Yahoo Tuning Groups Ultimate Backup mills-tuning-list XH 17: n-tet's and harmony

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Harmonic styles and n-tet's:
A multidimensional approach
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1. Introduction
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This paper grows out of two germinal articles in _Xenharmonikon_ 17:
Paul Erlich's far-ranging presentation on scale theory and 22-tone
equal temperament (22-tet) [1], and Brian McLaren's sweeping
chronicles of microtonalism in the 20th century [2].

Both articles, in different ways, raise the question of how one might
go about evaluating the harmonic possibilities of a given tuning
dividing the octave into n number of equal parts -- that is, an n-tone
equal temperament (n-tet).

While the term "harmonic" can mean many things to many people, I use
it here basically in the sense of the "vertical" dimension of music:
that is, the aspect of music concerned with sonorities of two or more
simultaneous tones, and progressions involving such sonorities.

Thus while both 13th-century and 18th-century Western European music
are harmonically oriented, they are "harmonic" in very different ways,
differing in their historical tunings, stable and unstable interval
categories, and directed cadences. Curiously, certain scales such as
53-tet can provide an excellent fit with _either_ harmonic system.

In evaluating the harmonic possibilities of n-tet's, we need ways of
mapping both the scales themselves and the harmonic systems they might
realize in whole or part. This paper represents one approach to this
multidimensional perspective.

Section 2 considers conceptual maps for n-tet scales themselves, while
Section 3 focuses on maps for some harmonic systems, and presents some
tentative observations about a few n-tet's including Erlich's
intriguing 22-tet. Section 4 proposes directions for further
analysis.

At the outset, I should offer a vital disclaimer: this paper focuses
mainly on Western European tunings and harmonic systems from around
1200 to the present, and thus presents at best a very partial and
biased sample of the musical possibilities. Other systems, for example
gamelan tunings approximating 5-tet or 7-tet, deserve equal
consideration, very possibly calling for a modification of the models
which follow. Being aware of the dangers of "universal" models of
language or music which prove to be more provincial than their authors
suspect, I would strongly caution that this paper is not exempt from
such dangers.

-------------------------------------------------------
2. Maps for scales: just intervals and meantone tunings
-------------------------------------------------------

In orienting ourselves to a given n-tet, there are at least two
familiar conceptual maps available. One approach is to compare the
intervals of an n-tet with those of a just intonation system based on
integer ratios, such as Pythagorean tuning or Zarlino's tuning for
vocal music. Another is to compare an n-tet with meantone systems
where all fifths (except possibly one) are tempered by the same
amount.

Both approaches have a long history. Using the just intonation
approach, for example, we can say that 53-tet offers excellent
approximations both for the true Pythagorean major third of 81:64 and
the Zarlinan major third of 5:4. Using the meantone approach, we can
say that 19-tet has a close kinship to 1/3-comma meantone, and 31-tet
to 1/4-comma meantone.[3]

---------------------------------------------------------
2.1. Intonational justice: limits, mixes, and curiosities
---------------------------------------------------------

In describing just intonation systems, it is common to refer to the
most complex prime number used in generating the intervals of a given
system as its "n-limit."

Thus in a 3-limit or quintal just intonation system -- often known as
Pythagorean tuning -- all interval ratios are derived from multiples
of 2 and 3, with the pure fifth (3:2) as the generating interval. In a
5-limit system such as Zarlino's, the pure major third (5:4) also
becomes a generative ratio. In a 7-limit system, the septimal minor
seventh (7:4) also becomes literally and figuratively a "prime factor"
in defining the scale, and so on.

In doing "intonational justice" to these systems, it is very important
to recognize that the prime limit of a given system is not necessarily
the "limit" of intervals perceived by users of the tuning to be
"concordant" or musically valuable. This point is of crucial
importance to the analysis which follows, so illustrations may be
helpful.

Thus in the complex 3-limit system of Gothic polyphony, the major
third (81:64) and minor third (32:27) are ranked as relatively
concordant in theory, and in practice play a vital role as mildly
unstable intervals in various sonorities. The just 9-based intervals
of the major second (9:8), minor seventh (16:9), and major ninth
(9:4), although regarded as somewhat more tense, are also credited
with some "compatibility" or "concord," adding harmonic color and
action to the music. Around 1400, the even more complex schismatic
form of major third (8192:6561) plays a pivotal role in keyboard
tuning systems and compositions.

While the 5-limit just intonation of the Renaissance tends to
correlate with less subtly shaded theoretical schemes of concord and
discord, nevertheless authors such as Zarlino recognize the vital role
of intervals such as the major second (9:8 or 10:9), minor seventh
(9:5), and major seventh (15:8) in gracing a music based mainly on
5-limit consonances. Composers of the Manneristic epoch around 1600
such as Monteverdi and Gesualdo exploit such intervals more boldly, as
well as the diminished fifth (45:32) already endorsed by Zarlino as
pleasing when aptly resolved.

Much 20th-century music intended for 12-tet provides an even more
dramatic example, where stable sonorities involving mixtures of thirds
and minor sevenths may suggest a "7-limit" concept, although the
tuning system offers no close approximation of a 7:4 ratio.[4]

Further, with just intonation systems as with others, the very
_deviations_ of certain intervals from the simplest or "purest"
possible ratios can have a musical value all its own. For example, the
complexity of a 3-limit major third (81:64) nicely fits the Gothic
concept of "imperfect concord," while the analogous tension of a
5-limit minor seventh (9:5) adds an edge to Renaissance suspensions
and the bolder formations of a Monteverdi. A just appraisal should
strive to recognize such intervals as features, not as the "flaws"
they might well be in the context of a different system.

Having considered some complications of the "n-limit" concept itself,
we face the further complication that a given n-tet may approximate a
mixture of intervals from various n-limit systems, sometimes a
curiously selective one.

For example, 53-tet has excellent 3-limit, 5-limit, and 7-limit
approximations, so that it might be used for any style of harmony from
Gothic to 7-limit just.. While 12-tet also offers a very good 3-limit
approximation, it fits 5-limit intervals rather less accurately, and
7-limit intervals still less so.

Other scales, however, may offer a "mixed bag" of approximations not
so easily described. Thus 19-tet and 22-tet offer fair approximations
of 3-limit intervals (with the fifth in either case about 7 cents from
a just 3:2), but much closer approaches to just 5-limit intervals. In
17-tet, we have a good 3-limit approximation of 3:2 and 4:3, and also
major and minor thirds not too far from 9:7 and 7:6 respectively --
but no real equivalents for 5-limit intervals.

Even more intriguingly, 11-tet has no real equivalents for 3-limit
fifths or fourths, but reasonable approximations for the 5-limit minor
third (5:4) and major sixth (5:3), as well as the 7-limit or 9-limit
major third (9:7) and minor seventh (7:4).

Brian McLaren notes that in scales such as 13-tet, 18-tet, and 23-tet,
"the thirds are reasonably good but there is nothing like a perfect
fifth."[5] Such scales are in fact an opportunity to realize
traditional harmonic systems in new ways, or indeed to construct new
systems, an opportunity which, as McLaren urges, should not be missed.

--------------------------
2.2. The meantone spectrum
--------------------------

A different paradigm for mapping n-tet's is the meantone spectrum,
ranging in historical terms from Pythagorean tuning (0-comma meantone)
to around 1/3-comma of tempering. This spectrum may be expanded in
both directions to embrace regular tunings with fifths wider than a
just 3:2, or narrower than just by more than 1/3 syntonic comma.

Certain points on the meantone spectrum coincide with just tunings.
Thus Pythagorean tuning is both a just intonation system and a
meantone tuning, although not a meantone _temperament_. Curiously, the
5-limit just tuning with a fifth equal to 16384:10935 (precisely a
schisma narrower than 3:2) very closely approximates 1/11-comma
meantone and 12-tet. Other points offer just ratios for certain
intervals: thus 1/4-comma meantone has pure 5-limit major thirds (5:4)
and minor sixths (8:5), while 1/3-comma has pure minor thirds (6:5)
and major sixths (5:3).

At first blush, it might seem both intuitive and easy to place a range
of n-tet's on the meantone spectrum based on the size of fifth. Thus
19-tet is much like 1/3-comma temperament with its quite narrow fifths
and pure minor thirds, and 31-tet like 1/4-comma with its somewhat
more moderate tempering of the fifth and pure major thirds. Less
familiarly, for example, 91-tet approximates 1/7-comma meantone, with
somewhat restful thirds (vis-a-vis Pythagorean or 12-tet).

This kind of orientation based on fifth size can work even for certain
tunings beyond the historically familiar range. Thus 17-tet, with a
fifth of around 705.88 cents, seems to behave as we might expect,
serving as a kind of "ultra-Pythagorean" tuning with "superactive"
thirds and sixths urgently inviting resolution to stable 3-limit
intervals.[6] Even here, however, the novel element of the neutral
third cautions that new systems are more than stimulating variations
on old ones.

A reading of Erlich's insightful article, even if one has (like
myself) only begun to digest the riches there offered, neatly reveals
how misleading it can be to judge a tuning in this way on the basis of
its fifth size. In 22-tet, the fifth at 709.09 cents is more than 7.13
cents wide, yet the scale offers nice approximations of 5-limit thirds
and sixths, as well as the 7-limit and 9-limit intervals present in
11-tet (a subset).[7]

In short, the meantone model is a "map" which works fairly well for
some n-tet's, but not for others. In this area as in others, 22-tet is
a tuning with lots of musical and conceptual surprises to upset such
models (at least when applied overbroadly) in a delightful way.

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3. Harmonic systems: trines, triads, tetrads, and beyond
--------------------------------------------------------

During the long interval of Western European composition from around
1200 to 1900, interestingly enough, harmonic systems were based on two
structures requiring a mininum of three voices: the medieval 3-limit
trine (string ratio 12:8:6, frequency ratio 2:3:4); and the
Renaissance to Romantic 5-limit triad (string ratio 15:12:10,
frequency ratio 4:5:6). Both threefold sonorities have inspired poetic
and even theological allegory.[8]

Since the time of Debussy, more complex sonorities have been accepted
as stable: for example, tetrads such as the "added sixth" (12:15:18:20
in one possible 5-limit tuning) and "minor seventh" (12:14:18:21 in a
9-limit tuning).

One obvious requisite of realizing a given harmonic system with a
given n-tet is that "reasonable" approximations of the intervals
involved in such stable structures be available. Thus 53-tet seems a
superb choice for either trinic or triadic harmony. While 12-tet
likewise would be a reasonable choice for either system, its almost
pure fifths but rather active thirds might make it more apt for trinic
than for triadic styles.

In contrast, 17-tet has good fifths and fourths for trinic sonorities,
but no intervals close enough to 5:4 or 6:5 to support a conventional
triadic system. With 11-tet, we have fair approximations for the
triadic concords of the minor third (6:5) and major sixth (5:3) -- but
no fifths or fourths of the kind required for either trinic or triadic
sonorities of a historical kind.[9]

With 22-tet, as Erlich rightly emphasizes, we have a set of intervals
which may not fit this or that specific integer ratio as well as some
other n-tet's, but which covers all bases, as it were, supporting a
very diverse range of trines, triads, and tetrads.[10]

-------------------------------------------------------
3.2. Broader views: quintal/quartal, tertian, and other
-------------------------------------------------------

To make our map of harmonic systems more inclusive, and thus more
accommodating to the special potentials of a variety of n-tet's, we
must note that the trinic-triadic-tetradic paradigm of composed
Western European and related music is only one possible approach.

More generally, we might view Gothic trinicism as one subset of
_quintal/quartal_ harmony where fifths and fourths serve as the most
euphonious simple intervals, and Renaissance-Romantic triadicism
likewise as a subset of _tertian_ harmony based on thirds and sixths
as the most favored intervals. A given n-tet might support some of
these intervals but not others, as in the case of 11-tet. We should
not ignore the tertian possibilities of this scale because it happens
not to have anything like a triadic fifth.

--------------------------------------------
3.3. Instability, system, and expressiveness
--------------------------------------------

Also, even in focusing on the historical trinic and triadic systems,
we should note the vital role of _unstable_ sonorities in defining the
qualities of a system. For example, in Gothic style, complex 3-limit
sonorities such as 4:6:9 or 9:12:16 may have a rather concordant
quality; in traditional 5-limit harmony, as Erlich notes [11], these
sonorites serve as "suspended" dissonances; and in 20th-century
quartal/quintal harmony they may form stable chords of the fifth and
fourth.

Similarly, in Gothic style, a combination of three superimposed thirds
(e.g. e-g-b-d') may resolve to a stable 3-limit sonority (e.g. f-c');
in triadic harmony, it may resolve to a 5-limit sonority; in 7-limit
harmony, it may serve as a stable tetrad.

Looking _only_ at the stable intervals of a given harmonic system, we
might overlook some crucial ways in which a "kindred" n-tet can alter
and distort the subtle "balance of power" between the intervals,
possibly in creative ways.

For example, while I have referred to 17-tet as a "neo-Gothic" tuning,
its distortions of the Gothic harmonic spectrum might well warrant, at
the least, an accent upon the _neo-_ portion of this description. In
theory and practice, Gothic music is premised on a tuning system where
major and minor thirds (81:64, 32:27) are somewhat more concordant
than bare major seconds and minor sevenths (9:8, 16:9) -- although
these intervals alike are often regarded as neither stable nor sharply
discordant.

In 17-tet, with its major thirds of around 423.53 cents, this order is
arguably reversed, altering the harmonic color of the system radically
although many of the standard trinic resolutions (e.g. those involving
M3-5 and M6-8 by stepwise contrary motion) remain quite practical. In
fact, these progressions may take on an extra quantum of force from
the increased level of harmonic tension and the "supernarrow"
70.59-cent leading tone; but we have moved into the realm of
Xeno-Gothic as opposed to Gothic style.[12]

If our purpose is a more or less faithful approximation of the actual
Gothic interval system, then Pythagorean just intonation would be
ideal, and 53-tet -- or, yes, 12-tet -- a wiser choice than 17-tet.

If our purpose is instead a _really_ novel distortion of the modified
Pythagorean system in vogue around 1400, with its prominent mixture of
regular and schismatic thirds and sixths, then Erlich's 22-tet has
some unique possibilities. While the fifth -- or "perfect seventh" in
Erlich's decatonic system -- is about 7.14 cents wide of 3:2, it is
closer to just than in 1/3-comma meantone or 19-tet, and might lend an
interesting flavor to stable trines.

The striking "neo-late-Gothic" feature of 22-tet is its drastically
contrasting versions of intervals I will here call in traditional
fashion thirds and sixths, although Erlich's decatonic nomenclature is
well worth studying and may give a better insight into the scale's
structure[13]:

---------------------------------------------------------------------
Late Gothic keyboard tuning 22-tet
---------------------------------------------------------------------
interval ratio cents steps cents approx
m3 (regular) 32:27 294.13 5 272.73 ~7:6
m3 (schisma) 19683:16384 317.60 6 327.27 ~6:5

M3 (schisma) 8192:6561 384.36 7 381.82 ~5:4
M3 (regular) 81:64 407.82 8 436.36 ~9:7

m6 (regular) 128:81 792.18 14 763.64 ~14:9
m6 (schisma) 6561:4096 815.64 15 818.18 ~8:5

M6 (schisma) 32768:19683 882.40 16 872.73 ~5:3
M6 (regular) 27:16 905.87 17 936.95 ~12:7
_____________________________________________________________________
---------------------------------------------------------------------

In either system, we have contrasting pairs of more active and more
blending thirds and sixths -- but with the "active" forms much more
dramatically so in 22-tet, to say the least. One attraction of n-tet's
is their ability to transform the familiar into the new, and Erlich's
tuning does so in ways a musician might especially relish.

-----------------------------
4. Conclusions and directions
-----------------------------

The above analysis should not invalidate any of the historically
realized harmonic possibilities, nor the xenharmonic possibilities
already being realized in practice and theory. For example, McLaren's
engaging chronicles sum up one outlook on n-tet's:

"Scales without fifths -- viz., 9, 11, or 13 tones per
octave -- favor a fast-paced contrapuntal style with
brisk percussive timbres, while scales with excellent
fifths, like 31, 41, 53, or 118 tones per octave, favor
a triadic homophonic style of composition."[14]

The concepts I have sketched might lead us to take this conventional
wisdom -- if one can use this term in such an experimental context --
and season it with some further questions. For example, does "triadic"
here mean specifically "5-limit" or "tertian," or might it have
Partch's wider sense of any three-tone combination (e.g. 1:3:9)? A
tuning such as 53-tet can nicely support trinic, triadic (in the
tertian sense), or tetradic harmony, and all these potentials merit
exploration.

Also, where would a tuning such as 17-tet fall in such a dichotomy? It
has an "excellent" fifth, closer to a just 3:2 than 31-tet, for
example -- but is hardly ideal for "triadic" harmony, at least in a
tertian sense. However, it might nicely support a "homophonic style
of composition" based on Xeno-Gothic progressions from unstable
7-limit or 9-limit sonorities for three or more voices to stable
3-limit sonorities -- and possibly "reverse suspensions" where a third
resolves to a fourth or a major second.[15]

Finally, although this may be a curious comment for one of Pythagorean
inclinations, I would like to join McLaren in questioning whether the
fifth need be treated as the only generative interval for scales or
harmonic systems.[16] If a tuning such as 11-tet or 13-tet happens not
to have any equivalent of 3:2, why not focus on the various 5-limit,
7-limit, 9-limit, and other intervals it _does_ approximate, and
develop a concept of harmonic style -- possibly in some cases
"homophonic" -- from there?

As McLaren shows in this and other portions of his history, timbre is
indeed a crucial factor. Personally, I have found that intervals near
9:7, for example, can take on a pleasant effect if given a certain
kind of "choirlike" voicing available on some synthesizers, lending
themselves to slower textures with resolutions to fifths in a
Gothic-like fashion.

Of course, one might add that "a fast-paced contrapuntal style" can
suit a wide variety of n-tet's, as it has often suited composers
of the Gothic, Renaissance, and Baroque traditions based more or less
on the ideal of some kind of just intonation.

Both McLaren and Erlich have helped show the way to new realms of
theory and practice, while very fruitfully inviting more open
questions.

-----
Notes
-----

1. Paul Erlich, "Tuning, Tonality and Twenty-Two Tone Temperament,"
_Xenharmonikon_ 17 (Spring 1998), 12-40.

2. Brian McLaren, "A Brief History of Microtonality in the Twentieth
Century," _Xenharmonikon_ 17 (Spring 1998), 57-110.

3. Here it may be wise to explain that meantone tunings are commonly
described by specifying the fraction of a syntonic comma (81:80, about
21.51 cents) by which 11 of the 12 fifths of a familiar 12-tone
chromatic scale are tempered. Conventionally, it is assumed that this
tempering is in the _narrow_ direction. Thus in 1/4-comma meantone,
each fifth is narrowed by 1/4 syntonic comma or about 5.38 cents,
being reduced from a just 3:2 (about 701.955 cents) to around 697.58
cents. In the special case of 12-tet, the 12th fifth (traditionally
often g#-eb, for example) is tempered (in theory) by precisely the
same amount as the other 11, all being equal to 700 cents.

4. Not so surprisingly, given its affinities to Pythagorean tuning,
12-tet offers a better approximation of just intonation for sonorities
built in fourths or fifths (e.g. d-g-c' or d-a-e', calling ideally for
3:2, 4:3, 16:9, 9:4, etc.) than for sonorities built with 5-limit or
higher thirds; 20th-century harmony draws on both possibilities.

5. McLaren, see n. 2 above, at p. 87.

6. Of course, such expectations assume a familiarity with the Gothic
harmonic styles of the 13th and 14th centuries.

7. Erlich, see n. 1 above, explores the potentials of 22-tet vis-a-vis
other n-tet's at some length; see especially pp. 13-22.

8. Thus the trine (outer octave, lower fifth, upper fourth) for
Johannes de Grocheio (c. 1300), and likewise the triad for later
theorists such as Johannes Lippius, Johannes Kepler, and Andreas
Werckmeister, was a "perfect harmony" representing in music the
harmony of the Trinity. In traditional theories based on string
ratios, these two sonorities represent the harmonic divisions of the
octave (12:8:6) and fifth (15:12:10) respectively. From the
perspective of frequency ratios, each presents three adjacent tones in
the harmonic series (2:3:4, 4:5:6) beginning with a tone equal to 2^n
of the fundamental.

9. For McLaren's comments on other n-tet's presenting a similar
situation of "no fifths but good thirds," see n. 7 above. Systems with
fifths varying from a just 3:2 by as much as around 7 cents on the
narrow side (1/3-comma meantone, 19-tet) or the wide side (22-tet) can
clearly support a "triadic" style of harmony; some systems of unequal
well-temperament narrow certain fifths in a triadic context by as much
as 1/3 Pythagorean comma, or about 7.82 cents.

10. Erlich, see n. 1 above, at p. 25, offers a very handy table
showing the intervals of 22-tet and the many integer ratios nicely
approximated by the tuning.

11. Ibid., at pp. 12-13 and 26.

12. "Xeno-Gothic" can refer to a specific just tuning based on a chain
of 23 pure fifths generating a 24-note octave, or to a set of Gothic
and "Gothic-like" progressions supported by this scale, or to a
general attitude. A Xeno-Gothic tuning includes the medieval 12-note
chromatic Pythagorean scale as a subset, but additionally introduces
new forms of unstable intervals a Pythagorean comma wider or narrower
than usual. The "Xeno-" aspect may relate especially to such
intervals, which for example can form unstable combinations resembling
7-limit or 9-limit tetrads (e.g. ~12:14:18:21) resolving to stable
3-limit trinic harmonies. A scale such as 17-tet or 22-tet, which
includes both reasonable approximations of 3:2 and 4:3 and intervals
resembling 9:7, 12:7, or 7:4, for example, can therefore elicit a
"Xeno-Gothic" response: "Let's try having that near-9:7 expand to a
fifth, and that near-12:7 to an octave -- or having that near-7:4
contract to a fifth, etc."

13. Erlich, see n. 1 above, e.g. at p. 25, Table 1. This scheme is
indeed a radically different "map" for the intervals: thus 17 steps is
a "minor 9th" (about 927.3 cents), while 18 steps is a "major ninth"
(about 981.8 cents). Much of the surrounding analysis is breathtaking,
the musical equivalent of walking around in some higher-level
geometry. Interestingly, as discussed in recent Tuning List posts,
22-tet with its decatonic scale may invite some Jewish allegories,
adding another view to the metaphors discussed in n. 8 above.

14. McLaren, see n. 2 above, at p. 81. Here the author's emphasis is
on the realization that "_all_ equal temperaments are equally useful
for composing beautiful music" (emphasis in original).

15. Here I am warmly indebted to John Chalmers, Jr. and Gary Morrison
for many discussions about the possibilities of systems such as
17-tet, but of course any dubious conclusions expressed in this paper
are solely my responsibility.

16. McLaren, see n. 2 above, at pp. 86-87, specifically raises
questions about models of "diatonicism" presented in classic works of
Joseph Yasser and Easley Blackwood as the best basis for approaching
tunings such as 13-tet, 18-tet, and 23-tet -- and also many of the
5*n-tet's (e.g. 10-tet, 15-tet, etc.) which violate Blackwood's
requirements for a "recognizable diatonic" tuning. It seems to me that
such questioning is in the best tradition of these two theorists.

Most respectfully,

Margo Schulter
mschulter@value.net

XH 17: n-tet's and harmony

🔗"M. Schulter" <mschulter@...>

🔗"John Loffink" <jloffink@...>