Nano-temperament

[Please note that this article is a draft which may have some mathematical
bugs and glitches, but I hope that the concepts at any rate are
interesting -- M.S.]

In digital music, as in the digital graphic arts, one must adapt
"continuous-tone" concepts to the technical realities of devices
limited to a set of discrete colors or tones. Thus a monochrome laser
printer can, strictly speaking, produce only black and white dots, not
intermediate shades. A synthesizer with 768 or 1024 steps per octave
can produce only these tones, not the precise intermediate values
required for a given tuning scheme.

In the case of the laser printer, we can gain the visual effect of
intermediate shades of gray by a process of _halftoning_: grouping
black and white dots into repeated patterns, or halftone cells,
produces an impression of gray shading.

Similarly, with a microtunable synthesizer, it is possible to mix
slightly different interval sizes in order to approximate an
intermediate value. Here I shall first give a simple example of this
technique in approximating Pythagorean tuning with a synthesizer such
as the Yahama TX81Z having 768 steps per octave, and then consider the
somewhat more intricate example of approximating 17-tone equal
temperament (17-tet) on a device with 768 or 1024 steps per octave.

A possible term for this technique is _nano-temperament_, the
deliberate variation of intervals by very small quantities, typically
on the order of one synthesizer tuning unit (e.g. 1.5625 cents on a
768-step device, or 1.171875 cents on a 1024-step device). Since
"microtonal" variations are often somewhat larger, for example the
5.38-cent tempering of the fifth in 1/4-comma meantone, the term
"nano-temperament" may be descriptive.

In fact, nano-temperament may be a useful technique in generating a
wide variety of temperaments, for example different shades of
historical and contemporary meantone. The following examples, which
focus on a Gothic or Xeno-Gothic worldview where fifths and fourths
are the primary harmonic intervals, should be taken as a reflection of
one side of the author's tastes, not of the scope of the technique
itself.


----------------------------------
1. Nano-temperament: A simple case
----------------------------------

The problem of approximating a classic Pythagorean tuning on a
synthesizer provides a ready introduction to nano-temperament.
Interestingly, this is a case where in practice nano-temperament seems
useful on a 768-step device but superfluous on a 1024-step device.

Let us first consider the latter device, where each discrete tuning
step is equal to 1200/1024 or 1.171875 cents. As it happens, 599 of
these steps produce an interval of 701.953125 cents, or only about
002 cents narrower than a pure 3:2 fifth (about 701.955 cents).

While one might suspect that the choice of 1024 steps per octave as a
standard has more to do with the binary status of this number as 2^10
(making the octave a convenient digital kilostep) than with a taste
for precise Pythagorean tuning, nevertheless this choice generates an
extraordinarily close approximation of a just fifth.

With the very popular standard of 768 steps per octave, however, the
approximation is not quite so close. Taking 449 steps of 1200/768 or
1.5625 cents, we produce a fifth of 701.5625 cents, or about .3925
cents narrow.

If we consider only a single fifth, this is a very reasonable
approximation, being about five times closer to just than 12-tet, in
which fifths are often considered "nearly pure."

If we consider other intervals generated from a series of such fifths,
however, then this small error becomes cumulative, possibly having
tangible effects on the aural quality of such intervals.

Let us consider, for example, the Pythagorean major third. In Gothic
and related contexts, this is an active interval, ideally having a
ratio of 81:64 (about 407.82 cents). Four fifths of 701.5625 cents,
however, produce a slightly narrow major third of 406.25 cents, or
about 1.57 (.3925 x 4) cents smaller than our ideal size.

Is this a distinction with a musical difference? At least one author,
Easley Blackwood, suggests a demarcation line at around 406 cents as
the point where major thirds become too active to serve pleasingly as
points of harmonic repose -- corresponding in regular tunings to a
fifth of around 701.5 cents.[1] Thus if we want the extra bit of
"edge" that would make our major third unequivocally Pythagorean and
active, some adjustment might not be out of place.

Fortunately, it is possible to make this adjustment simply by making
one fifth out of four an extra tuning step large: 450 steps rather
than the usual 499. This yields a fifth of 703.125 cents, or about
1.17 cents wide -- almost exactly correcting the error of the other
three fifths, each .3925 cents narrow.

Nano-tempering one fifth out of each four in our tuning chain thus
produces a major third equal to three 449-step fifths plus one
450-step fifth. From this sum of 1797 steps, we factor out two octaves
(1536 steps), arriving at a size of 261 steps, or 407.8125 cents, only
about 0.0075 cents from an ideal 81:64.

Curiously, this is the same result we get with four equal fifths of
599 steps on a 1024-step device, which yields a major third with a
size of (2396 - 2048) or 348 steps, also 407.8125 cents. Such an
outcome is not so surprising, since three steps on a 768-step device
are equal to four steps on a 1024-step device. Both results, in fact,
are equivalent to 87 steps in 256-tet.


---------------------------------------------
1.1. Pythagorean nano-temperament in practice
---------------------------------------------

Applying these techniques to an actual chain of fifths, we might
choose the following scheme for 768-step devices, with "s" and "l"
meaning small and large fifths respectively:

l s s s l s s s l s s
768 Eb Bb F C G D A E B F# C# G#

s = 449#768 = 701.56 cents
l = 450#768 = 703.125 cents

A nano-temperament of this kind involves a slight compromise in the
quality of the fifths intentionally tuned a bit further from just in
order to compensate for cumulative errors in the other direction. Such
compromises, fortunately, are typically much milder than those
encountered in traditional kinds of temperament. Here each wide fifth
of 703.125 cents, although three times as far from pure as the other
very slightly narrow fifths, is still only about 1.17 cents or about
1/20 Pythagorean comma from just. At c'-g' (the fifth on middle C),
this tempering would produce one beat each 1.88 seconds or so.

It should also be noticed that while all major thirds will have as
generating intervals three small fifths and one large fifth, thus all
sharing the same nearly-ideal Pythagorean size, other intervals will
vary somewhat.

For example, major sixths or minor thirds may result either from three
small fifths, or from two small fifths plus one large one. The former
arrangement results in a slightly "subdued" interval about 1.17 cents
narrower than 27:16 in the case of M6, or wider in the case of m3. The
latter arrangement produces an even more slightly "super-vibrant"
interval just .3925 cents narrower than a precise Pythagorean tuning
in the case of M6, and the same amount narrower for m3.


------------------------------

Topic No. 3
2. Nano-temperament for 17-tet
------------------------------

Topic No. 4

One way of viewing 17-tet is as a kind of exaggerated Pythagorean
tuning. Fifths are equal to ten scale steps of about 70.588 cents, or
705.88 cents. Major thirds are equal to six steps, or 423.53 cents,
roughly midway between the Pythagorean 81:64 (407.82 cents) and the
9:7 form of some extended just intonation systems (435.08 cents).
While the question of what to do with the latter interval might have
many answers, an obvious one from a medievalist viewpoint is "Let it
expand to a fifth by conjunct contrary motion."

This scale also provides a minor seventh (14 steps) of 988.24 cents,
about a third of the way from the Pythagorean 16:9 (996.09 cents) to a
7:4 (968.83 cents). The major second (3 steps) of 211.76 cents is
likewise about a third of the way from a Pythagorean 9:8 (203.91
cents) to 8:7 (231.17 cents).

Translating a scale step in 17-tet into sythesizer tuning units, we
find that one scale step equals about 45.176 units on a 768-step
device, or 60.235 units on a 1024-step device. Thus the closest
approximation to a scale step is 45 tuning units on a 768-step device
or 60 units on a 1024-step device, in either case 70.3125 cents (about
276 cents narrow).[2]

To correct for this error, we might add an extra tuning step for every
four scale steps on a 1024-step device, or every six steps on a
768-step device:


----------------------------------------------------------------------
17-tet 768-tet 1024-tet
interval steps cents steps cents +/- steps cents +/-
----------------------------------------------------------------------
Unison 0 0.000 0 0.000 0.000 0 0.00 0.000
m2 1 70.588 46* 71.875 +1.287 61* 71.484 +0.896
A1 2 141.176 91 142.188 +1.012 121 141.797 +0.621
M2 3 211.765 136 212.500 +0.735 181 212.109 +0.346
m3 4 282.353 181 282.813 +0.460 241 282.422 +0.069
A2 5 352.941 226 353.125 +0.184 302* 353.906 +0.965
M3 6 423.529 271 423.437 -0.092 362 424.186 +0.657
4 7 494.118 317* 495.313 +1.195 422 494.531 +0.413
d5 8 564.706 362 565.625 +0.919 482 564.844 +0.138
A4 9 635.294 407 635.938 +0.644 543* 636.328 +1.034
5 10 705.882 452 706.250 +0.368 603 706.641 +0.759
m6 11 776.471 497 776.563 +0.092 663 776.953 +0.482
d7 12 847.059 542 846.875 -0.184 723 847.266 +0.207
M6 13 917.647 588* 918.750 +1.103 784* 918.750 +1.103
m7 14 988.235 633 989.063 +0.827 844 989.063 +0.827
A6 15 1058.824 678 1059.375 +0.551 904 1059.375 +0.551
M7 16 1129.412 723 1129.688 +0.276 964 1129.688 +0.276
8 17 1200.000 768 1200.000 0.000 1024 1200.000 0.000
-----------------------------------------------------------------------

For both 768-tet and 1024-tet devices, an asterisk (*) indicates that
the last scale step has an extra tuning unit: 46 units (71.875 cents)
in 768-tet, and 61 units (71.484 cents) in 1024-tet.

Adding what might be called an "intercalary" tuning unit every six
steps in 768-tet or four steps in 1024-tet (rather like a leap year)
gives a close approximation of 17-tet.[3] However, this simple pattern
does cause some intervals to vary from their ideal size by more than
half of a tuning unit. Making a correction only when an error of at
least half a tuning unit would otherwise occur results in a scheme
like this:

----------------------------------------------------------------------
17-tet 768-tet 1024-tet
interval steps cents steps cents +/- steps cents +/-
----------------------------------------------------------------------
Unison 0 0.000 0 0.000 0.000 0 0.00 0.000
m2 1 70.588 45 70.313 -0.275 60 70.313 -0.275
A1 2 141.176 90 140.625 -0.551 120 140.625 -0.551
M2 3 211.765 136* 212.500 +0.735 181* 212.109 +0.346
m3 4 282.353 181 282.813 +0.460 241 282.422 +0.069
A2 5 352.941 226 353.125 +0.184 301 352.734 -0.207
M3 6 423.529 271 423.437 -0.092 361 423.047 -0.482
4 7 494.118 316 493.750 -0.368 422* 494.531 +0.413
d5 8 564.706 361 564.063 -0.643 482 564.844 +0.138
A4 9 635.294 407* 635.938 +0.644 542 635.156 -0.138
5 10 705.882 452 706.250 +0.368 602 705.469 -0.413
m6 11 776.471 497 776.563 +0.092 663* 776.953 +0.482
d7 12 847.059 542 846.875 -0.184 723 847.266 +0.207
M6 13 917.647 587 917.188 -0.459 783 917.578 -0.069
m7 14 988.235 632 987.500 -0.735 843 987.891 -0.344
A6 15 1058.824 678* 1059.375 +0.551 904* 1059.375 +0.551
M7 16 1129.412 723 1129.688 +0.276 964 1129.688 +0.276
8 17 1200.000 768 1200.000 0.000 1024 1200.000 0.000
-----------------------------------------------------------------------


-------------
3. Conclusion
-------------

The method of nano-temperament seems useful in a range of musical
contexts, for example in approximating various shades of meantone.
Since a meantone major third is derived from four fifths, it is
possible by varying the size of _some_ of these fifths by one tuning
unit to achieve shades of _average_ temperament only 1/4 tuning unit
apart. The major thirds will vary by an amount four times this size: a
single tuning unit.

In many ways, nano-temperament presents a kinder and gentler version
of usual tuning dilemmas: some intervals are might slightly less pure
in order to avoid larger discrepancies or inconsistencies. As the use
of synthesizers for historical and new scales and temperaments becomes
more common, this technique invites further exploration.


---------------------
Notes
---------------------

1. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_
(Princeton: Princeton University Press, 1985), pp. 202-203. Blackwood
sets "the largest permissible fifth" for music where thirds are stable
intervals at about 701.5 cents.

2. Pythagorean tuning enthusiasts may take note that a 17-tet step of
70.588 cents is not far from three Pythagorean commas of around 23.46
cents, or 70.38 cents. As it happens, both 768-tet and 1024-tet give
an excellent approximation of this comma with 15 and 20 tuning units
respectively, either yielding 23.4375 cents.

3. This "intercalary" metaphor is much indebted to Guido d'Arezzo, who
compares an octave to a week: "Just as when seven days have elapsed we
repeat the same ones, so that we always name the first and eighth days
the same; so we always represent and name the first and eighth notes
the same way..." See Guido's _Micrologus_ (c. 1030?), translated in
Warren Badd, tr., Claude V. Palisca, ed., _Hucbald, Guido, and John on
Music: Three Medieval Treatises_ (Yale University Press: New Haven,
1978), at p. 61. Guido's metaphor has a special appeal because the
term "octave" is indeed used in the liturgical calendar to signify the
interval of a week. Given the complexities of calendar reform,
possibly comparing an interval tuned slighter wider (or narrower) than
the closest approximation on a digital device to a "leap year" is not
so inapposite.

Margo Schulter
mschulter@value.net
22 July 1998