- Hello group, on 29 november 2002 Margo Schulter, on the topic "Re:
Unison vectors (Peppermint, 72-EDO, etc.)" wrote us :
<For my own purposes of optimization, if I were using notes taken from
a single chain of fifths in a regular temperament around this region,
I might choose a generator of around 704.160 cents>
<In this scheme, the representations of 2-3-7-9-11-13 and 14:17:21 are
much as in Peppermint 24, where an arbitrary "quasi-diesis" of ~58.680
cents is used to achieve a pure 7:6.
<Note, however, that from your 17-limit perspective, there's an
important disadvantage in this approach: the noncontiguous chains
don't include the excellent approximation of 4:5 as 21 fifths up
(augmented second plus natural 12-diesis).>
<From my own point of view, I might prefer Peppermint 24, because ...>
<From my practical perspective as a keyboardist,... >
<The main advantage of a regular and contiguous temperament such as
e-based tuning (Blackwood's R=e) is that we have free
ratios of 11:12:13:14 are overall more accurately represented. Each
scheme has its own charms.
Indeed, many of us are looking for "a better tuning" or "a better
scale", and are taking an optimization approach, as some look for the
My question is: has one of us, or another friend of music, reason or
beauty, attempted to list or sort out optimization criteria ? Has one
also tried to optimize the search for such criteria, and what could
be the criteria for that new optimizing process? Are there scales of
With deep interest and cordiality
- --- In tuning@y..., "domilare" <Dominique.Larre@w...> wrote:
> My question is: has one of us, or another friend of music,reason or
> beauty, attempted to list or sort out optimization criteria ? Hasone
> also tried to optimize the search for such criteria, and whatcould
> be the criteria for that new optimizing process?these, if i understand you correctly, have been major concerns of
the tuning-math list. i refer you to that list, its archives, and its
members, for more info . . .
> Are there scales ofi'm not sure how to understand that sentence, though "criterion
> criterion futility?
futility" sounds like a good name for a james bond movie or
something . . . :)
- Hello, there, Dominique Larre, and thank you for raising a question in
response to some recent comments of mine which is a most happy one:
"What is the basis for optimizing a tuning, or more generally a
musical style or outlook?"
While a group like tuning-math might be the ideal place to discuss the
mathematical fine points of finding a precise "ideal" size of fifth
for a given genre of temperament, for example, I consider the question
of _why_ one prefers a given type of tuning for a given style very
much a characteristic topic here. Of course, this is not to suggest
that this kind of philosophical discussion would be out of place for a
more specialized kind of forum either, only to say that "optimization"
is a question which invites considering "Why?" as well as "How?"
For example, the meantone temperaments associated with the Renaissance
music of 16th-century Europe seem nicely to fit the style: thirds with
ratios at or near 5 (5:4, 6:5) fit with a smoothly flowing texture of
mostly homogenous and saturated concords based on these ratios, while
the rather wide diatonic semitones comport with the gentleness of the
style. This isn't to say that there isn't an element of tension and
contrast -- the suspension, especially, beautifully fulfilling this
role -- but that the tuning system and style seem in harmony.
Of course, musicians and mathematicians can debate the precise degree
of meantone temperament which ideally serves a given taste, for a
specific piece: thus Mark Lindley prefers 1/4-comma (pure major thirds
at 4:5) for some pieces, but 2/7-comma (major and minor thirds equally
impure) for slower-moving Venetian organ pieces, and 1/5-comma (major
thirds a bit larger than pure) for some English keyboard music around
1600 where the narrower diatonic semitones or other nuances are more
"sprightly" in effect.
A theorist might come up with some "ideal" meantone from the viewpoint
of some mathematical function or formula -- and this has been done
more than once -- but different people can have different preferences,
now as in the 16th century, and listeners such as Lindley suggest that
the "optimal" answer might be different for one piece than another.
With this prelude on a more familiar kind of tuning system, I'll try
to share a few aesthetic considerations for what I call a "common
temperament" (_participatio communis_) where fifths around about two
cents wider than pure -- about the same amount of temperament as in
12-tone equal temperament, but in the opposite direction.
At one level, the "common music" for which such a temperament is
favored is largely based on a "classical" European background -- that
is, the European tradition in the era from Perotin to Machaut, or
about 1200-1400. From this viewpoint, the basic 12-note temperament on
which a _participatio communis_ tuning is based can be seen as a
somewhat accentuated form of medieval Pythagorean tuning.
At the same time, however, the aesthetic of this _common music_ also
focuses on categories of intervals not discussed in the medieval
European tradition, but recognized in medieval Near Eastern traditions
and used in certain modern Near Eastern musics also. Some of these
type of ratios and intervals, interestingly, also arise from the logic
of experimentation in a European-type style with the new sizes of
intervals made available by a 21st-century tuning with wide fifths.
One criterion is that the tuning should produce a regularly arranged
and very attractive diatonic scale. With fifths of around 704 cents,
for example, whole-tones are around 208 cents, about 4 cents larger
than the Pythagorean 9:8 (~203.91 cents), while diatonic semitones are
close to 22:21 (~80.54 cents). These semitones are rather more narrow
and incisive than the 100-cent steps of 12-equal, or even the classic
256:243 steps of medieval Pythagorean tuning (~90.22 cents).
At the same time, two other sizes of melodic steps often play a
prominent role. Chromatic semitones at around 14:13 (~128.30 cents)
add a contrasting "color" to the music; while in a usual "common
temperament" of 24 notes, a small semitone or "thirdtone" often a bit
smaller than 28:27 (~62.96 cents) is often used in cadences involving
resolutions of intervals with ratios at or near 7.
In a 24-note system such as Peppermint 24, possibly the quintessential
"common temperament," there are other steps closely approximating
medieval Near Eastern ratios such as 12:11 and 13:12, often described
as neutral seconds. Here the "variety of steps" is deemed a musical
virtue, with a special elegance resulting when the steps closely
approach superparticular ratios (n+1:n).
While sonorities involving fifths and fourths mark the standard of
stable concord and richness, as in 13th-14th century European music,
unstable intervals such as thirds, sixths, major seconds, and minor
sevenths are often used in "relatively blending" sonorities and
textural passages, sometimes involving prolonged parallel motion with
an eventual cadence to a stable sonority. The fauxbourdon of the early
15th century provides a kind of model for some 21st-century forms of
parallelism involving sonorities such as a tempered 7:9:12 or
Regular or usual diatonic ratios for major and minor thirds are
generally around 14:11 (~417.51 cents) and 13:11 (~289.21 cents) or
33:28 (~284.45 cents), and these usual ratios do much to set the tone
and color for "common music" in such a temperament.
Augmented seconds at around 17:14 (~336.13 cents) and diminished
fourths at around 21:17 (~365.83 cents) provide a contrasting color,
with the "Four Convivial Ratios" (14:11, 13:11, 17:14, 21:17) serving
both as especially esteemed ideals for the imperfect concords, and as
friendly landmarks to describe a given fine degree of temperament. In
Peppermint, all four ratios are within 1.5 cents of pure, a
distinction giving it a special place on the musical map.
The regular major thirds around 14:11, and major sixths around 56:33
or 22:13, are often praised as having a "bright," "active," or "sunny"
quality, and as lending a certain "buzz" to directed cadential
progressions. They are also beloved intervals for passages in
conventional 15th-century fauxbourdon, where the observation by
Johannes Boen (1357) about parallel thirds and sixths before a cadence
is often cited: these pleasing intervals are like "forerunners and
handmaidens" of the stable concords which typically follow.
In a 24-note system such as Peppermint, a related kind of effect but
with a somewhat color is achieved by playing in parallel sonorities
such as approximate 7:9:12 ratios (~0-435-933 cents in just
intonation, and around 0-437-933 cents in Peppermint, with 7:12 pure
but 7:9 about 2.14 cents wide, the same amount by which the fourth is
narrow). These 7:9:12 sonorities are available on 10 of the 12 scale
steps of the upper keyboard manual: one plays the lower note on the
upper manual, and the notes visually a fourth and minor seventh higher
on the lower manual.
While in a conventional late medieval European style or related genre
of improvisation involves a variety of intervals, and even strict
15th-century fauxbourdon with little ornamentation involves a routine
contrast between major and minor types of thirds of sixths
(e.g. A3-C4-F4 vs. G3-B3-E4, with C4 showing middle C), a texture of
parallel 7:9:12 sonorities involves _literal_ parallelism, with all
voices moving by identically-sized steps. The effect might recall both
an organ mixture stop and some of the textures of Debussy at the turn
of the 20th century.
Another Debussyian effect possible in Peppermint is the use of
parallel sonorities with ratios near 11:14:18, with a usual 11:14
major third and a large 7:9 major third forming an outer neutral sixth
at 11:18. This feels to me very rich and "different," with a floating
quality that could suggest some sonorities of Debussy involving more
notes. In "common music" there is often a preference for relatively
mild timbres where complex ratios will have an effect or richness or
subtlety rather than "dissonance."
As already mentioned, a "common temperament" such as Peppermint
includes many intervals characteristic of medieval or modern Near
Eastern music, and this "bicultural" aspect of style further enriches
the music. A Near Eastern scale might be used in a primarily melodic
context, or as a source for vertical progressions introducing some
curious variations on "standard" medieval European harmonies and
Having described some especially valued features of "common music," I
should add that a "common temperament" or _participatio communis_ is
only one type of tuning system that might be used for realizing this
kind of music. For example, a 17-note system, either equal, or
well-tempered (as in George Secor's superb 17-note well-temperament),
or based on integer ratios, could be used to play much of the music we
are discussing very pleasantly.
There is also an awareness that a "common temperament" is indeed a
_modern_ tuning, in contrast to the classic Pythagorean intonation of
medieval Europe -- although the medieval Near East provides its own
classic setting for many of the interval types and sizes which
A final "optimization" feature might bear mentioning: in a tuning
system like Peppermint, while each keyboard is arranged in a standard
diatonic fashion, transposing or mixing notes from the two keyboards
makes possible many small nuances and distinctions between step sizes
and modes. Although a fixed-pitch instrument like a keyboard cannot
fully achieve the flexibility of ensemble performers either as
suggested by the medieval European theory of Marchettus of Padua
(1318) or reported for Near Eastern performances of _maqaam_ music,
the variety of step sizes and intervals at least makes it possible to
approximate some of these distinctions.
Here I have attempted to suggest some general aspects of
"optimization" -- tastes and preferences as to the best sizes for
intervals such as major and minor thirds in a given style -- along
with considerations such as variety of step sizes. I've picked a
rather conventionalized, although possibly not so widely familiar,
kind of 21st-century tuning to illustrate some of these points.
- --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> A theorist might come up with some "ideal" meantone from the viewpointIt's easier to do a calculation than wait for 100 years of practice and testing to suggest an answer. The calculation gives an immediate and practical answer to the question "How should I tune in this system?" Since there is more than one way of doing such a calculation, it also suggests we not take our answer as engraved in stone.
> of some mathematical function or formula -- and this has been done
> more than once -- but different people can have different preferences,
> now as in the 16th century, and listeners such as Lindley suggest that
> the "optimal" answer might be different for one piece than another.
--- In firstname.lastname@example.org, "M. Schulter" <MSCHULTER@V...> wrote:
> Of course, musicians and mathematicians can debate the precise
degree of meantone temperament which ideally serves a given taste,
for a specific piece: thus Mark Lindley prefers 1/4-comma (pure major
thirds at 4:5) for some pieces, but 2/7-comma (major and minor thirds
equally impure) for slower-moving Venetian organ pieces, and 1/5-
comma (major thirds a bit larger than pure) for some English keyboard
music around 1600 where the narrower diatonic semitones or other
nuances are more "sprightly" in effect.
***Thank you, Margo, for posting this. This is terrifically
interesting... I guess I had a rather *hazy* idea about these
distinctions, but not much of one. I note that all of Mark Lindley's
books on Amazon.com are currently out of print, as has been about
every book I've wanted to get for about the last two years... Some
eventually come in, others don't...