Re: Friendly introduction to hypermeantones (2 of 2)

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A friendly introduction to hypermeantones:
Regular temperaments beyond Pythagorean
(Part 2 of 2)
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3. Neo-Gothic hypermeantones: a plethora of prime limits
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Our hypermeantones with pure 9:7 or 7:6 thirds, and the closely
related 22-tet and 27-tet, occupy a fascinating realm situated at
the farther end of the neo-Gothic spectrum and the middle region of
the larger hypermeantone spectrum from Pythagorean to 5-tet. The
following chart illustrates this overview:

0 ~+7.14 ~+9.15 ~+18.04
~701.96 ~709.09 ~711.11 720
|---------------------|--------|--------------------------------|
Pyth 22-tet 27-tet 5-tet
|----------------------...........? ? ?
Neo-Gothic spectrum

Tempering our fifths and fourths by 1/4 or 1/3 septimal comma involves
a modest adjustment in comparison to many world musics favoring
intonations closer to 5-tet, but a rather dramatic compromise from a
European medievalist perspective centered on Pythagorean ratios of 3:2
and 4:3.

For many timbres, 22-tet (fifths ~7.14 cents wide) may approach the
upper limit of the neo-Gothic spectrum; 27-tet (fifths ~9.15 cents
wide) may require special "care and feeding" in a neo-Gothic context,
but very convincingly succeeds with a somewhat gamelan-like timbre.

Just as 7-limit thirds provide "prime attractions" (pun intended) in
the far neo-Gothic zone reaching out to 22-tet and beyond, so other
regions offer their own prime attractors. Here I would like very
partially to survey the near neo-Gothic zone between Pythagorean and
17-tet, suggesting a musical universe with a plethora of prime limits
and commas.

Starting at Pythagorean tuning, we find that the 81:64 major third is
smaller than 19:15 (~409.24 cents) by only a small comma of 1216:1215,
or ~1.42 cents. As it happens, 41-tet (fifths ~702.44 cents, ~0.48
cents wide) slightly exceeds the tempering required for pure 19:15
thirds (fifths ~702.31 cents, ~0.35 cents wide), making these
intervals about 0.51 cents wide.

To specify the hypermeantone with pure 19:15 major thirds, we might
use either of these notations:

1/4-(1216:1215)-hypermeantone
1/4-(19:15/81:64)-hypermeantone

Either notation shows that the fifth in this hypermeantone -- roughly
approximated by 41-tet -- is tempered by 1/4 of the 1216:1215 comma by
which 19:15 exceeds the Pythagorean 81:64.[15]

Suppose that we desire a hypermeantone with pure 13:11 minor thirds,
which are narrower than the usual Pythagorean 32:27 by a comma of
352:351 (~4.93 cents). Our tuning has fifths at ~703.59 cents, or
about 1.64 cents wider than pure; 29-tet at ~703.45 cents (fifths
~1.49 cents wide) features minor thirds at ~289.66 cents, or ~0.45
cents wider than 13:11 (~289.21 cents).

We may therefore say that 29-tet approximates a meantone notated in
either of these ways:

1/3-(352:351)-hypermeantone
1/3-(32:27/13:11)-hypermeantone

Continuing our journey from Pythagorean toward 17-tet, we might seek a
major third at a pure 14:11 (~417.51 cents), which exceeds the
Pythagorean 81:64 by a comma of 896:891 (~9.69 cents). Our tuning has
a fifth of ~704.38 cents (~2.42 cents wide), almost identical to
46-tet with a fifth of ~704.35 cents (~2.39 cents wide) and a major
third of 417.39 cents -- a difference of only ~0.12 cents.

We accordingly find 46-tet almost identical to the tuning alternately
notated as

1/4-(896:891)-hypermeantone
1/4-(14:11/81:64)-hypermeantone

While we have so far focused on major and minor thirds, other types of
intervals may also serve as the basis for hypermeantone tunings and
approximations. Suppose that we wish to make the "odd" fourth of a
12-note tuning chain -- actually an augmented third, e.g. Eb-G# --
have a pure 11:8 ratio (~551.32 cents).

Our 11:8 interval exceeds a usual Pythagorean augmented third or "Wolf
fourth" at 177147:131072 by a comma of 180224:177147, or ~29.81 cents,
an amount slightly larger than the septimal comma we encountered in
Section 2. Here, however, we can spread this comma over 11 fifths or
fourths, producing a fifth of ~704.67 cents (~2.71 cents wide).

As it happens, an "e-based hypermeantone" I have proposed with a ratio
between the whole-tone and diatonic semitone equal to Euler's _e_
(~2.71828) closely approximates this result with a fifth at ~704.61
cents (~2.65 cents wide), and an augmented third at ~550.68 cents, or
about 0.64 cents narrower than 11:8.

We might say that the e-based tuning approximates our pure 11:8 tuning
notated by the forms

1/11-(180224:177147)-hypermeantone
1/11-(11:8/177147:131072)-hypermeantone

Arriving at 17-tet, we encounter two general issues of tuning theory:
do very high prime-limits have a distinctive musical flavor, and if
so, how large is the "zone of attraction" within which a tempered
interval should fall in order to communicate this flavor for a given
limit?

The 17-tet minor third at ~282.35 cents is rather close to a pure
20:17 ratio at ~281.36 cents -- a difference of ~0.99 cents. The
17-tet fifth at ~705.88 cents (~3.93 cents wide) compares with the
fifth at ~706.21 cents (~4.25 cents wide) needed to produce 17:10
minor thirds. Such a tuning would disperse the comma by which the
Pythagorean 32:27 exceeds 20:17 -- 136:135, ~12.78 cents -- over a
chain of three fourths. We could notate this tuning by the forms

1/3-(136:135)-hypermeantone
1/3-(32:27/20:17)-hypermeantone

If we grant that 17-odd-limit ratios are musically distinctive, then
the question remans as to whether a 17-tet minor third with its
variance of about 1 cent from 17:10 has a flavor of "17-ness."

The 17-tet major third at ~423.53 cents comparably approximates a
ratio with a yet higher prime-limit, 23:18 (~424.36 cents), with a
variance of ~0.83 cents. A hypermeantone with a pure 23:18 major third
would have a fifth at ~706.09 cents (~4.14 cents wider than pure),
dispersing over four fifths the comma of 736:729 by which 23:18
exceeds the Pythagorean 81:64. We could notate this tuning:

1/4-(736:729)-hypermeantone
1/4-(23:18/81:64)-hypermeantone

Here we confront the issues of whether a pure ratio such as 23:18 has
a distinctive "23-ness" -- and, if so, whether the 17-tet major third
with its variance of ~0.83 cents approaches its "zone of attraction"
closely enough to share this flavor.

As these examples may suggest, neo-Gothic hypermeantones can bring
into play a plethora of higher prime ratios or approximations. The
musical possibilities invite exploration in practice and theory.

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4. A coda: hypermeantones and hypomeantones
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In Section 2, we focused on hypermeantones with pure 9:7 and 7:6
thirds, analogous to the familiar 1/4-comma and 1/3-comma meantone; in
Section 3, we more generally surveyed a range of hypermeantones
featuring or approximating a variety of higher-prime ratios.

To this point, however, we have hardly brought hypomeantones into the
picture, other than to conclude that, like hypermeantones, they
involve different stylistic and aesthetic considerations than the
3-limit/5-limit balance of historical meantones.

However, connections between all three categories of regular tunings
arise when we consider the matter of "alternative thirds" formed by
diminished fourths or augmented seconds.

These relationships are most familiar between meantones and
hypermeantones. Thus Pythagorean tuning and its immediate neighbors
(e.g. 53-tet, 41-tet) feature diminished fourths or "schisma thirds"
very close to the regular major thirds of 1/4-comma meantone at a pure
5:4. In turn, 1/4-comma meantone offers diminished fourths at 32:25
(~427.37 cents), comparable to the regular major thirds of a
neo-Gothic hypermeantone such as 17-tet (~423.53 cents).

At other points in the hypermeantone spectrum, however, we encounter
intriguing links with the alternative universe of hypomeantone.

In 22-tet, for example, the diminished fourth of 6 (9 - 3) steps,
equal to a fourth minus a chromatic semitone, is ~327.27 cents, an
interval somewhat larger than the pure 6:5 at ~315.64 cents. This
identical interval is the regular minor third of 33-tet, formed from a
whole-tone of 5 steps plus a diatonic semitone of 4 steps. Here we have
have a regular fifth (19 steps) of ~690.91 cents (~11.04 cents narrow
of 3:2), placing us well into the hypomeantone region.

In the e-based hypermeantone we considered in Section 3, the
diminished fourth is ~363.14 cents, fairly close to 21:17 (~365.83
cents). In 33-tet, the regular major third (10 steps, or two
whole-tones of 5 steps each) is an almost identical ~363.64 cents.

Musically, of course, these tunings suggest quite different contexts
for these almost identical sizes of thirds. In the hypermeantone
setting, our near-21:17 can serve as a "submajor" or "subditonal"
third inviting an expansion to the fifth with one the voices moving by
a striking chromatic semitone (~132.25 cents). This "special effects"
interval contrasts with our usual near-14:11 major third, expanding to
a fifth by way of the usual compact diatonic semitone (~76.97 cents).

In the hypomeantone setting of 33-tet, our near-21:17 is the regular
major third, likely playing a more pervasive role; and similarly the
regular minor third at around 327 cents, a "special effects" interval
in a Pythagorean interpretation of 22-tet where the regular minor
third is rather close to 7:6 (~272.73 cents).

Suppose we seek a regular tuning featuring thirds at a pure 11:9, a
common flavor of "neutral third." The hypomeantone with a fifth at
~686.85 cents (~15.10 cents from 3:2) will produce this interval as a
regular major third -- that is, from a chain of four fifths. Here we
are approaching 7-tet; we have whole-tones of ~173.70 cents, diatonic
semitones of ~165.74 cents, and chromatic semitones of ~7.96 cents.
This tuning might have a somewhat pelog-like flavor.

In the hypermeantone portion of the spectrum, we can obtain 11:9 as an
augmented second (formed from a chain of 9 fifths) by choosing a fifth
of ~705.27 cents (~3.31 cents wide), about midway between our e-based
hypermeantone and 17-tet. Using the notation introduced in Section 3,
we might describe this tuning in terms of the difference by which 11:9
exceeds the Pythagorean augmented second 19683:16384 (~317.60 cents,
very close to 6:5), a comma of 180224:177147 or ~29.81 cents[16]:

1/9-(180224:177147)-hypermeantone
1/9-(11:9/19683:16384)-hypermeantone

Another choice would be to seek a diminished fourth at 11:9, calling
for a fifth at ~706.57 cents, or ~4.61 cents wide, in the region
between 17-tet and 39-tet. Here a chain of 8 fifths disperses the
difference between the Pythagorean diminished fourth 8192:6561
(~384.36 cents, very close to 5:4) and 11:9, a comma of 8192:8019 or
~36.95 cents. This suggests the notations

1/8-(8192:8019)-hypermeantone
1/8-(8192:6561/11:9)-hypermeantone

Having found a few "bridges" between the universes of hypermeantone
and hypomeantone, we may have some sense of the musical richness
awaiting us in both these less-travelled portions of the continuum of
regular tunings.

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Notes to Part 2
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15. For 3-limit/5-limit and 3-limit/7-limit interactions, we often
speak of a single syntonic comma (81:80) or septimal comma (64:63).
For interactions between the 3-limit and higher prime limits, however,
the relevant commas may vary from interval to interval. For example,
while a 19:15 major third is larger than the Pythagorean 81:64 by a
comma of 1216:1215, a 19:16 minor third is also _larger_ than the
Pythagorean 32:27 by a comma of 513:512 (~3.38 cents). The sum of
these two commas, 361:360 (~4.80 cents), is the amount by which a
fifth formed from a pure 19:15 and a pure 19:16 (361:240, ~706.76
cents) exceeds 3:2. Because of such asymmetries, I propose a notation
specifying either the ratio of the comma itself or the ratios of the
intervals defining this comma. The latter style can often be more
"user-friendly," since it explicitly identifies the relevant
Pythagorean interval and the higher-limit "target" interval of the
temperament, either of which may prompt easier recognition than a
comma ratio (especially a less familiar one).

16. This comma is identical to the one we met in Section 3 between
11:8 and the Pythagorean "Wolf" fourth or augmented third at
177147:131072. To explain this equivalence we might note that an 11:9
is smaller than an 11:8 by a pure 9:8; and likewise a Pythagorean
augmented second at 19683:16834 is smaller than an augmented third by
a regular whole-tone, the same 9:8 ratio.

Most respectfully,

Margo Schulter
mschulter@value.net