--------------------------
Temperament extraordinaire
--------------------------
A _temperament extraordinaire_ is a kind of _fifth-circulating_
temperament (all fifths within 7 cents or so of 3:2) in 12 notes.
Eight fifths (F-C#) are tempered in a regular meantone somewhere in
the range from about 1/4-comma to 2/7-comma or possibly a bit beyond;
and the other four (C#/Db-G#/Ab-Eb/D#-Bb/A#-F) tempered equally wide,
with a degree of impurity comparable to that of the narrow fifths.
Thus all 12 fifths, narrow or wide, are about 5-7 cents from just.
Thirds in the nearer part of the circle are identical to those of the
regular shade of meantone in use, and at or very close to the just
5-limit or pental ratios of 5:4 and 6:5, with the outlying Bb-D and
E-G# at around 395-397 cents, considerably more impure but still less
so than in 12-EDO. The more remote portion of the circle has five
major thirds in the range from Pythagorean to septimal, somewhat
resembling in this aspect a 17-tone well-temperament such as George
Secor's 17-WT.
This scheme is designed to support usual Renaissance and Manneristic
styles of the meantone era (say 1450-1650) in the nearer portion of
the circle (Bb-G#), and medieval (e.g. 13th-14th century) or related
neomedieval styles in the more remote portion with its Pythagorean to
septimal range of thirds.
Thus the temperament extraordinaire is an example of a tuning system
explicitly designed to support two distinct musical styles, here
Renaissance and medieval/neomedieval, in different portions of the
tuning. Possibly this situation has some affinity to the concept of
polymicrotonality, or the use of different microtonal systems for a
single piece, although the two "subsystems" of Renaissance meantone
and neomedieval wide-fifth temperament share the same 12-note chain of
fifths, and thus flow seamlessly one into another.
The temperament extraordinaire derives its name from the _temperament
ordinaire_ popular in 17th-18th century France, where typically most
of the fifths are tuned in some shade of regular meantone, and a few
are tempered wide of pure; it may be viewed as a special instance of
this category.
Here is a piece in a 2/7-comma temperament extraordinaire based mainly
on a 16th-century meantone style and focusing on the nearer portion of
the circle, with a few more remote sonorities also heard, for example
in the final cadence:
An improvisation drawing on medieval European styles of around 1200
focuses on the remote portion of the circle and features some septimal
approximations:
---------------------------
1. Definitional fine points
---------------------------
From an artistic point of view, the distinguishing characteristic of a
temperament extraordinaire is its goal of embracing the full range of
major and minor thirds from pental (5:4, 6:5) to septimal (7:6, 9:7)
in a circulating (i.e. fifth-circulating) 12-note system. A 1/4-comma
version has major thirds at 386-427 cents, and minor thirds at 279-310
cents; with 2/7-comma, we have 383-434 cents and 275-313 cents. The
goal is to achieve a circulating system with this colorful range while
altering as little as possible the usual meantone sound of the nearer
portion of the circle (Bb-G#), which can accommodate many 16th-century
pieces much like a regular meantone.
This involves deftly balancing three commas: the ditonic or
Pythagorean; the syntonic or Didymic; and the septimal or Archytan.
The eight narrow meantone fifths are more than sufficient to disperse
the Pythagorean comma (531441:524288, 23.46 cents), and in fact
overshoot this mark by something on the order of another such comma, a
harmonic "excess" or "surplus" which is in turn dispersed by the four
wide fifths, so that the circle balances.
In the nearer portion of the circle, just as in a regular meantone, a
chain of three or four narrow fifths (or wide fourths) temper out the
syntonic or Didymic comma (81:80, 21.51 cents) so that major and minor
thirds are at or very close to pental ratios of 5:4 and 6:5.
In the most remote portion of the circle, much as in regular tunings
such as 39-EDO or 22-EDO, a chain of two, three, or four wide fifths
(or narrow fourths) tempers out the septimal or Archytan comma (64:63,
27.26 cents), yielding major and minor thirds approximating ratios of
9:7 and 7:6, and also minor sevenths approximating 7:4. A temperament
extraordinaire thus has one major third (C#/Db-F) closest to 9:7; two
minor thirds closest to 7:6 (F-G#/Ab, Bb-C#/Db); and three minor
sevenths closest to 7:4 (Eb-C#/Db, F-Eb, Bb-G#/Ab).
As in 39-EDO or 22-EDO, or an irregular system like Secor's 17-WT, for
example, dispersing the rather large 27-cent septimal comma in a chain
of two, three, or four fifths no more than about 7 cents wide often
requires compromises in the accuracy of the septimal approximations.
The following table compares the just sizes for 7:6, 9:7, and 7:4 with
their best approximations in temperament extraordinaire (TE) systems
at 1/4-comma and 2/7-comma, and also Secor's 17-WT, 39-EDO, and 22-EDO.
------------------------------------------------------------------
Just ratio/ 1/4-comma 2/7-comma 17-WT 39-EDO 22-EDO
cents TE (best) TE (best) (best)
------------------------------------------------------------------
9:7 435.08 427.37 (32:25) 433.52 428.88 430.77 436.36
-7.71 (225:224) -1.56 -6.20 -4.31 +1.28
------------------------------------------------------------------
7:6 266.87 279.47 274.86 278.34 276.92 272.73
+12.60 +7.99 +11.47 +10.05 +5.86
------------------------------------------------------------------
7:4 968.83 986.31 983.24 985.56 984.62 981.82
+17.48 +14.41 +16.73 +15.79 +12.99
------------------------------------------------------------------
The quest for "reasonable" septimal representations for these
intervals thus suggests that the underlying meantone temperament for a
temperament extraordinaire should be at least around 1/4-comma, with
2/7-comma yielding somewhat closer approximations but also calling for
a greater compromise in the fifths (narrow and wide).
Mark Lindley has suggested that the zone of maximal euphony for a
Renaissance meantone may be around 1/4-2/7 comma, so that someone
sharing this estimate and giving weight to the goal of optimizing the
system for a usual 16th-century style might have mixed feelings about
any temperament heavier than 2/7-comma. However, regular meantones
such as Wilson's Metameantone, Harrison/Lucy temperament, and 88-EDO
could also be used, and would permit closer septimal approximations.
The 88-EDO implementation may be near the limits of what might be
considered full fifth-circulation, and thus may approach an upper
bound for the temperament extraordinaire genre. In this interesting
system, the eight meantone fifths are at 51/88 octave (695.45 cents),
or 6.50 cents narrow; and the four wide fifths at 52/88 octave,
identical to the regular fifth of 22-EDO (709.09 cents), 7.14 cents
wide. Curiously, the result is almost identical to a system based on
Harrison/Lucy (regular meantone major thirds at 1200/pi cents, or
381.97 cents, as compared with 381.82 in 88-EDO), with meantone fifths
at 695.49 cents (6.46 cents narrow) and wide ones at 709.01 cents
(7.05 cents wide).
To sum up, the essence of the temperament extraordinaire is the
management or juggling of the three commas so as to optimize meantone
euphony in the nearer portion of the circle and seek good septimal
approximations in the most remote portion, at the same time keeping
all fifths, narrow and wide, reasonably agreeable. Different points on
the spectrum running roughly from 1/4-comma to Harrison/Lucy tuning or
88-EDO have their own charms and compromises in fulfilling these
sometimes delicately counterpoised goals.
---------------------------------------------------------
2. Comparison with well-temperaments and regular meantone
---------------------------------------------------------
A temperament extraordinaire is an example of a fifth-circulating
system distinct from a well-temperament. While both types of systems
have all fifths reasonably close to pure, a well-temperament
additionally keeps the sizes of major and minor thirds within certain
bounds which a temperament extraordinaire exceeds -- or rather,
deliberately transcends, seeking to survey the spectrum from pental to
septimal.
Thus in a 12-note well-temperament, fifths are generally either pure
or tempered narrow (as in a meantone or 12-EDO), with thirds ranging
roughly from a pure pental 5:4 to a Pythagorean 81:64 (386-408 cents).
Some definitional leeway should be allowed. Neidhardt and others have
published well-temperaments where at least one fifth is wide of pure,
but the circle is arranged so that all major thirds remain Pythagorean
or smaller. Also, the most remote major thirds might go a cent or two
beyond 81:64, as in a system of Aaron Johnson using some 19:15 thirds
(409 cents). However, if interpreted flexibly, the pental-Pythagorean
concept remains a useful guideline.
In a typical 17-note well-temperament such as Secor's 17-WT, all
fifths are tempered wide of pure (here by 2.422 cents or 5.265 cents),
with all usual major and minor thirds thus in the range from
Pythagorean to septimal: respectively 408-435 and 267-294 cents. If
the circle includes some fifths around 22-EDO size (7.14 cents wide),
arguably about the limit for full fifth-circulation, then some major
thirds may exceed 9:7 (435.08 cents) by a cent or two. Again, if
interpreted with some leeway, a Pythagorean-septimal concept nicely
conveys the nature of the tuning.
A temperament extraordinaire, in contrast, mixes narrow and wide
fifths so as to produce a spectrum of major and minor thirds ranging
from roughly pental to septimal, with the nearer meantone portion of
the circle like that of a 12-note well-temperament, and the more
remote part like a 17-note well-temperament.
In this aspect, the temperament extraordinaire might be said to
resemble a regular meantone in the 1/4-2/7 comma range, which likewise
offers thirds with both pental and septimal sizes. As Paul Poletti has
emphasized, designers of classic 12-note well-temperaments were
concerned not only with the meantone Wolf fifth, but especially with
the "bad thirds" as judged from a pental point of view, e.g. the
diminished fourths at sizes like 427 or 434 cents. Such thirds, of
course, are the pride and joy of a temperament extraordinaire. Thus
one might be tempted to say that the ethos is simply "12-note meantone
without the Wolf fifth."
In a regular meantone, however, there are only two sizes of major or
minor thirds: for example 262/313 cents and 383/434 cents in 2/7-comma.
The temperament extraordinaire embraces this pental-septimal range,
but adds the element of _gradation_ with its many intermediate sizes:
thus major thirds at 383, 396, 408, 421, and 434 cents in a system
based on 2/7-comma; and minor thirds at 275, 287, 300, and 313 cents.
This arrangement admittedly dilutes some of the dramatic contrast of
regular meantone, but may facilitate subtler transitions from pental
to Pythagorean to septimal color when this effect is desired. Some
opportunities for direct jumps from the smallest to largest sizes
remain, as in this piece in a 2/7-comma system:
Another regular meantone feature somewhat compromised, but still
largely retained, is the often striking contrast between diatonic and
chromatic semitones: 117/76 cents in 1/4-comma, and 121/71 cents in
2/7-comma. In a 2/7-comma system, for example, C-C#-D and F-F#-G
retain their divisions of 71-121 cents; G-G#-A and Bb-B-C have a
smaller but still notable contrast of 83-108 cents; and D-Eb-E has an
equal division into two 96-cent steps.
It is only fair to emphasize that each genre of tuning has its
advantages, and also its limitations: some of the latter being obvious
from within the worldview of the system design itself, and others
becoming evident only when a comparison is made with the possibilities
opened by other systems.
For example, a 12-note well-temperament makes it possible to play
conventional 18th-19th century tonal music in any key with more or
less reasonable interval sizes for this style -- unlike a regular
12-note meantone or temperament extraordinaire, where earlier modal
styles are central to the design; or a 17-note well-temperament, which
unlike any of these systems offers a wealth of neutral intervals.
----------------------------------------
3. Interval regions and musical contexts
----------------------------------------
The eight narrow and four wide fifths of a temperament extraordinaire,
whether based on a meantone in the arguable zone of ideal euphony
between around 1/4-comma and 2/7-comma, or on a heavier degree of
temperament ranging out to around Harrison/Lucy tuning or 88-EDO,
generate a characteristic musical structure meant to accommodate both
Renaissance meantone and neomedieval styles.
In a tuning system with only 12 notes per octave, there is a
delicately balanced sharing of resources between these two styles.
Thus we find seven meantone-range major thirds, five pure or near-pure
as in the regular shade of meantone on which the tuning is based, and
two around 395-397 cents (Bb-D, E-G#); and five neomedieval major
thirds ranging from around Pythagorean to septimal.
A regular 12-note meantone (Eb-G#) has usual near-pure major thirds at
the same seven locations, plus also at Eb-G -- a third tuned around
Pythagorean in the temperament extraordinaire! When playing in a
16th-century style, some discretion is required with respect to this
third. If it occurs prominently in the sonority Eb-G-Bb, where the
tension of a Pythagorean major third can disturb the smooth flow of
meantone concords, then one might consider either tastefully
ornamenting to alleviate this impression, or transposing to avoid the
problem -- for example, from G Dorian to D Dorian, where this sonority
becomes the milder Bb-D-F with a major third around 395-397 cents.
However, at least to some modern tastes, other sonorities involving
Eb-G such as C-Eb-G or Eb-G-C tend expressively and delightfully to
"color" rather than disturb the flow of the music in a Renaissance or
Manneristic style, and are indeed a special attraction for a mode such
as G Dorian.
While Eb-G-Bb may be problematic in a 16th-century style, it is a fine
neomedieval sonority; the other four large major thirds present no
problem in either style, since in a regular Eb-G# meantone they are
expected to be very wide diminished fourths often at around a septimal
size (spelled in this context as B-Eb, C#-F, F#-Bb, and G#-C).
Minor thirds might be placed in three groups: six usual meantone
thirds at the same size as in a regular tuning; two thirds (G-Bb,
G#-B) at 300 cents, which might be viewed as a kind of bridge between
the meantone and neomedieval realms, often occurring in either style;
and four neomedieval thirds in the Pythagorean-septimal range.
Just as Eb-G is a usual meantone third in a regular tuning that
becomes roughly Pythagorean in a temperament extraordinaire, so C-Eb
shifts from usual meantone size to somewhat smaller than Pythagorean.
In practice, the result can be more charming than disruptive in a
16th-century context -- at least to some modern tastes. The other
three small minor thirds are augmented seconds in a regular meantone,
so that usual Renaissance or Manneristic intonational expectations are
met (with spellings in this context of Eb-F#, F-G#, and Bb-C#).
In a Renaissance/Manneristic context, one has a "safe" range of Bb-G#,
a typical range of accidentals for untransposed modes without an
accidental signature. Transposed modes with a Bb signature have a
typical accidental range of Eb-C#, where caution may be indicated if
Eb-G-Bb occurs prominently, as noted above, with transposition down a
fourth or up a fifth as one possible solution and ornamentation
another.
In a neomedieval context, conversely, the remote portion of the circle
is prize territory: we would like to use major and minor thirds
between Pythagorean and septimal, or also 300-cent minor thirds which
are not too far from the Pythagorean 32:27 (294 cents).
Normal interval spellings may shift as we move from one style to the
other. Thus in a 16th-century context, C#-F in the vicinity of 9:7
makes sense as a diminished fourth rather than a usual third; in a
neomedieval context, it is a fine regular major third, Db-F. Likewise
we have E-G#-C as a "special effects" sonority in meantone with a
diminished fourth G#-C, but Ab-C-Eb-F as an outstanding neomedieval
cadential sonority resolving to G-D-G with the major third Ab-C
expanding to the fifth G-D.
Similar shifts quite properly occur in semitone spellings. Thus the
meantone chromatic semitone C-C#, for example in the striking melodic
progression C-C#-D, would be spelled C-Db in a neomedieval modality
such as C Phrygian, where it plays the role of a small diatonic
semitone.
In a 13th-century variety of neomedieval style where often only the
seven diatonic steps of a given mode plus Bb -- or their transposed
equivalents -- are required, it is often possible to find locations in
a temperament extraordinaire where major thirds are always or
generally around Pythagorean or larger, and minor thirds 300 cents or
smaller (e.g. Eb Dorian, C Phrygian, Ab Mixolydian). However,
especially in a 14th-century style calling for various routine
cadential inflections, some meantone-range thirds will inevitably
occur: the circle is not large enough for it to be otherwise.
Such an effect may recall the contrasts of modified Pythagorean
keyboards tunings evidently popular in the earlier 15th century, where
regular Pythagorean thirds are often juxtaposed with altered ones very
close to 5:4 or 6:5. Here the effect may be even more pronounced, with
major thirds typically varying from around 5:4 to 9:7, the extreme
sizes differing by the regular meantone diesis (about 41 cents in
1/4-comma, 50 cents in 2/7-comma, and 55 cents in 88-EDO).
From an historical perspective, one must prudently caution that even
within its usual meantone range of Bb-G#, a temperament extraordinaire
will significantly alter the sound of a regular meantone tuning:
prominent major thirds at 395-397 cents and minor thirds at 300 cents
are very likely to occur (e.g. G-Bb-D, Bb-D-F, E-G#-B).
In a neomedieval style, where Pythagorean-to-septimal thirds (and
300-cent minor thirds with their Pythagorean affinity) are the norm,
the limited size of the system will often introduce striking
juxtapositions with meantone thirds near 5:4 and 6:5 -- a touch of
"modal color" often pleasant and sometimes captivating, but a matter
of necessity rather than free choice. However, the most serious
aesthetic concern might be the pervasive tempering of fifths and
fourths, the stable concords in this style, by about 5-7 cents -- as
compared to their purity in Pythagorean intonation!
An awareness that the temperament extraordinaire supplements rather
than replaces regular meantones and neomedieval tuning systems can
place this alternative in better perspective.
--------------------------------------------
4. Interval regions and resources: a summary
--------------------------------------------
One strategy for getting an overview of the interval regions and
categories discussed in the last section is a table showing the
sizes of major and minor thirds available in some representative
shadings of the temperament extraordinaire along the spectrum from
1/4-comma meantone to 88-EDO.
Here it may be helpful quickly to describe the degrees of meantone on
which the five systems shown in the table are based.
The 1/4-comma temperament was evidently described by Pietro Aaron in
1523, who urges the student to tune a major third as "sonorous and
just" as possible, and then divide it into four equally impure fifths,
then continuing with the other fifths and thirds. This account seems
to imply, but does not explicitly specify or mathematically define, a
regular 1/4-comma. Zarlino's definition of 1571 may be the earliest.
Thorvald Kornerup's Golden Meantone tuning (1935) is based on the
Golden Ratio, Phi (~1.618034), here the ratio between the logarithmic
sizes (as measured in cents) of the whole tone and diatonic semitone
(about 192.43 and 118.93 cents), and likewise the diatonic and
chromatic semitones (about 118.93 and 73.501 cents). It is located
near the middle of the range between 1/4-comma and 2/7-comma, Mark
Lindley's range of maximum meantone euphony.
The 2/7-comma temperament, described by Gioseffo Zarlino in 1558, is
the first known meantone to be specified mathematically. Major and
minor thirds are equally impure, which can make it both an attractive
compromise and rather difficult to tune accurately by ear in
comparison to 1/4-comma with its pure 5:4 major thirds, as Zarlino's
remarks of 1571 may suggest.
Erv Wilson's Metameantone (or Meta-meantone), described in a paper of
1995, is designed to produce proportional triads, and tempers somewhat
more heavily than 2/7-comma.
The 88-EDO temperament extraordinaire takes advantages of the two
fifths reasonably close to 3:2 available in this system. The smaller
fifth of 51 steps can be used to generate a regular meantone with
fifths about 6.50 cents narrow, and minor thirds only about 2 cents
narrow of a just 6:5. In the temperament extraordinaire, eight of
these meantone fifths are combined with four fifths of 52 steps at
about 7.14 cents, identical to those of 22-EDO, which yield fine
approximations of a just 7:6, about 5.86 cents wide.
For each system, the table shows the two sizes of fifths and nine of
thirds (five major, four minor); and the locations for these sizes.
For each type of major or minor third, the number of meantone (MT) and
wide (W) fifths in the chain generating it is also shown: thus a major
third of the Pythagorean type formed from two meantone and two wide
fifths is (2 MT/2 W). Following the charts are explanations of the
suggested interval categories or "zones."
----------------------------------------------------------------------------
Varieties of thirds in temperament extraordinaire systems
----------------------------------------------------------------------------
TUNING 1/4-comma Kornerup 2/7-comma Wilson 88-EDO
MT Golden MT MT MetaMT
----------------------------------------------------------------------------
Meantone 5ths (F-C#) 696.58 696.21 695.81 695.63 695.45
Tempering from 3:2 -5.38 -5.74 -6.14 -6.32 -6.50
8 locations F-C, C-G, G-D, D-A, A-E, E-B, B-F#, F#-C#
............................................................................
Wide 5ths (C#/Db-F) 706.84 707.57 708.38 708.74 709.09
Tempering from 3:2 +4.89 +5.62 +6.42 +6.78 +7.14
4 locations C#-G#, G#/Ab-Eb, Eb-Bb, Bb-F
----------------------------------------------------------------------------
MAJOR THIRDS
----------------------------------------------------------------------------
Regular meantone, 5:4 zone 386.31 384.86 383.24 382.52 381.82
5 locations (4 MT 5ths) F-A, C-E, G-B, D-F#, A-C#
............................................................................
Outlying meantone zone 396.58 396.21 395.81 395.63 395.45
2 locations (3 MT/1 W) Bb-D, E-G#
............................................................................
Pythagorean, 81:64 zone 406.84 407.57 408.38 408.74 409.09
2 locations (2 MT/2 W) Eb-G, B-D#
............................................................................
14:11-to-17-EDO zone 417.11 418.93 420.95 421.85 422.73
2 locations (1 MT/3 W) Gb-Bb, Ab-C
............................................................................
Septimal, 9:7 zone 427.37 430.28 433.52 434.96 436.36
1 location (4 W 5ths) Db-F
----------------------------------------------------------------------------
MINOR THIRDS
----------------------------------------------------------------------------
Regular meantone, 6:5 zone 310.26 311.36 312.57 313.11 313.64
6 locations (3 MT 5ths) D-F, A-C, E-G, B-D, F#-A, C#-E
............................................................................
Transitional, 300 cents 300.00 300.00 300.00 300.00 300.00
2 locations (2 MT/1 W) G-Bb, G#-B
............................................................................
13:11-33:28 zone 289.74 288.64 287.43 286.89 286.36
2 locations (1 MT/2 W) C-Eb, Eb-Gb
............................................................................
Septimal, 7:6 zone 279.47 277.29 274.86 273.78 272.73
2 locations (3 W 5ths) F-Ab, Bb-Db
____________________________________________________________________________
----------------------------------------------------------------------------
Among the major thirds, the first three categories should be fairly
self-explanatory. Regular meantone thirds in this portion of the
spectrum are quite close to 5:4 (386.31 cents), while the outlying
thirds are about 9-10 cents wide of this ratio. The Pythagorean zone
centers around the ratio of 81:64 (407.82 cents), with 88-EDO giving
a fine near-just version of the slightly larger 19:15 (409.24 cents).
The 14:11-to-17-EDO zone starts around 14:11 (417.51 cents), the
classic mediant between the simpler major third ratios of 5:4 and 9:7,
and extends to the neighborhood of 17-EDO (423.53 cents). Somewhere
around 422-423 cents may mark what David Keenan has termed the region
of Noble Intonation (NI), or area of "metastability" and maximum
harmonic complexity between 5:4 (386.31 cents) and 9:7 (435.08 cents)
as estimated by a Phi-weighted mediant. The 1/4-comma system yields a
near-just 14:11 (0.40 cents narrow), with Kornerup's Golden Meantone
also close (1.41 cents wide), while 2/7-comma or beyond has more of an
affinity to Keenan's suggested NI region.
The septimal zone approaching 9:7 (435.08 cents) may begin around the
32:25 of 1/4-comma (7.71 cents narrow), with Wilson's Metameantone
interestingly yielding a virtually just 9:7 (0.13 cents narrow). More
generally, if one seeks an especially accurate approximation of this
septimal ratio, then temperaments in the range from around 2/7-comma
to 88-EDO produce sizes within 2 cents of pure.
Among the minor thirds, the regular meantone sizes closely approximate
6:5 (315.64 cents), with the transitional 300-cent thirds a kind of
bridge between the meantone and Pythagorean-septimal worlds, often
seeing use in either a neomedieval or Renaissance/Manneristic style.
These transitional thirds are quite close to 19:16 (297.51 cents), and
might lend an extra "rootedness" or "solidity" to a closing sonority
such as G-Bb-D (e.g. G Dorian) in a style of around 1500 where pieces
sometimes conclude with a minor third above the modal final.
Moving into neomedieval Pythagorean-septimal territory, we next
encounter a zone with the ratios of 33:28 (284.45 cents) and 13:11
(289.21 cents) as possible landmarks; the latter is the classic
mediant between the simple ratios for a minor third of 6:5 and 7:6
(266.87 cents). Both 1/4-comma and Kornerup's Golden Meantone are very
close to 13:11, respectively 0.53 cents wide and 0.57 cents narrow.
The other systems are in the middle region between 13:11 and 33:28,
with 2/7-comma leaning a bit toward the former; Wilson's Metameantone
almost precisely equidistant from both (2.32 cents narrow of 13:11,
2.44 cents wide of 33:28); and 88-EDO leaning a bit toward the latter.
Around 33:28 we may be close to Keenan's Noble Intonation or NI region
of maximum complexity between 6:5 and 7:6, possibly centered somewhere
around 283.6 cents.
The smallest minor thirds approximate the septimal ratio of 7:6, with
interesting variations in shading. From 1/4-comma to 2/7-comma, with
sizes of around 275-279 cents, we are in what might be termed the
"Monzian" region, after the composer Joe Monzo. In writing a piece, he
desired to tune an interval near but not precisely at 7:6, choosing
279 cents by ear, and then deciding in the total context on the
complex just ratio of 75:64 (274.58 cents). As it happens, 1/4-comma
almost exactly matches the first size, and 2/7-comma the second, with
Kornerup's Golden Meantone roughly in the middle (277.29 cents). These
sizes can pleasantly evoke a septimal color, but inhabit suburban
precincts about 8-13 cents from 7:6 itself. To approach within 7 cents,
we must proceed to Wilson's Metameantone, with a size of 273.78 cents,
or 6.91 cents wide. Pressing on yet further to 88-EDO improves our
accuracy by another cent: 272.73 cents, or 5.86 cents wide of 7:6.
In any of these systems, the five varieties of major thirds and four
of minor thirds open broad intonational prospects; and each point on
the spectrum of temperament has its own delicate tradeoffs and charms.
-----------------------------------------------
5. Appendix: Harmonic recycling and fifth sizes
-----------------------------------------------
An engaging feature of the temperament extraordinaire is that the
regular meantone and wide fifths are comparably impure. Here we will
look more closely at the mechanism of "harmonic recycling" governing
this relationship, first more informally, and then, for those who
might be curious, in more general mathematical terms.
A basic attribute of any 12-note circle is that the fifths must have a
net temperament of one Pythagorean comma narrow (531441:524288, about
23.46 cents): the amount by which 12 pure fifths exceed seven pure
octaves. In some classic well-temperaments, something like four to six
of the fifths are tempered narrow, using up this comma and leaving the
circle balanced with the others pure.
In the temperament extraordinaire, eight of the fifths are tempered by
somewhere between around 1/4-comma and 2/7-comma or even 3/10-comma,
considerably exceeding the one Pythagorean comma needed to balance out
the circle -- and indeed creating a "harmonic surplus" which is then
distributed equally among the four wide fifths, restoring the balance.
The narrow fifths give rise to usual meantone intervals, the wide
fifths to some approximately septimal intervals, and their mixture to
a delightfully variegated spectrum of intermediate sizes and "modal
colors."
A complication in understanding this harmonic "ecosystem" is that
meantone temperaments are often measured in terms of the syntonic
comma (21.51 cents), while the larger Pythagorean comma (23.46 cents)
is vital in determining how much of a surplus the wide fifths will
recycle. For now, we can appreciate some aspects of the process simply
by using cents as a common denominator, and bearing in mind the size
of each comma. There is also a method for converting fractional commas
from one measure to the other, presented later for those who may be
interested.
Let us consider a temperament extraordinaire based on 1/4-comma, where
fifths are tempered about 5.38 cents narrow of pure. Thus means that
eight such fifths will do a total of about 43.04 cents of narrowing --
as compared with the 23.46 cents sufficing to balance the circle. This
leaves us with a surplus of about 19.58 cents for the four wide
fifths, each thus about 4.895 cents wider than pure. As it happens,
our 1/4-comma fifths are more precisely about 5.3766 cents narrow, and
the wide ones so by about 4.888 cents.
How about 2/7 comma, where fifths are wide by about 6.14 cents? Eight
of these fifths will do about 49.12 cents of narrowing, which less the
Pythagorean comma leaves about 25.66 cents for recycling, or about
6.415 cents for each wide fifth. More precisely, the two types of
fifths are about 6.145 cents narrow and 6.424 cents wide.
The same approach can apply to a system based on some meantone defined
otherwise than in fractions of a syntonic comma: for example, 88-EDO,
where our narrow fifth is equal to 51/88 octave, or 695.455 cents --
about 6.500 cents narrow of 3:2. Eight such fifths will thus do about
52 cents of narrow tempering -- which, less the Pythagorean comma,
leaves about 28.54 cents for harmonic recycling, or 7.135 cents for
each of the four wide fifths; a more precise amount is 7.1359 cents.
From these examples, we may note a certain apparent pattern. At
1/4-comma, the low end of our range of temperaments, the narrow fifths
are tempered a bit more than the wide ones (5.38/4.89 cents). At
2/7-comma, however, the temperament of the narrow fifths is not quite
as great as that of the wide ones (6.14/6.42 cents). Moving to the yet
heavier meantone temperament of 88-EDO, we find that this disparity
becomes somewhat greater (6.50/7.14 cents).
Is there a point somewhere within this range where the two types of
fifths would be tempered by precisely the same amount, albeit in
opposite directions?
Let us consider a meantone temperament of 1/4 Pythagorean comma, a
measurement less common for meantones than the syntonic comma, but
perfectly practical. Our meantone fifth is about 696.090 cents, or
smaller than 3:2 by 1/4 Pythagorean comma (about 5.865 cents).
Abbreviating "Pythagorean comma" as "PC," we can conveniently call
this 1/4-PC meantone, as distinct from the more familiar "1/4-comma"
(i.e. syntonic comma) temperament.
Since each fifth is 5.865 cents narrow, eight will do 46.92 cents of
narrow temperament, which, less the 23.46-cent Pythagorean comma,
leaves 23.46 cents of recycling -- making each of the remaining four
fifths 5.865 cents wide. Thus each of the 12 fifths is tempered by the
same amount -- eight narrow and four wide!
We can also reason this out in commas, without needing to use cents.
Each regular meantone fifth is 1/4-PC narrow, so eight must do a total
of 2 Pythagorean commas of narrow tempering. Subtracting the one
Pythagorean comma needed to balance the circle, this leaves a harmonic
surplus of one Pythagorean comma -- or 1/4 PC of wide temperament for
each of our four remaining fifths. Thus the circle is balanced, with
each fifth tempered either 1/4-PC narrow or 1/4-PC wide.
More generally, for temperament extraordinaire shadings based on
meantones of less than 1/4-PC, the narrow fifths will be tempered more
than the wide ones. Precisely at 1/4-PC, the two types of fifths will
be tempered by identical amounts -- in opposite directions, of course.
Beyond 1/4-PC, the wide fifths will be tempered more heavily than the
narrow ones: they have a lot of harmonic recycling to do, but can
translate this extra effort into benefits such as more accurate
approximations of septimal ratios such as 7:6, 9:7, and 7:4. Of
course, this also means some extra beating for the narrow and wide
fifths alike, with tastes varying as to the ideal compromises or
"sweet spots."
To summarize the discussion so far, here is a quick table of the
amount of temperament of narrow and wide fifths in the systems we have
considered to this point. Fractions of a comma refer to the syntonic
comma unless otherwise noted -- i.e., the "PC" notation.
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TEMPERAMENT 1/4-comma 1/4-PC 2/7-comma 88-EDO
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Narrow fifth - 5.377 - 5.865 - 6.145 - 6.500
Wide fifth + 4.888 + 5.865 + 6.424 + 7.136
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One point illustrated by this table is that increasing the tempering
of the narrow fifths by a given amount increases the tempering of the
wide fifths by twice that amount. Intuitively, we might explain this
by observing that increasing the temperament of each narrow fifth by a
given quantity will add eight times that quantity to the harmonic
surplus, with only four wide fifths to do the recycling, so that each
must absorb twice that quantity.
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5.1. The mathematics of harmonic recycling
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Translating some of the concepts we have considered into a more
concise mathematical approach, we may begin with the following formula
to calculate the temperament of the wide fifths from that of the
narrow fifths, both expressed in Pythagorean commas:
8t - 1
w = ------
4
Here w is the tempering of a wide fifth, and t the tempering of a
narrow one. For example, let us consider the simplest example of
1/4-PC meantone. Here t is 1/4, and 8t is equal to 2, the total amount
of narrow tempering done. Subtracting a Pythagorean comma from this
amount -- "8t - 1" -- yields the harmonic surplus, here (2 - 1) or one
Pythagorean comma. Dividing this amount by 4 yields the amount of
harmonic recycling done by each wide fifth, 1/4 Pythagorean comma: in
other words, the amount by which it is tempered wide to balance the
circle.
This formula can be simplified to:
w = 2t - 1/4
Applying this version to our 1/4-PC temperament, we simply take twice
the temperament of a narrow fifth, getting 1/2, and subtract 1/4,
arriving again at a temperament of 1/4 PC for our wide fifths.
How about meantones measured in the traditional manner using fractions
of a syntonic comma? One practical modern approach is simply to use
cents as a common unit and convert both syntonic and Pythagorean
commas to this convenient measure. This is what we did in much of the
informal discussion above. Our formula, with both "t" and "w" measured
in cents, is then:
w = 2t - 5.865
Let us try, for example, a 1/4-comma (i.e. syntonic comma) meantone.
Here t is about 5.377 cents, and 2t about 10.754 cents, which less the
5.865 cents of 1/4 PC is about 4.889 cents for w, the temperament of
each wide fifth -- very close to the actual 4.8881 cents or so.
If for some reason one wishes to calculate directly in syntonic and
Pythagorean commas without using cents, we can take advantage of a
near-exact equation: a syntonic comma (about 21.51 cents) is virtually
equal to 11/12 of a Pythagorean comma. Thus if t is measured in
syntonic commas, we can calculate w in Pythagorean commas by using
this version of the formula:
w = 11/6 (t) - 1/4
For example, let us consider 2/7-comma meantone. Using the formula
calls for some patient arithmetic with fractions. Multiplying the
2/7-comma (i.e. syntonic comma) temperament of each narrow fifth by
11/6, getting 11/21, and then subtracting 1/4, we find that each wide
fifth is tempered by 23/84 PC. This is equivalent to 6.424 cents, and
also to 23/77 syntonic comma, for the curious -- so that, as it
happens, the wide fifths are tempered almost precisely by 1/77 of a
syntonic comma more than the narrow ones at 2/7 (or 22/77) of this
comma.
For a 12-note circulating system such as the temperament
extraordinaire, there is another unit of measure often very convenient
to use: John Brombaugh's _temperament unit_ or TU, equal to precisely
1/720 of a Pythagorean comma. Thus there are 720 TU in a Pythagorean
comma, and almost precisely 660 TU in a syntonic comma. Since the
number 720 is conveniently divisible evenly by various factors, we are
often able to set up a temperament or do other calculations using
simple integers rather than fractional commas or decimal cents.
Thus with the tempering of both narrow and wide fifths expressed in
TU, we have:
w = 2t - 180
Let us consider, for example, a temperament extraordinaire based on
3/10-comma meantone. Since a syntonic comma is 660 TU, multiplying
this by 3/10 gives an even 198 TU for t, the temperament of each
narrow fifth. We multiply by 2, getting 396 TU, and subtract 180 TU,
arriving at 216 TU of tempering for each of the four wide fifths.
Converting these temperings from TU to cents, if we so desire, will
still involve some decimal math: a TU is about 0.03258 cents, so that
there are roughly 30.6905 TU in a cent. In our 3/10-comma system,
almost identical to John Harrison's pi-based meantone (1775)
popularized and extended in recent times by Charles Lucy (where the
regular major third is equal to 1200/pi cents), the eight regular
meantone fifths are narrowed by about 6.452 cents, and the wide ones
made wide by about 7.038 cents. These values, as it happens, are
equivalent to 3/10 syntonic comma for the meantone fifths (as
advertised), and 3/10 PC for the wide ones -- respectively 695.503
cents and 708.994 cents.
When thinking mainly in TU, we might now and then simply want to get a
rough idea of how quantities might measure up in cents, without
needing an exact conversion. If so, we find that 30 TU at 1/24 PC are
almost but not quite equal to a cent, while 60 TU or 1/12 PC are equal
to the familiar 1.955 cents by which each fifth is narrowed in 12-EDO.
Thus in the last example based on 3/10-comma meantone, we can guess
that the meantone fifths at 198 TU narrow are tempered by something
like 6.5 cents, and the wide ones at 216 TU wide by something on the
order of 7 cents. Dividing the number of TU by 30, and remembering
that the accurate result in cents "is actually a bit less," can be a
handy approach for casual and friendly approximations.
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5.2. Septimal intervals: an amusing reckoning
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Having focused mainly on how much harmonic recycling is done in a
given temperament extraordinaire, and thus how much each of the four
wide fifths must be tempered, we might briefly the consider the
harmonic efficacy of that recycling in achieving one principal goal of
the system: reasonably accurate septimal approximations in the most
remote portion of the circle.
In addition to the syntonic and Pythagorean comma we have so far used
informally and more formulaically to measure temperings of meantone
and wide fifths, this concern brings into play the septimal comma,
64:63 or about 27.26 cents. This is the amount of harmonic recycling
that must be done in an interval's chain of wide fifths (or narrow
fourths) to make it a just 9:7 major third, 7:6 minor third, or 7:4
minor seventh. Here we focus on intervals generated from chains where
all fifths are wide (or fourths are narrow), since these are the
intervals in a temperament extraordinaire which can yield reasonably
accurate septimal approximations.
Using temperament units, where the Pythagorean comma or PC is
precisely 720 TU and the syntonic comma for all practical purposes
equivalent to an even 660 TU, we find that the larger septimal comma
is almost exactly 836.75 TU. For casual "ballpark estimates," the
convenient approximation of 840 TU exaggerates the size of the comma
by about 0.105 cents; here we shall use a "middle-precision" rounding
to the nearest integer, 837 TU, with an error of about 0.008 cents.
Let us briefly consider our previous example of a 3/10-comma meantone
system, where narrow fifths are tempered by 198 TU and wide ones by
216 TU. We will assume the usual arrangement where the eight fifths
from F to C# are tempered narrow, and the other four wide.
How close will the largest major third Db-F (spelled in a meantone
context also as the diminished fourth C#-F) come to a just 9:7? This
third has all four wide fifths in its chain: C#/Db-Ab-Eb-Bb-F. Each
fifth, with its harmonic recycling, contributes 216 TU to this good
cause, as well as balancing the circle in the process. In all, we find
that 864 TU of wide tempering are done -- actually overshooting our
comma at 837 TU by about 27 TU (for purists, almost exactly 27.25 TU).
We can thus guess that Db-F will be larger than 9:7 by something
approaching a cent. In fact its size is about 435.975 cents, or around
0.891 cents wide.
How about the minor thirds F-Ab and Bb-Db, each derived from a chain
of three wide fifths or narrow fourths (F-Bb-Eb-Ab, Bb-Eb-Ab-Db)? How
close will they come to 7:6? Here, again, each fifth or fourth does
216 TU of tempering in the desired direction -- but this time there
are only three of these generators in a chain, or 648 TU of recycling
done in all. This falls short of our desired 837 TU by 189 TU, or
something on the order of six cents. In fact, these thirds are at
about 273.019 cents, or 6.148 cents wide of 7:6. Note that falling
short of a full septimal comma of recycling or tempering causes a
minor septimal interval to be wide of a just ratio like 7:6, while
overshooting the comma causes the same result for a major septimal
interval, as with the 9:7 major third in our last example.
Finally, how about the minor sevenths F-Eb, Bb-Ab, and Eb-Db, each
from a chain of two wide fifths or narrow fourths (e.g. F-Bb-Eb)?
How closely will these sevenths approach 7:4? Here each wide fifth or
narrow fourth again contributes 216 TU, but with only two generators,
the total recycling is limited to 432 TU, only a bit more than half of
our 837 TU comma. The difference is 405 TU, or something on the order
of 13 cents, the amount by which we expect these sevenths to be wide
of the beloved 7:4. In fact, at about 982.012 cents, they are wide by
around 13.187 cents.
These calculations illustrate how, even for a meantone like 3/10-comma
near the upper end of the temperament extraordinaire spectrum, the
process of dispersing the septimal comma in a chain of only two,
three, or four fifths can achieve the highest accuracy only for 9:7,
and must inevitably fall somewhat short for 7:6 and yet more so for
7:4. This need not discourage us from enjoying the many shadings of
septimal color available at different points along the spectrum,
surveyed for certain representative tunings in Section 4 above.
Most appreciatively,
Margo Schulter
23 October 2007
Corrected
19 January 2009