Yahoo Tuning Groups Ultimate Backup tuning Re: Friendly introduction to hypermeantones (1 of 2)

------------------------------------------------
A friendly introduction to hypermeantones:
Regular temperaments beyond Pythagorean
(Part 1 of 2)
------------------------------------------------

In a recent article on "Neo-Gothic tunings and temperaments: Meantone
through a looking glass"[1], I described "reverse meantone" tunings
with fifths somewhat _wider_ than the pure 3:2 ratio of Pythagorean
tuning. Familiar examples of such tunings might include 41-tone equal
temperament (41-tet), 29-tet, 46-tet, 17-tet, and 22-tet.

From an artistic point of view, such temperaments offer accentuated
variations on standard medieval Pythagorean tuning both for 13th-14th
century Western European music of the Gothic era, and for allied
"neo-Gothic" styles of composition or improvisation. Usual Pythagorean
traits such as active thirds and sixths inviting efficient resolutions
to stable 3-limit concords, and narrow diatonic semitones for
expressive melody and incisive cadential action, are heightened in a
form of intonational mannerism.

By describing these regular temperaments beyond Pythgorean as
"neo-Gothic meantones," however, I elicited some predictable and
constructive controversy. In a prompt response, Paul Erlich[2]
questioned whether tunings with fifths wider than pure could be
"meantones" in even "the most inclusive sense."

In a germinal paper surveying the entire spectrum of regular tunings
with fifths from 685 cents to 721 cents, David C. Keenan (1998)
acknowledged that "some authors" would refer to all such tunings as
meantones, only to reject this usage "on historical grounds."[3]

As the result of a very helpful private dialogue with Paul Erlich, in
which he introduced the word "hypertone"[4], I would like to propose
the term "hypermeantone" to describe regular temperaments with fifths
larger than pure. This term may be taken in at least two senses:

(1) The size of the fifths goes "beyond" the range of conventional
meantone temperaments, a range with Pythagorean tuning ("zero-comma
meantone," pure fifths) as one possible upper limit.

(2) The regular major second or whole-tone, larger than 9:8, serves as
a "hypermeantone" for some regular major third with a size going
"beyond" the Pythagorean 81:64.

If we adopt this definition of "hypermeantone," then the term
"hypomeantone" might apply to regular tunings with fifths_smaller_
than in historical meantones, with 1/3-comma meantone or 19-tet as a
possible line of demarcation. One example might be 26-tet, with fifths
at ~692.31 cents, or about 9.65 cents narrower than pure.[5]

From this perspective, the continuum of regular diatonic tunings with
fifths ranging between the limits of 7-tet (~685.71 cents) and 5-tet
(720 cents) would invite a conceptual map[6] like this.

-~16.24 -~7.22 0 +~18.04
~685.71 ~694.74 ~701.96 720
|-----------------|-------------|------------------------------|
7-tet 19-tet Pyth 5-tet
|-----------------|-------------|------------------------------|
hypomeantone meantone hypermeantone

Here signed (+/-) numbers show the tempering of the fifth in the
negative (narrow) or positive (wide) direction from the pure 3:2 ratio
of Pythagorean tuning ("Pyth").

As the term "hypermeantones" may suggest, regular tunings beyond
Pythagorean are at once distinct from traditional meantones and yet
may share kindred aspects of structure and artful compromise. This
paper seeks to explore some contrasts and parallels, while touching
here and there on hypomeantones and hopefully encouraging a full
exploration of these tunings also.

Section 1 offers an approach to meantones as regular tunings involving
a trade-off between 3-limit and 5-limit concords, and thus having a
range from about 1/3-comma (pure 6:5 minor thirds) or 19-tet to
Pythagorean (pure 3:2 fifths). Hypomeantones with fifths narrower than
in 19-tet, and hypermeantones with fifths wider than Pythagorean,
evidently involve other kinds of compromises and balances.

Section 2 explores how hypermeantones may bring into play interactions
and balances between prime limits _analogous_ to meantones, involving
for example the septimal rather than syntonic comma, as in the case of
22-tet as a near approximation of "1/4-septimal-comma hypermeantone"
with pure 9:7 major thirds.

Section 3 shows how, more generally, hypermeantones may achieve or
approximate higher-prime-limit ratios for various intervals, thereby
facilitating an intriguing encounter between the intonational systems
and musical ideals of the 14th and 21st centuries.

Section 4 considers how "alternative thirds" -- diminished fourths and
augmented seconds -- can serve as bridges between the hypermeantone
and hypomeantone portions of the spectrum.

-----------------------------------
1. The meantone equation and beyond
-----------------------------------

Any exploration of the "meantone" concept might aptly begin with the
recognition that this term can mean many different things to different
people, or indeed to the same person at different times.[7]

For the purposes of this paper, I would like to present a possible
perspective centering on "meantone" as the region of a tradeoff or
compromise between 3-limit and 5-limit concords, and "stylistic
meantone" as the slightly narrower region where these concords are
deemed to be in "acceptable balance" for tertian styles of music where
both 3-limit and 5-limit intervals participate in stable sonorities.

From an historical point of view, the advent of meantone temperaments
around 1450 represents an effort to achieve thirds and sixths at or
near pure 5-limit ratios (M3 5:4, m3 6:5, M6 5:3, m6 8:5) while
keeping fifths reasonably close to their ideal 3-limit ratio of 3:2.

Such a "meantone" equation or dialectic focuses on the syntonic comma
by which the active Pythagorean thirds and sixths of traditional
Gothic style differ from their pure 5-limit counterparts avidly sought
by the mid-15th century. This comma of 81:80, or ~21.51 cents, is thus
a logical as well as traditional measure of meantone temperaments.

Taking the trade-off between 3-limit and 5-limit intervals as the
essence of the meantone equation, we find that this equation suggests
two limiting conditions which may define boundaries of the meantone
spectrum.

When fifths are narrowed by 1/3-comma (~7.17 cents), minor thirds are
at a pure 6:5 and major sixths at a pure 5:3. Any further tempering
would compromise these intervals as well as moving other 5-limit as
well as 3-limit concords further from their ideal ratios. In 19-tet, a
minutely greater amount of tempering (~7.22 cents) may be motivated by
a desire for precise mathematical closure and symmetry; this tuning
thus serves as a convenient lower limit.

When fifths and fourths are pure (Pythagorean tuning), any tempering
in the _wide_ direction would compromise these 3-limit intervals as
well as further accentuating the full comma by which thirds and sixths
differ from the ideal 5-limit ratios implied by a "meantone" frame of
discourse. Thus Pythagorean tuning or "zero-comma meantone" serves as
one logical upper limit to the meantone spectrum.

Therefore 1/3-comma meantone or 19-tet with pure or virtually pure
minor thirds, and Pythagorean tuning with pure fifths, represent the
mathematical limits or boundary conditions of the 3-limit/5-limit
meantone tradeoff. Hypomeantones with fifths narrower than 19-tet, and
hypermeantones with fifths wider than Pythagorean, evidently reflect
other tradeoffs and aesthetic possibilities.

---------------------------------------------------
1.1. Stylistic meantone and 5-limit "acceptability"
---------------------------------------------------

From a mathematical point of view, Pythagorean tuning with pure fifths
is at once the upper limit of the meantone spectrum and the lower
limit of the hypermeantone spectrum.

Musically, however, this quintessential Gothic tuning and the almost
identical 53-tet have a strong affinity to neo-Gothic hypermeantones.
Their active and dynamic thirds and sixths superbly fit a medieval or
neo-medieval style, in contrast to the Renaissance and later 5-limit
styles usually associated with the term "meantone."

Given this aesthetic reality, and the origin of meantone temperaments
around 1450 as a calculated departure from Pythagorean tuning, the
term "meantone" often implies a regular temperament where the fifths
are narrowed sufficiently to bring thirds and sixths appreciably
closer to 5-limit ratios, so that they may serve comfortably as full
concords.

"Stylistic meantone" in this sense thus implies an "acceptable
balance" between 3-limit and 5-limit intervals for tertian styles
where both types of intervals participate in fully concordant
sonorities.

Easley Blackwood[8] places an upper limit of "acceptability" on the
size of major thirds for 5-limit music at around 406 cents, with
regular fifths at around 701.5 cents (~0.46 cents narrower than pure,
or ~1/47-comma meantone). This is the point where such thirds become
just restful enough to form stable triads -- or, from another point of
view, where they are just approaching a Gothic/neo-Gothic level of
activity and energy.

If we adopt Blackwood's limit as a useful conceptual guide, then the
spectrum of "stylistic meantones" ranges from our lower limit of
19-tet or 1/3-comma meantone to an upper limit slightly below
Pythagorean (with fifths at around 701.5 cents).[9]

From this perspective, the rather narrow border zone between 701.5
cents and 53-tet or Pythagorean is technically still within the
meantone region but musically is more of a portal or antechamber to
the world of Gothic and neo-Gothic intonations.[10]

Transition zones and fuzzy boundaries of this kind should be seen not
as a flaw but as an enticing feature of maps, whether geographical or
conceptual, especially as we focus on finer levels of detail.[11]

------------------------
1.2. Meantone and beyond
------------------------

For the purposes of this paper, we have defined the meantone spectrum
as the region of compromise between 3-limit and 5-limit intervals, or
the slightly narrower region where an "acceptable balance" is struck
between these intervals for Renaissance and later tertian European
styles.

Hypomeantones with fifths narrower than in 19-tet or 1/3-comma
meantone, and hypermeantones with fifths wider than Pythagorean,
belong to different although related musical universes on the larger
continuum of regular tunings from 7-tet to 5-tet.

"Stylistic meantones" in our rather broad sense (including 12-tet),
and also the irregular well-temperaments favored for European keyboard
music during the era of around 1680-1850, reflect a musical imperative
to make both 3-limit and 5-limit intervals acceptable stable concords.

With hypomeantones and hypermeantones, a variety of other styles,
balances, and artful compromises may apply, in some cases analogous to
those of meantones. The remainder of this paper focuses primarily on
hypermeantones in the range from Pythagorean to 22-tet or 27-tet as
used for Gothic or neo-Gothic styles of music, with hypomeantone
connections briefly considered in Section 4.

------------------------------------------------------
2. New balances: hypermeantones and the septimal comma
------------------------------------------------------

In Gothic and neo-Gothic styles of music based at once on a 3-limit of
stability (with fifths and fourths the most complex stable concords)
and a subtle scale of relative concord/discord, Pythagorean tuning
serves as a classic starting point for various forms of intonational
mannerism. Unstable intervals may be "attracted" to various ratios
based on higher prime limits.

In such a setting, neo-Gothic tunings including hypermeantones may
typically be subject to the following basic constraints:

(1) Stable 3-limit concords should remain within "acceptable" limits;

(2) Other regular intervals should remain "recognizable" as variants
on their standard Pythagorean forms, e.g. with wide major thirds and
sixths distinguishable from fourths or minor sevenths[12]; and

(3) At least in more "classical" styles, these intervals should fit
13th-14th century orderings of relative concord/discord, with a bare
major third, for example, being _relatively_ blending and somewhat
more "concordant" than a relatively tense bare major second.

In the complex 3-limit setting of Gothic or neo-Gothic music, one very
viable approach to this intonational equation is to avoid temperament,
instead extending Pythagorean just intonation to obtain new flavors of
intervals while retaining all of the traditional medieval ones.

For example, suppose we feel an attraction toward an extra-wide
cadential major third at 9:7 (~435.08 cents) resolving to a stable
fifth. In a Xeno-Gothic tuning (24-note Pythagorean mapped to two
12-note keyboards a Pythagorean comma apart), the interval formed by
16 fifths in the upward direction will serve nicely for this
purpose. At ~431.28 cents, this interval is only about 3.80 cents
narrower than a pure 9:7 -- and precisely a Pythagorean comma wider
than the usual major third at 81:64 (~407.82 cents).

Using a "@" symbol to show a note lowered by a Pythagorean comma, and
a MIDI-like notation with C4 as middle C, we could therefore notate
such a Xeno-Gothic cadence as follows:

B3 -- +67 -- C@4
(431) (702)
G@3 -- -204 -- F@3

Here the numbers in parentheses show vertical intervals in cents,
while signed numbers show ascending or descending melodic intervals in
cents.

Both the "superwide" major third at ~431 cents, and "supercompact"
diatonic semitone in the upper voice at ~66.76 cents (a Pythagorean
comma narrower than the usual 256:243 at ~90.22 cents) are "special
effects" intervals. At the same time, the melodic motion in the lower
voice by a pure 9:8 whole-tone (~203.91 cents), and the resolution to
a pure 3:2 fifth, are normal features of Pythagorean just intonation
which we retain in our extended tuning.

The obvious complication as well as artistic opportunity of such a
system is that it calls on the performer to negotiate comma
distinctions. Suppose we desire a simpler system where regular major
thirds (formed by a chain of four fifths) are at or near a pure 9:7.

In order to make four regular fifths add up to a 9:7 major third, we
must disperse the septimal comma of 64:63 (~27.26 cents) which defines
the difference, for example, between a usual Pythagorean major third
at 81:64 and a 7-limit major third at 9:7 (or 81:63). Each fifth must
therefore be tempered in the wide direction by 1/4 of this comma, or
~6.82 cents, resulting in a size of ~708.77 cents.

In this tuning, each regular major second serves as a "hypermeantone"
of ~217.54 cents, equal to precisely half of a pure 9:7 major third,
and to the geometric mean of the unequal 9:8 and 8:7 whole-tones
forming a 9:7 third.[13] These hypermeantones are 1/2 septimal comma
(~13.63 cents) wider than the Pythagorean 9:8, so that two of them
produce a major third a full septimal comma wider than 81:64 (i.e. our
desired 9:7).

The following diagram shows this hypermeantone temperament scheme,
with "SC" standing for the septimal comma:

+ 1/4 SC + 1/4 SC + 1/4 SC + 1/4 SC
|-----------|-----------|----------|----------|
C G D A E
|-----------------------|---------------------|
+ 1/2 SC + 1/2 SC
|---------------------------------------------|
+ 1 SC (9:7)

We might call this tuning "1/4-septimal-comma hypermeantone," closely
analogous in structure to the familiar 1/4-(syntonic)-comma meantone
with its pure 5:4 major thirds. The aesthetic and musical qualities
of these temperaments are, of course, radically different.

The most obvious compromise of our pure 9:7 hypermeantone tuning is
the tempering of the fifths (and fourths) by about 6.82 cents, an
amount approaching that in 1/3-comma meantone (~7.17 cents) or 19-tet
(~7.22 cents), and also the tempering of major seconds and minor
sevenths by a full 1/2 septimal comma in the context of such
relatively concordant sonorities as 4:6:9, 6:8:9, 8:9:12, or 9:12:16.

Just as minor thirds in 1/4-comma meantone are tempered in the narrow
direction by the same amount as the fifths (~5.38 cents in relation to
6:5 and 3:2 respectively), so in our 1/4-septimal-comma temperament
the minor thirds are 1/4 septimal comma (~6.82 cents) wider than a
pure 7:6, or about 273.69 cents. Major sixths are narrower than a pure
12:7 by the same amount.

While the tempering of our prime concords by almost 7 cents may be a
less than ideal compromise, this tuning in an appropriate timbre can
indeed meet typical constraints of neo-Gothic style. Stable 3-limit
intervals remain within acceptable limits, and other intervals remain
"recognizable" variants of the usual Pythagorean intervals. Both the
7-limit and near-7-limit intervals of this tuning, and its very narrow
diatonic semitones at ~56.14 cents, contribute to its flavor.

Interestingly, there is a well-known equal temperament quite close to
1/4-septimal-comma hypermeantone: 22-tet, with fifths tempered by
~7.14 cents, and major thirds just slightly larger than 9:7 (~436.36
cents, or ~1.28 cents wide). Here the diatonic semitone of 1 step
(~54.55 cents) is literally a "quartertone" in relation to the major
second of 4 steps (~218.18 cents).[14]

Another equal temperament, albeit one calling for special timbral
conditions in a neo-Gothic setting, very closely approximates
1/3-septimal-comma hypermeantone with pure 7:6 minor thirds: 27-tet,
with fifths at ~711.11 cents (~9.15 cents wider than pure). Here the
minor thirds at ~266.67 cents are only ~0.20 cents narrower than 7:6,
and the major sixths wider than 12:7 by the same amount.

Thus our situation is very close to that shown in the following
diagram, where three upward fourths each narrowed by 1/3 septimal
comma (~9.09 cents) produce a regular minor third at a pure 7:6, a
septimal comma narrower than the usual Pythagorean 32:27 (~294.13
cents).

- 1/3 SC - 1/3 SC - 1/3 SC
|-----------------|-----------------|-----------------|
D G C F
|-----------------------------------------------------|
- 1 SC (7:6)

Just as in the usual 1/3-comma meantone we have pure 6:5 minor thirds
and major thirds 1/3 comma narrower than 5:4, so in 1/3-septimal-comma
hypermeantone the major thirds are 1/3 septimal comma wider than a
pure 9:7 (~444.17 cents). In 27-tet, they are ~444.44 cents.

With a somewhat gamelan-like timbre, 27-tet can be an enchanting
neo-Gothic tuning, with the fifths and fourths quite acceptable as
stable concords, and the very wide major thirds not only
"recognizable" but quite pleasant. Near-pure ~7:6 minor thirds
contracting to unisons, and ~12:7 major sixths expanding to octaves,
lend a special flavor to standard Gothic progressions, along with the
diatonic semitone (literally a fifthtone) of ~44.44 cents.

As these examples of 1/4- and 1/3-septimal-comma tunings (approximated
by 22-tet and 27-tet) may suggest, neo-Gothic hypermeantones may
involve some of the same interactions and compromises between prime
limits as conventional meantones. Such relationships may extend to a
variety of intervals and prime factors.

---------------
Notes to Part 1
---------------

1. Part 1 in Tuning Digest (TD) 704:2 (8 July 2000); Part 2 in
TD 705:8 (9 July 2000).

2. TD 704:5 (8 July 2000).

3. David C. Keenan, "Harmonic errors in single-chain-of-equal-fifths
tunings" (1998; updated 1999), available on World Wide Web at
http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm. See
the section "Familiar regions" on the issue of defining meantones.

4. Erlich remarked that a major second in a regular tuning with fifths
wider than pure would not be a "mean-tone" of some major third which
"comes out closer to 5:4" than the Pythagorean 81:64, but rather a
"hypertone." He suggests a necessary condition that a meantone tuning
have a major second which is a "mean-tone" in the sense of being
somewhere _between_ the 9:8 and 10:9 whole-tones forming a pure 5:4 in
5-limit just intonation. (Often the term "mean-tone" refers uniquely
to the major second of 1/4-comma meantone, the exact geometric mean of
9:8 and 10:9.) In contrast, the major second of a tuning with fifths
wider than pure exceeds 9:8, and is thus Erlich's "hypertone" -- or my
"hypermeantone of some major third greater than 81:64."

5. The term "hypermeantone" might just as logically refer to tunings
with fifths _smaller_ than in historical meantone tunings -- that is,
tunings where the meantone process of narrowing the fifths is carried
to a degree "beyond" historical practice. I am indebted to Paul Erlich
both for this point and for the example of 26-tet.

6. For a different and very engaging map, see Keenan, note 3 above.
Exploring and comparing a variety of conceptual maps of the spectrum
of regular tunings may make us aware of landmarks which might be
overlooked in any one view.

7. Thus I find it quite natural to ask whether a German organ piece
from around 1450 "sounds better in Pythagorean or meantone," or
whether a Spanish lute piece from around 1540 "sounds better in
meantone or 12-tet." Here the term "meantone" suggests the range of
historical temperaments from about 1/3-comma to 1/6-comma favored as
keyboard tunings during the era of around 1450-1680. In other
contexts, however, I might include 12-tet ("1/11-comma meantone") or
even Pythagorean ("zero-comma meantone") as meantone tunings.

8. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_
(Princeton: Princeton University Press, 1985), pp. 202-203.

9. For Blackwood's limit as a possible upper bound of meantone -- my
"stylistic meantone" -- I am much indebted to Paul Erlich. A regular
tuning at Blackwood's limit might be the best solution for
compositions inviting a "Pythagorean-like" approach where diatonic
semitones are noticeably narrower than chromatic semitones, and sharps
higher than corresponding flats (e.g. Ab-G#), but where 5-limit
triadic sonorities are common. Following Johnny Reinhard's thesis on
the "Pythagorean" intentions of Charles Ives for some of his works,
this tuning would fit such intentions while permitting some degree of
cohesion for 5-limit triads built from regular thirds.

10. This border region is also the domain of "schismic" or "skhismic"
temperaments emulating 5-limit just intonation, where the diminished
fourth is tuned and remapped as a pure 5:4 major third, or the
augmented second as a pure 6:5 minor third. These temperaments are
often measured by the schisma (32805:32768, ~1.95 cents) by which a
Pythagorean diminished fourth at 8192:6561 (~384.36 cents) falls short
of a pure 5:4 (~386.31 cents), and a Pythagorean augmented second at
19683:16834 (~317.60 cents) exceeds a pure 6:5 (~315.64 cents). Thus a
1/8-schisma temperament (fifths ~701.71 cents, ~0.24 cents narrow)
yields pure 5:4 diminished fourths, while a 1/9-schisma temperament
(fifths ~701.74 cents, ~0.22 cents narrow) yields pure 6:5 augmented
seconds.

11. Such border regions may lend themselves to various perspectives.
For example, a wider "semi-meantone/semi-Gothic" zone from around
12-tet to Blackwood's limit (fifths ~700-701.5 cents) might better
capture the ambivalence of this territory, with 12-tet featuring
"Pythagorean" as well as "stylistic meantone" qualities. Keenan, n. 3
above, places the boundary between his "meantone" and "Pythagorean"
regions at 12-tet.

12. The deliberate bending of such categories is a feature of
"ultra-Gothic" music, where an interval such as 19 steps in 24-tet
(950 cents) or 23 steps in 29-tet (~951.72 cents) can serve as either
a very wide major sixth inviting expansion to an octave or a very
narrow minor seventh inviting contraction to a fifth. In these
tunings, a major third one step larger than usual -- 9/24 octave (450
cents), 11/29 octave (~455.17 cents) -- expanding to a fifth has a
similar effect of "blurring" or "warping" familiar categories, the
latter term in an ultra-Gothic context carrying the pleasant
connotation of "travelling through a new space."

13. Note that in our Xeno-Gothic example of a near-9:7 major third
resolving to a fifth, this third G@-B can likewise be divided into two
unequal whole-tones of 9:8 and ~8:7: G@-A@ and A@-B, or G@-A and A-B.
In each division, one of the whole-tones exceeds the usual 9:8 by
precisely a Pythagorean comma (531441:524288, ~23.46 cents), and has a
size of 4782969:4194304 (~227.37 cents, ~3.80 cents narrow of 8:7).

14. As I emphasize in my paper on neo-Gothic tunings (n. 1 above),
this description of whole-tones and semitones in 22-tet assumes a
Pythagorean interpretation of the tuning; for another outlook
producing very different interval definitions, see Paul Erlich,
"Tuning, Tonality, and Twenty-Two-Tone Temperament," _Xenharmonikon_
17 (Spring 1988), pp. 12-40.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Friendly introduction to hypermeantones (1 of 2)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗hstraub64 <straub@...>