Quasi-17-tet by just intonation

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Quasi-17-tet by 5-limit just intonation:
A "wide double schisma fifth" approximation
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1. Introduction
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Recently I posted a just intonation tuning very closely approximating
12-tone equal temperament (12-tet), using the schisma fifth
16384:10935 -- which had, in fact, been proposed by Johann Philipp
Kirnberger as early as 1766.

>From a xenharmonic perspective, this interval (about 700.00128 cents)
and the fourth of 16384:10935 (its octave complement) occur in various
musical systems and tunings, including Alexander Ellis's variant on
Helmholtz's 1/8-schisma temperament[1].

It is only fair to add that while Kirnberger and other developers and
exponents of this interval (Lambert, Marpurg, and later Farey) were
indeed seeking a just intonation approach for tuning the virtual
equivalent of 12-tet, the schisma fourth or fifth is for Ellis
actually a "defective" interval in a tuning scheme striving for pure
fifths and thirds -- albeit hardly a "Wolf" .[2]

Apart from teaching me once again that it can indeed be hard to come
upon a really "new" tuning, my encounter with Kirnberger's
quasi-12-tet scheme has suggested an even more intricate scheme with
even larger integer ratios for tuning an approximation of one of my
favorite scales: 17-tet.

To get this idea, I needed only to realize that a 17-tet fifth of
about 705.882 cents is very close to a pure fifth at 3:2 (about 701.955
cents) plus _two_ schismas of about 1.95372 cents -- or 705.862 cents.

While such an approximation of 17-tet might be even more unlikely than
Kirnberger's quasi-12-tet for anyone actually to tune, I found it a
charming exercise in musico-mathematical poetry. Each interval in the
chain seems full of historical associations lending their flavor to
the intrigue of the whole.


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2. Of fifths, thirds, and a curious semitone
--------------------------------------------

Our object is to arrive at a generating interval for our quasi-17-tet
scale: a fifth two schismas wider than pure. This turns out easy to
arrange on paper, although even less easy than quasi-12-tet to tune in
practice. If one accepts Barbour's description of Kirnberger's scheme
for the latter task as "roundabout,"[3] I'm not quite sure what term
might apply to this scheme -- possibly "labyrinthine"?

We begin with a chain of 9 pure fifths up, or fourths down where
called for to keep within the desired octave, arriving at a schisma
minor third of 19683:16384, e.g.

c-g-d-a-e-b-f#-c#-g#-d#

This interval of c-d#, a Pythagorean augmented second, is precisely
one schisma wider than a pure m3 at 6:5 (about 315.64 cents), or
about 317.595 cents.

If we add another chain of nine pure fifths, we get a kind of curious
"diminished fifth" equal in size to precisely two such schisma thirds:

19683:16384
----------------------
c-g-d-a-e-b-f#-c#-g#-d#-a#-e#-b#-f##-c##-g##-d##-a##-e##
-----------------------------------
19683:16384

The interval c-e## might be described in two ways. From a Pythagorean
viewpoint, it consists of two augmented seconds each a Pythagorean
comma (531441:524288, about 23.46 cents) wider than a usual m3 at
32:27 (about 294.13 cents).

>From a 5-limit perspective, it could also be described as being formed
from two minor thirds each a schisma wider than 6:5, producing an
interval _two_ schismas wider than a diminished fifth of 36:25 (about
631.13 cents).

Either calculation gives us a ratio for c-e## of 387420489:268435456,
or the square of our schisma m3 ratio 19683:16384 -- about 635.190
cents.

To get from this curious interval to our quasi-17-tet fifth, we move
first a just 5:4 major third up, and then a just 6:5 minor third
down. The difference of these two intervals defines a 5-limit
chromatic semitone of 25:24, or about 70.672 cents. Adding this 25:24
interval to our "double minor schisma third" c-e##, we arrive at the
not inconsiderable ratio of 3228504075:2147483648, or about 705.862
cents.

While conventional Pythagorean notation nicely expresses all of our
chain of fifths from c to e##, representing these last steps invites a
notation showing variances from a Pythagorean tuning in syntonic
commas of 81:80 or about 21.506 cents. Thus we might write the
intervals up a major third and then down a minor third as:

e##-g###(-1)
g###(-1)-e###(-2)

and the final quasi-17-tet fifth (two schismas wider than just) as:

c-e###(-2).

Here each sharp raises a note by a Pythagorean apotome of 2187:2048
(about 113.69 cents), while each parenthetical negative number lowers
a note by that number of syntonic commas. Starting with a regular
Pythagorean major third of c-e (81:64, about 407.82 cents), adding an
apotome for each of the three sharps to obtain c-e###, and then
subtracting two syntonic commas to obtain c-e###(-2), brings us to the
expected fifth-plus-two-schismas.

It is understandable why tuning 18 pure fifths, a pure major third,
and then a pure minor third down to arrive at a quasi-17-tet fifth may
be more enchanting in theory than in practice.

>From a theoretical and contemplative viewpoint, however, each step has
its significance. The familiar Pythagorean intervals of the 18-fifth
chain c-e## represent the classical Gothic style from Perotin and
earlier to Machaut. The schisma minor thirds c-d# and d#-e##, also a
part of this chain, recall the era around 1400 when such thirds
heralded a new approach to keyboard tunings and music. The 25:24
semitone, and the pure 5:4 and 6:5 third together defining it, evoke
for me the tertian just intonation of the Renaissance.


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3. How accurate is quasi-17-tet?
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Comparing a true 17-tet fifth of 10/17 octave (about 705.8824 cents)
with our quasi-17-tet fifth-plus-two-schismas (about 705.8624 cents),
we have a difference of about 0.0200 cents, or 1/50 cent.

If we generate a quasi-17-tet scale, seeking to close the circle at
the octave by tuning 17 such fifths, then we will fall slightly short
of this octave (17 fifths - 10 octaves) by a comma having a ratio
equal to

(3228504075:2147483648)^17
--------------------------
2^10

or the rather large integer ratio

449803428507699070078142388657801512055553043909284604089805711
333684881797231175590710351961955330881848802903918457871062446
517174209283548407256603240966796875:
449891379454319638281053850768598185886969711830191663310075557
261183758067148787031904068610389085714992091063352089512320826
605549429968900851518086516385513472

It will be seen that the two almost equal terms of this ratio are each
somewhat larger than a googol, or 10^100. Since 17-quasi-tet fifths
yield an interval slightly _smaller_ than an octave, our ratio is
slightly _less_ than unity, or about 0.99984775469834538259397238309.

Converting to cents, we find that the quasi-17-tet comma is equal to
about -0.33847849865247873430151573027 cents, with the negative sign
again showing that in this tuning 17 fifths is a bit _smaller_ than 10
octaves.

This approximation of 17-tet thus involves an error factor about 20
times larger than in Kirnberger's quasi-12-tet[4], but still somewhat
smaller than that of 1/3-comma meantone vis-a-vis 19-tet, or 1/4-comma
meantone vis-a-vis 31-tet. These two meantone tunings have commas of
about .9385 cents wide (19 fifths/11 octaves) and 6.0687 cents narrow
(31 fifths/18 octaves) respectively.

If we assume a truncated quasi-17-tet tuning with a pure 2:1 octave,
then the 17th fifth would be _larger_ than the others by the comma of
about.338 cents, thus being equal to about 706.201 cents, as compared
to the other fifths at 705.86 cents, or 705.88 cents in 17-tet
proper. Such a small divergence should hardly have a tangible effect
on the character of the tuning vis-a-vis true 17-tet.

Other divergences within the scale would be even smaller: thus the
true 17-tet major third (6/17 octave) is about 423.53 cents, as
compared to its quasi-17-tet counterpart of about 423.45 cents. This
interval thus retains its characteristic flavor and its penchant for
expanding to a fifth with a definite neo-Gothic "zonk."

Incidentally, a scale step in either 17-tet or quasi-17-tet -- that
is, the diatonic semitone of about 70.59 cents in either tuning --
happens to be almost identical to the 25:24 chromatic semitone of
tertian just intonation (about 70.67 cents). Thus it might be
appropriate that the just tuning chain for quasi-17-tet should combine
a chain of 18 pure fifths (suggesting the "neo-Pythagorean" qualities
of the tuning) with a 25:24 ratio.[5]


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4. Naming caution and dedication
--------------------------------

Having confirmed once again by my quasi-12-tet adventure that almost
any tuning may have a long documented history (in that case going back
at least to Kirnberger), I would hesitate to propose _any_ name for
this quasi-17-tet tuning or its comma, the latter likely involving
some of the larger integers encountered in just intonation theory.

However, I would not hesitate to dedicate this essay itself to one
whose career on this planet has immensely advanced xenharmonics, just
intonation, and the recognition of 17-tet as an honored member of the
diverse n-tet family: Ivor Darreg.


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Notes
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1. Hermann L. F. Helmholtz, with translation, notes, and Appendices by
Alexander J. Ellis, _On the Sensations of Tone as a Physiological
Basis for the Theory of Music_, 6th ed. (New York: Peter Smith, 1948),
p. 316, n. at *. Ideally, the Helmholtz tuning would call for making
each fifth 1/8-schisma narrow to produce pure major thirds; Ellis,
allowing for the limited accuracy of human hearing, instead "dumps"
the full schisma on one fifth or fourth out of the chain of 8
producing a pure M3.

2. Ibid. Ellis remarks that such fifths or fourths "are defective,
.. being 700 cents down or 500 up, in place of 702 and 498 as all the
others." Lest any reader be tempted to take this "defect" too
seriously, it might be well to recall the view of Helmholtz, ibid. at
p. 315, that "[t]he principal fault of our present tempered intonation
[i.e. 12-tet], therefore, does not lie in the Fifths; for their
imperfection is really not worth speaking of, and is scarcely
perceptible in chords." At issue, of course, was rather the tuning of
major and minor thirds.

3. J. Murray Barbour, _Tuning and Temperament: A Historical Survey_
(East Lansing: Michigan State College Press (1953), p. 64.

4. As Ellis notes in Helmholtz, see n. 1, p. 316 n. at *, the schisma
fifth (used in Kirnberger's quasi-12-tet) is larger than a true 12-tet
fifth by about 0.001280 cents. Thus the comma (12 fifths/7 octaves)
is 12 times this variance, or about 0.01536 cents.

5. It is interesting that an interval of about 70 cents serves as a
_diatonic_ semitone (e.g. e-f, c#-d, eb-d) in 17-tet or quasi-17-tet,
but as a _chromatic_ semitone (e.g. c-c#, eb-e) in tertian just
intonation, where the diatonic semitone of 16:15 is equal to about
111.73 cents. Curiously, the usual 17-tet or quasi-17-tet M3 of about
423.5 cents is quite close to the 32:25 diminished fourth (about
427.37 cents) of tertian just intonation. Here we see a dramatic case
of Mark Lindley's observation associating active thirds with keen
diatonic semitones (as holds more moderately in Pythagorean and
12-tet), and restful thirds with large diatonic semitones (as holds
also in 19-tet and 31-tet, and the kindred 1/3-comma and 1/4-comma
meantone tunings).

Most respectfully,

Margo Schulter
mschulter@value.net