET,
with some interesting symmetry properties, as well as in the more
typical neo-Gothic setting of 29-tET.

To sum up, this is a very notable Pythagorean bike, although with a
different 3-speed set than the one we're looking for here.

--------------------------------------------------
4. All three flavors -- a super-Monzonian solution
--------------------------------------------------

Only within the last couple of weeks or so did a solution to the
Pythagorean 3-speed problem occur to me -- and one with a touch I find
especially elegant, although lovers of _small_ integer ratios may take
another view.

In our previous two 24-note Pythagorean systems, while each keyboard
featured a chain of pure 3:2 fifths or 4:3 fourths, we used an
interval between the two keyboards involving factors such as 7 (64:63)
or 17 (459:448).

Here, however, we'll build a 3-speed bike using _only_ Pythagorean
ratios available in a 53-note chain of fifths or fourths -- that is,
everything will be built from powers of 2:1 and 3:2 or 4:3.

As it turns out, the solution is tune our two 12-note Pythagorean
keyboards at the distance of three Pythagorean commas, an interval
I'll call the "tricomma," equal to 36 pure fifths up less 21 pure
octaves (a single comma is 12 fifths up less 7 pure octaves). This
interval is about 70.38 cents.

The tricomma is also precisely defined by the rather large integer
ratio of 150094635296999121:144115188075855872.

Here is our 24-note tricomma tuning in rounded cents:

184 365 682 886 1066
C#*3 Eb*3 F#*3 G#*3 Bb*3
C*3 D*3 E*3 F*3 G*3 A*3 B*3 C*4
70 274 478 568 772 976 1180 1270
---------------------------------------------------------
114 294 612 816 996
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 204 408 498 702 906 1110 1200

Again, both keyboards are in Eb-G# Pythagorean tunings, so the regular
intervals are all there -- but how about the 7-flavor and 17-flavor?

Test driving this bike, we notice a change in the "user interface" --
the way we get these other speeds. Let's see how 7-flavor and
17-flavor intervals are spelled and played, before explicitly letting
the Monzonian cat out of the bag.

To get 7-flavor intervals, we use spellings like these:

F4 F*4 E*4 E4
D*4 C*4 D4 E4
C4 C*4 A*3 A3
G*3 F*3 G3 A3

Here, for example, in the first sonority G*3-C4-D*4-F4, the near-9:7
major third G*3-C4 is actually spelled as a fourth-less-tricomma, and
the near-9:7 major sixth G*3-F4 is a minor-seventh-less-tricomma.

For the second sonority G3-A*3-D4-E*4, the near-7:4 minor seventh
G3-E*4 is spelled as major-sixth-plus-tricomma, and the near-7:6 minor
third G3-A*3 as major-second-plus-tricomma.

In these progressions, a small semitone step such as C4-C*4 or F4-F*4
in the first cadence, or A*3-A3 or E*4-E4 in the second, is equal to
the 70-cent tricomma itself.

We call this kind of cadential semitone "metachromatic": it involves
moving from a note on one keyboard to the corresponding note on the
other keyboard, with the step equal to the distance between the
manuals, here 70.38 cents.

Most optimizations tend to involve compromises, and we find here that
our 7-flavor intervals are not especially accurate, varying somewhat
from the pure or near-pure ratios we have seen in some other tunings:

------------------------------------------------------
Pure ratio/cents tricomma tuning variation
------------------------------------------------------
9:7 435.08 427.66 -7.42
14:9 764.92 772.34 +7.42
......................................................
7:6 266.87 274.29 +7.42
12:7 933.13 925.71 -7.42
......................................................
7:4 968.83 976.25 +7.42
8:7 231.17 223.75 -7.42
------------------------------------------------------

Our 17-flavor is superb, also involving a different kind of spelling
than in our previous tunings featuring this flavor:

B*3 C#3
G#3 F#3
E*3 F#3

Here our near-17:14 supraminor third, E*3-G#3, is actually a usual
Pythagorean major third less a tricomma, an interval of around 337.44
cents, while the near-21:17 submajor third G#3-B*3 is a minor third
plus a tricomma, or about 364.52 cents. These intervals differ by only
about 1.31 cents from our 17-based ratios.

The large semitone or 2/3-tone steps characteristic of these
resolutions, here E*3-F#3 and B*3-C#4, are realized by the interval of
whole-tone-less-tricomma, or about 133.53 cents, quite close to the
133.33 cents of 36-tET or the 132.22 cents of our Pythagorean-based
pure 14:17:21 tuning.

Thus our 3-speed tricomma model is a success, at least to my ears in
the right timbre: we get the regular Pythagorean intervals on either
keyboard, plus a rather active 7-flavor and an ideal 17-flavor.

As "test driving" these different bikes may have suggested, we can use
the same symbols -- the 12 notes from Eb to G# plus Vicentino's dot --
to specify notes and intervals in any of these 24-note systems. The
musical meaning of these symbols, however, may vary from tuning to
tuning.

For example, we spell a 7-flavor minor third (at or near 7:6) as
G*3-Bb3 in 36-tET or the Pythagorean septimal comma tuning, but as
G3-A*3 in the tricomma tuning. In the pure 14:17:21 tuning with the
two keyboards 42 cents apart, either spelling would yield an interval
not far from 250 cents, or about midway between a regular major second
and a regular minor third by either Pythagorean or 36-tET standards.

In this way, the notation is rather like a tablature: it tells which
keys to play, with the resulting intervals depending on the tuning
system in effect for a given piece or section.

Our bike tour may also have suggested that "optimization" in a
neo-Gothic setting can often be very different than in a typical JI
setting: in addition to simple ratios such as 3:2 or 9:7, we are often
seeking _complex_ ratios, whether built up from Pythagorean chains of
fifths and fourths or derived from some special spacing between the
two manuals.

------------------------------------------------------------
3.1. The Monzonian cat: Out of the bag and across the bridge
------------------------------------------------------------

Now for the super-Monzonian (or Monzian?) aspect. Some months ago,
when the Monz discussed using a 75:64 minor third in place of a
simpler 7:6 in order to accommodate the "5-prime" structure of the
overall composition, I was drawn to the idea of using this ratio in a
neo-Gothic tuning. However, I couldn't come up with any immediately
compelling application, although I told myself that sooner or later I
should tune it on general principle.

Then, just within the last two weeks or so, looking at the "3-speed"
optimization problem, I came up with the tricomma tuning as a solution
for getting both a 7-flavor and a 17-flavor in a 24-note Pythagorean
tuning -- using only ratios of 2 and 3.

At about 70.38 cents, the tricomma just happens to be only 0.29 cents
or so smaller than a 25:24 semitone (~70.67 cents), Monz, in your
beloved 5-prime-limit tuning, which Paul Erlich describes as "RI"
(rational intonation).

As a result, the 7-flavor intervals of the tricomma tuning differ only
by this small amount from a complex 5-limit ratio such as the
Monzonian 75:64 minor third (~274.58 cents), here realized at ~274.29
cents. Likewise our 7-flavor major third is almost identical to 32:25,
our minor seventh to 225:128, and so forth.

In other words, we have what could be taken as a pure 3-prime-limit RI
emulation of Monz's 5-prime-limit RI emulation of a 7-based ratio.

Here my intention wasn't to emulate the Monz or to use a 75:64, only
to come up with a pure Pythagorean tuning with a good 17-flavor as
well as a 7-flavor. Looking at a 53-note Pythagorean tuning in Manuel
Op de Coul's Scala program for MS-DOS, I soon realized that a tricomma
tuning could be a neat solution, not-so-accurate 7-flavor and all.

Then, however, I realized that this tuning included that Monzian 75:64
(or a very close facsimile) I had been meaning to use for months --
maybe making this emulation all the more an act of ultimate flattery
to a connoisseur of "xenharmonic bridges" who richly deserves it.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Pythagorean 3-speed bikes: for Joseph Pehrson

🔗mschulter <MSCHULTER@VALUE.NET>

🔗monz <joemonz@yahoo.com>

🔗jpehrson@rcn.com