Cartesian tunings (same as planar?)

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Cartesian Tunings and Regularized Keyboards
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Hello, everyone, and one approach to devising a tuning system is what
I call a "Cartesian tuning," possibly synonymous with the "planar
temperaments" sometimes discussed here.

A typical Cartesian tuning consists of two or more "chains of fifths,"
and can be defined completely by these specifications or parameters:

(1) The number of notes or generators in a chain;
(2) The number of chains;
(3) The size of the fifth, or generator;
(4) The size of the interval between successive chains; and
(5) The "interval of equivalence" (e.g. a 2:1 octave).

Such a tuning might be expressed compactly in a notation like this,
with an explanation in user-friendly English (I hope) following:

1200 <12,2> (704.096, 58.680)

The "1200" indicates an octave of 1200 cents, or a pure 2:1, as the
interval of equivalence.

The <12,2>, written in angle brackets, specifies that each chain of
fifths consists of 12 notes; and that there are two such chains.

The (704.096, 58.680), written in parentheses, specifies a generator
or "fifth" of 704.096 cents; and an interval between chains of 58.680
cents.

When mapped to two 12-note keyboard manuals, forming what is called a
_regularized keyboard_ as we shall see, the result is as follows:

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
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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

Our example of a Cartesian tuning is "Peppermint 24," made up of two
12-note chains in the Wilson/Pepper temperament with a ratio between
the sizes of the chromatic semitone (e.g. C-C#) and diatonic semitone
(e.g. C#-D) equal to Phi, or the Golden Section, approximately 1.618.
The required generator is approximately 704.096 cents.

The interval between the two chains is equal to the difference between
a regular major second, e.g. C-D (about 208.191 cents), and a pure 7:6
minor third (about 266.871 cents), or approximately 58.68 cents.

As our example illustrates, a Cartesian tuning of the <12,2> variety
-- with chains or "rows" of two notes, and two such rows or chains --
lends itself to a "regularized keyboard" formed from two standard
Halberstadt keyboard manuals.

More generally, a "regularized keyboard" has two or more "manuals"
with identical arrangements of steps and intervals on each manual; and
identical intervals between two corresponding keys or levels on two
adjacent rows (here 58.680 cents).

A couple of more examples may further illustrate this concept. Let us
first consider

1200 <12,6> (700, 16.667)

This Cartesian tuning, based on a 1200-cent octave, would consist of
12-note chains or rows, with six such chains; a 700-cent generator
(7/12 octave) for each chain; and a distance of 16.667 cents (or 1/72
octave) between chains.

In other words, we have a regularized 72-EDO keyboard using six
12-note manuals (Halberstadt, Janko, or whatever) at 1/12-tone apart.

A different kind of Cartesian tuning I find quite attractive is this:

1200 <17,2> <705.882, 55.106)

Here, with the interval of equivalence again a 2:1 octave, we have
chains of 17 notes, with two such chains; a generator of 705.882 cents
(10/17 octave) for each chain; and a distance of 55.106 cents between
chains.

This tuning consists of two complete 17-EDO circles at the distance of
55.106 cents, the difference between a regular 17-EDO whole-tone at
3/17 octave (~211.765 cents) and a pure 7:6.

Such a Cartesian tuning would map neatly, for example, to a
regularized keyboard consisting of two 17-note manuals, each with five
flats and five sharps.

In the special case where each chain or row forms a closed circle or
approximately closed "loop" in itself, we have a "bike chain" tuning
or the like. Either of our last two examples illustrate such a
system: the first defines a complete 72-EDO tuning, while the second
consists of two complete 17-EDO circles but does not itself define a
closed tuning.

An example of an "approximate bike chain" tuning would be:

1200 <31,2> (697.578, 5.377)

This system consists of two 31-note chains or "loops" of 1/4-comma
meantone at 1/4 syntonic comma apart -- basically Vicentino's likely
system of adaptive just intonation (factors of 2-3-5).

This is an "approximate bike chain" because a 31-note meantone chain
is approximately or musically rather than precisely or mathematically
closed: 31 meantone fifths fall short of 18 pure octaves by about 6.07
cents, so that "closure" is a choice rather than a mathematical
necessity.

While the examples above all involve temperament, it is also quite
possible to have Cartesian just or rational tunings for example:

2:1 <12,2> (3:2, 64:63)

This system has two 12-note Pythagorean chains at a 64:63 (~27.26
cents) apart, yielding some pure ratios of 2-3-7-9 such as
12:14:18:21.

Most appreciatively,

Margo Schulter
mschulter@value.net