Greetings,
Several have expressed interest in some further delving into the history
of temperaments, and Margo Schulter, an awesome author and authority on the
subject of real early tuning posted the following to the "Tuning" list,(this
is a truncated repost, there is more!). I hope it may be of some interest.
Regards,
Ed Foote RPT
------------------------------------------------
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]
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