I can't swear to this, but I remember reading once that a "mean tone" is created
by taking an average (mean) of the major and minor whole tones that exist in a
pythagorian diatonic scale created by tuning pure fifths. In such a scale, the
do-re and re-mi intervals, both whole tones, are not equal, the do-re whole tone
is larger than the re-mi whole tone. This, of course, creates a problem if you
want to modulate (which is why we temper in the first place) and in the mean tone
system of tempering the goal was to create one "mean tone" which is neither a
major whole tone or a minor whole tone but a "mean tone" or an average between
the two. Correct me if I'm wrong.
John Gunderson, Belmar, NJ
440A@AOL.COM wrote:
> 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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