Yahoo Tuning Groups Ultimate Backup tuning Re: Middle paths

Hello, there, everyone, and it is with some trepidation that I find
myself writing at a time of more bombings and violence in the Middle
East. May these words, however trivial they may seem, serve to
encourage understanding among us as a musical community, possibly
providing a model for a larger world community of peace with music as
a force for communication and active reconciliation.

Mainly I would like to suggest that there are a range of "middle
paths" between classic multiprime just intonation (JI) systems and
equal temperaments or divisions of the octave (n-tET/n-EDO).

Further, tunings with octaves stretched or narrowed by a considerable
amount from a pure 2:1, or where an "octave" may not be deemed a
relevant concept (e.g. Bohlen-Pierce tuning in JI and tempered
versions based on a pure 3:1 "tritave"), provide other musical paths,
as do "spectrum tunings" based on the timbre of a specific type of
sound generator (e.g. acoustical or electronic).

Here I would like to present three types of "middle path" tunings, a
list possibly more suggestive than exhaustive.

-------------------------------
1. Pythagorean-based JI systems
-------------------------------

Pythagorean tunings featuring a chain of pure 3:2 fifths or 4:3
fourths have the property of being at once just and regular.

By combining two 12-note Pythagorean chains at some convenient
distance, for example a septimal comma of 64:63 (~27.26 cents), we can
achieve a JI system with the usual Pythagorean intervals within each
chain, plus intervals mixing notes from the two chains at pure ratios
of 2-3-7 prime (or 2-3-7-9 odd) such as 9:7, 7:6, 12:7, 7:4, and 8:7.

We thus get a 24-note system combining elements of regularity and
purity with free transposibility: in effect, we have a regular chain
of 23 fifths, with the one minor compromise that a single fifth is
stretched by about 3.80 cents, the 3-7 schisma, to obtain pure ratios
of 2-3-7-9.

Of course, like a regular 24-note Pythagorean tuning with 23 pure
fifths, this is an open rather than circulating system; for
circulation, we'd need a cycle of 53 notes in pure fifths. However, a
24-note chain provides lots of transpositions, much like an open
meantone or other eventone tuning (see next section) of this size.

For the kind of neo-medieval European style that the arrangement of
two 12-note Pythagorean keyboards a 64:63 apart might suggest, both
the traditional Pythagorean intervals and the new 7-based ones are
likely to be highly useful and largely interchangeable.

While other distances between the two Pythagorean chains can be very
attractive also, free transposibility seems an element of "middle
path" approaches, so it is the 64:63 system that I mention.

-------------------
2. Eventone tunings
-------------------

In addition to the meantone tunings discussed in this dialogue on
"middle" paths, I would mention other "eventone" tunings with regular
fifths all of the same size, and usual major thirds formed from four
of these fifths up less two octaves.

Meantones are eventones where these regular thirds more specifically
are at or near ratios of 5:4 and 6:5, with fifths narrower than the
pure 3:2 of Pythagorean tuning. Other eventones may seek to realize or
approximate ratios such as 9:7 and 7:6, or 14:11 and 13:11, with
fifths tempered in the _wide_ direction.

Meantones and other eventones can sometimes very closely approach
equal divisions of the octave, with the latter category possibly taken
as a subset of the former. However, identical or near-identical tuning
systems may be viewed from quite different musical perspectives.

For example, it is well-known that we can derive a circulating tuning
of 31 notes per octave from either 1/4-comma meantone with pure 5:4
thirds, or from 31-tET. These two tunings can both outstandingly serve
Renaissance and related styles of music, as well as some later styles.

As it happens, similarly, an eventone tuning with just 14:11 major
thirds is almost identical to 46-tET.

However, either of these mathematically synonymous tunings may be used
for some quite different musical purposes.

Someone who speaks of a "14:11 eventone" is likely focusing on the
regular and rather complex major third of this ratio (~417.51 cents)
formed from four fifths up, analogous to the 5:4 major third of
1/4-comma meantone, for example. Such a third nicely can serve a
neo-medieval European kind of style.

Many advocates of 46-tET, however, regard the "usual" major third as
the best approximation of 5:4, 15/46 octave or ~391.30 cents, an
interval formed from 21 fifths up.

For a 14:11 eventone enthusiast, this interval in 46-tET or its close
counterpart of ~391.92 cents in a just 14:11 tuning is a rather
"exotic" one, occurring at three positions in a 24-note tuning --
actually an augmented-second-plus-diesis.

As someone with a penchant for eventone tunings, I would say that the
elements of regularity and transposibility within a given range of
accidentals, and the diverse types of intervals generated as one
carries the system to more notes, are strong attractions.

----------------------------
3. Unequal well-temperaments
----------------------------

Both traditional 12-note well-temperaments of the kind often favored
in the era of European music around 1680-1850, and well-temperaments
based on tuning circles of other sizes such as 17 or 19 notes, offer
the advantages of a subtle shading of intervals within a system
featuring transposibility and circulation.

George Secor has published a superb 17-note well-temperament[1], while
Paul Erlich and others have devised new 12-note systems.

While eventone tunings may also have the property of circulation
(e.g. 1/4-comma meantone in a 31-note cycle, or a 14:11 tuning in a
46-note cycle), an unequal temperament often achieves such circulation
in fewer notes by varying the size of fifths in the tuning chain.

This technique, at the same time, causes variation in the sizes of
thirds and other intervals as one moves around the tuning circle. In
either a conventional 12-note scheme of the 17th-19th century type, or
a 17-note system such as Secor's, this gradation of intervals provides
an impressive musical resource.

----
Note
----

1. George Secor, "17-Tone Well Temperament," _Interval_ Volume 1,
Number 1 (Spring 1978), pp. 4-5, quoted in article by Brian McLaren,
"Secor well temperament," as posted by Gary Morrison, "Post from Brian
McLaren," 13 July 1994.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> -------------------------------
> 1. Pythagorean-based JI systems
> -------------------------------
>
> Pythagorean tunings featuring a chain of pure 3:2 fifths or 4:3
> fourths have the property of being at once just and regular.
>
> By combining two 12-note Pythagorean chains at some convenient
> distance, for example a septimal comma of 64:63 (~27.26 cents), we
can
> achieve a JI system with the usual Pythagorean intervals within each
> chain, plus intervals mixing notes from the two chains at pure
ratios
> of 2-3-7 prime (or 2-3-7-9 odd) such as 9:7, 7:6, 12:7, 7:4, and
8:7.
>
> We thus get a 24-note system combining elements of regularity and
> purity with free transposibility: in effect, we have a regular chain
> of 23 fifths, with the one minor compromise that a single fifth is
> stretched by about 3.80 cents, the 3-7 schisma, to obtain pure
ratios
> of 2-3-7-9.
>
> Of course, like a regular 24-note Pythagorean tuning with 23 pure
> fifths,

It is rather like it. But for the specific definition of "middle
path" I had in mind, one would want an MOS -- since for me the
prototypical JI scales are periodicity blocks, and ETs are of course
tempered periodicity blocks. One could sharpen the fifths by a
fraction of a cent, in effect stretching the Pythagorean comma close
to the septimal comma value, and a 17- or 29- tone chain would then
exemplify the "middle path" quite well as well as embodying the
harmonic relationships you've outlined. However, from the point of
view of a performer/improviser working with two 12-tone keyboards, I
can understand the modus operandi for your specific arrangement quite
well.
>
> While other distances between the two Pythagorean chains can be very
> attractive also, free transposibility seems an element of "middle
> path" approaches, so it is the 64:63 system that I mention.

And even more so when the septimal comma becomes equal to the
Pythagorean comma through (micro)tempering.
>
> George Secor has published a superb 17-note well-temperament[1],
while
> Paul Erlich and others have devised new 12-note systems.

Margo -- I also devised a 22-tone well-temperament in my paper, which
allows for 5-limit-oriented Indian scales in some keys, and my 7-
limit-oriented decatonic scales in others. You might want to look at
my paper again -- I suspect you'll like this.

>
> While eventone tunings may also have the property of circulation
> (e.g. 1/4-comma meantone in a 31-note cycle, or a 14:11 tuning in a
> 46-note cycle), an unequal temperament often achieves such
circulation
> in fewer notes by varying the size of fifths in the tuning chain.
>
> This technique, at the same time, causes variation in the sizes of
> thirds and other intervals as one moves around the tuning circle. In
> either a conventional 12-note scheme of the 17th-19th century type,
or
> a 17-note system such as Secor's, this gradation of intervals
provides
> an impressive musical resource.

Indeed.

By the way, Margo, Joe Monzo lists you as an advocate of 17-tone
equal temperament . . . is that your "favorite" ET?

🔗genewardsmith@juno.com

🔗mschulter <MSCHULTER@VALUE.NET>

Hello, there, Paul and everyone.

Thank you for your explanation of the "middle path" as meaning an MOS
system. You clarify some points, and also remind me of a very
important omission which I may now correct before addressing some of
your distinctions and questions, which maybe suggest a new category of
tuning (or a subcategory of some approach you've discussed?).

Of course I should have mentioned your 22-tone well-temperament for
decatonic music, indeed combining this tonal system (your main theme
in the article) with elements of the sruti system of India.

Why did I not recognize it as a "well-temperament," as well as a
"cross-cultural" kind of tuning system? What sticks with me especially
is the use of some fifths tempered just slightly less than in 17-tET,
a topic about which you ask and I'll reply a bit later in this
article.

Anyway, from now on, I'll certainly mention your 22-tET
well-temperament when I discuss this category, along with 17 and 19
and so forth.

Moving to the MOS concept, I might propose a concept like "dual-MOS
tuning" or maybe "multiple-MOS tuning" to include things like my
24-note Pythagorean schemes where each 12-note keyboard indeed defines
an MOS, but the overall tuning does not fit this category.

Your interestingly point to two attributes a "microtempered 2-3-7-9
JI" tuning would need to quality as an MOS: first, all fifths of equal
size (e.g. a temperament by 1/15 of a 3-7 schisma); and secondly, the
right number of notes, such as 17 or 29.

I'm noting that such a microtemperament -- or I'm tempted to say
nanotemperament, although that would be a different use of the term
than the more specific concept of adjusting for synthesizer tuning
units or the like -- in 24 notes would be strictly regular, but not an
MOS.

What I'm looking for is a term to communicate the idea of two MOS
tunings, with identical 12-note keyboard arrangements, for example,
which do not together form a larger MOS.

I'm tempted to speak of a "two-dimensional" tuning system, whether
based on Pythagorean (3-prime) JI or some temperament, since the
"rows" are MOS systems of a certain size, and the "columns" are pairs
of notes at a certain interval by which the rows are spaced.

Here maybe, as with consistency, we have the question: does an overall
MOS organization maybe become less important as the number of steps
and intervals increases?

In my view, a significant feature of a "two-dimensional" system with
each row a 12-note MOS, for example, is that lots of melodies can be
played and vertical progressions can take place within an MOS row.

For example, an 11-note Pythagorean tuning of the kind which may have
been common on keyboards around 1300 (Eb-C#) seems to invite expansion
to 12, especially once G# becomes a common accidental in progressions
from E-G# to D-A (major third to fifth in its most decisive form with
an ascending semitone).

However, does the distinction between 24 and 29, or between 36 and 41,
have a comparable significance? At this point we have what might be
described from a pragmatic viewpoint as "multiple versions of the same
note or interval category," whether the system is an MOS or otherwise.

The idea of a "multi-MOS" system may explain some of the appeal of
24-note systems with two 12-note keyboards (Pythagorean, meantone, or
neo-Gothic eventone, etc.)

Such system may represent a certain kind of compromise because it
permits one either to pursue regularity and symmetry within one of the
keyboards, or to seek new intervals, comma or diesis shifts, and so
forth, by mixing notes from the two MOS "rows." It is at once, one
might say, both regular and irregular, maybe with some of the appeal
of both alternatives.

You ask about my favorite equal temperament, and certainly I would say
that 17 expresses a lot of what I'm about for a very large portion of
my musicmaking. It was the first non-12 equal temperament that I
tuned, and I'm very happy to be associated with it.

Curiously, nowadays, I find myself also a passionate advocate for
13-tET as a neo-Gothic tuning: it seems very "modernistic" and
21st-century, with the implied process of Chowningization or the like
to obtain a 3:2-like effect for the 8/13-octave interval.

Here's a question: does being an "advocate" for an n-tET mean using
and favoring it specifically as a complete system, or simply using
some subset, and maybe being drawn to the division of the whole-tone
into equal fractional parts or similar features?

For example, I'm very enthusiastic about 29-tET, which I might guess
that I've used rather more than 17-tET -- but 12-out-of-29, or
24-out-of-29, rather than the full set.

Am I advocating "29-tET as a system," or simply "a regular temperament
around 703.45 cents"?

With 13, 17, 20, 22, and 24, I often do tune the full set, and do
encounter "system properties" such as the availability of any interval
from any note.

However, with 29, it seems more to me like one pleasant point on the
pleasant portion of the continuum roughly from 703.45 cents to
something like the e-based temperament at 704.61.

One very humorous side of this question is the kind of system I would
propose to carry 17 further, and get some new interval categories.
Lots of people advocate 34-tET, an excellent solution if one wants a
near-just 5-limit system, but own option is rather different.

Some months ago, it occurred to me that two 17-tET tunings at an
arbitrary distance of around 55.106 cents would yield some pure and
near-pure ratios of 2-3-7-9 (pure 7:6 and 12:7, with 9:7, 14:9, 7:4,
and 8:7 impure by the same amount as the wide fifth, ~3.93 cents).

Also, we supplement the fine neutral intervals of 17 with some
supraminor and submajor thirds and sixths, and add some interesting
near-Pythagorean intervals, including a minor third close to 19:16
(available for near-16:19:24 sonorities).

Maybe my problem is that it's hard to pick a single "favorite" n-tET
because each has its own attractions -- and this includes 22-tET,
where I find the 55-cent diatonic semitone quite satisfactory,
although not everyone agrees.

A curious thing is that with 17-tET or 29-tET, I can easily articulate
a defining quality of the "equalness" involved -- equal dieses or
fifthtones fifthtones in 29, and equal thirdtones in 17.

With 46, I'm more likely to say something like this: "This tuning is
almost identical to a 14:11 eventone, and has the same features:
regular thirds near 14:11 and 13:11, very nice supraminor/submajor
thirds, and some not-so-precise approximations of 2-3-7-9."

While I do clearly recognize 17 for its symmetry and "17-ness," this
might apply even more to 13, "the division of the octave into 13 equal
limmas," since 1/13 octave is very close to the Pythagorean limma at
256:243 (~90.22 cents).

With 17, there is for me a kind of meeting of styles, with those
beautiful 70.59-cent semitones and 423.53-cent major thirds for usual
14th-century cadential progressions -- and in the right timbres, those
thirds can be gentle and efficient at the same time -- and all those
neutral intervals to lend a "Romantic" or "Impressionist" air.

It's a bit like moving between a slightly alternative version of the
Gothic Era itself, say around the time of Guillaume de Machaut
(c. 1300-1377), and some curious conception of the "Gothic" in an
alternative 19th-century Romanticism.

Also, I should mention something that maybe seems a bit more familiar,
or at least a single keyboard within it -- 24-of-36, one of the best
optimizations I've seen for both 12:14:18:21 and 14:17:21.

Curiously, I haven't seen a lot of raving about 36-tET, or this
subset; maybe it's mainly a question of what one seeks to optimize.

Of course, while we're at this, I should mention a "near-ET" that I
use an immense amount: 1/4-comma meantone, which in 24 notes has many
diesis steps and progressions available. Here, while I know that this
isn't precisely 24-of-31-tET, I tend to count fifthtones as if it
were; carrying to a full 31, one could also circulate around the
system as if it were.

Anyway, for n-tET's which I tune as n-tET's (complete sets or
subsets), I would say that 17-tET and 12-or-24-of-29-tET are my
favorites, with 13-tET a favorite "unconventional" choice, and 20-tET
also in this category as slightly less unconventional.

Thank you for the clarifications, questions, and ideas, as usual.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
> Hello, there, Paul and everyone.
>
> Thank you for your explanation of the "middle path" as meaning an
MOS
> system.

It doesn't necessarily, Margo. If we're talking about, for example, a
planar temperament, the result won't be MOS.

> What I'm looking for is a term to communicate the idea of two MOS
> tunings, with identical 12-note keyboard arrangements, for example,
> which do not together form a larger MOS.

In many cases two MOSs interlace to form a larger MOS, but in this
case you don't. So whatever term you choose should reflect this.
>
> I'm tempted to speak of a "two-dimensional" tuning system, whether
> based on Pythagorean (3-prime) JI or some temperament, since the
> "rows" are MOS systems of a certain size, and the "columns" are
pairs
> of notes at a certain interval by which the rows are spaced.

Well, yes, it is a two-dimensional tuning system -- specifically, JI
with a 3-axis and a 7-axis . . . though the "chunk" of this plane
you're taking is not quite a periodicity block (well, it probably is,
for some odd choice of unison vectors).

> Here maybe, as with consistency, we have the question: does an
overall
> MOS organization maybe become less important as the number of steps
> and intervals increases?

Kraig Grady has argues, based largely on Erv Wilson's work, that such
features continue to be important. For me, they simply contribute to
a sense of "closure" -- a logical place to stop adding more notes.

> In my view, a significant feature of a "two-dimensional" system with
> each row a 12-note MOS, for example, is that lots of melodies can be
> played and vertical progressions can take place within an MOS row.
>
> For example, an 11-note Pythagorean tuning of the kind which may
have
> been common on keyboards around 1300 (Eb-C#) seems to invite
expansion
> to 12, especially once G# becomes a common accidental in
progressions
> from E-G# to D-A (major third to fifth in its most decisive form
with
> an ascending semitone).
>
> However, does the distinction between 24 and 29, or between 36 and
41,
> have a comparable significance?

If one is writing very sophisticated microtonal music along Wilsonian
lines, perhaps.
>
> Some months ago, it occurred to me that two 17-tET tunings at an
> arbitrary distance of around 55.106

Which is the same thing as two 17-tET tunings at a distance of 15.482
cents, right?

> Also, I should mention something that maybe seems a bit more
familiar,
> or at least a single keyboard within it -- 24-of-36, one of the best
> optimizations I've seen for both 12:14:18:21 and 14:17:21.
>
> Curiously, I haven't seen a lot of raving about 36-tET, or this
> subset; maybe it's mainly a question of what one seeks to optimize.

Yes -- I suspect most people are hung up on 36-tET being just as bad
as 12-tET when it comes to ratios of 5.

Thanks for your fascinating discussion, as always!

🔗mschulter <MSCHULTER@VALUE.NET>

Hello, there, Paul, and thank you for another very engaging response.

Maybe we could distinguish between "the middle path," meaning as you
explain actually a number of approaches based on MOS or periodicity
block structures (if I now follow correctly), and "middle paths" more
generally based on some kind of eventone tuning, for example.

The latter type of "middle paths" category doesn't necessarily imply
either an MOS for the overall system, although it's very likely to
have MOS subsets, or a periodicity block structure, which you've
commented wouldn't apply to at least some of my 24-note tunings.

Your response to my mention of 36-tET raises a point that could lead
into a tuning that occurred to me early this year for a style of music
that 36-tET doesn't really optimize.

Suppose that we want to find an optimal tuning for a style mixing
neo-Gothic _and_ Xeno-Renaissance elements in the same piece. This
means that we want not only sonorities like 12:14:18:21 and 14:17:21,
for which 36-tET has a really outstanding sum of squares, but also
fairly close approximations of 4:5:6 and 10:12:15.

As you observed, of course, the regular 12-tET intervals of 36-tET
give a not-so-close approximation of 5-limit. For a neo-Gothic style
alone, this might be described as a virtue, since these same intervals
can serve as not too far from Pythagorean or 3-limit.

However, let's suppose we really want a Renaissance-style 5-limit. An
excellent solution, it turns out, is a Renaissance-style meantone --
in fact, the first regular meantone tuning known to us with a precise
mathematical definition.

Yes, it's Gioseffo Zarlino's 2/7-comma meantone of 1558, carried to 24
notes.

The regular major and minor thirds, of course, are only 1/7 syntonic
comma from a pure 5:4 and 6:5 -- about 3.07 cents.

The diminished fourths are very close to 9:7 (~435.08 cents), at about
433.52 cents, or ~1.57 cents narrow; and the augmented seconds quite
close to 7:6 (~266.87 cents), at around 262.29 cents, or ~4.58 cents
narrow.

Thus Zarlino's 2/7-comma meantone yields a very nice approximation for
6:7:9 or 7:9:12, although the near-7:4 is much less accurate than in
1/4-comma meantone. Here 2/7-comma gives us an augmented sixth at
around 958.10 cents, or ~10.72 cents narrow -- a variation on what
might be the more curious side, possibly with a resemblance to the
20-tET interval of 960 cents.

For a supraminor third at around 17:14 (~336.13 cents), we use an
interval of a major third (~383.24 cents) less the enharmonic diesis
of ~50.28 cents, at about 332.97 cents (~3.16 cents narrow),
e.g. C*3-E3, with the ASCII * showing a note raised by a diesis. The
submajor third near 21:17 (~365.83 cents) is equal to a minor third
(~312.57 cents) plus a diesis, or ~362.84 cents (~2.98 cents narrow).

Maybe the most dramatic difference between 2/7-comma and 1/4-comma as
24-note or larger tunings is that in 1/4-comma, the chromatic semitone
of ~76.05 cents gets divided into two near-equal dieses of the kind
called for in Vicentino's enharmonic styles -- a large fifthtone at
around 41.06 cents, and a small fifthtone at around 34.99 cents.

In 2/7-comma, in contrast, we have a division of the chromatic
semitone at precisely 25:24 (~70.67 cents) into a diesis at around
50.28 cents, plus a kind of comma (the "19-comma," the difference
between 19 fifths up and 11 pure octaves) at around 20.40 cents.

Thus 1/4-comma, or 31-tET, is the appropriate choice for Vicentino's
enharmonic progressions based on a conceptually equal division of the
chromatic semitone into two diesis steps.

However, 2/7-comma is a fascinating system to carry to 24 notes for a
mixed kind of "neo-Gothic/Xeno-Renaissance" style, and, of course,
each 12-note manual provides a beautiful Renaissance meantone tuning.

Finally, Paul, you very correctly point out that my tuning with two
17-tET chains (or 12-of-17-tET subsets thereof) at ~55.106 cents apart
is equivalent to a distance of ~15.482 cents apart -- the sum of these
intervals being the step of 1/17 octave, or ~70.588 cents.

My 55.11-cent arrangement is "metachromatic," with 7:6 minor thirds
played as major-second-plus-diesis, for example (C3-D*3), with lots of
resolutions involving melodic motion from a key on one manual to the
corresponding key on the other manual:

A*3 A3 F4 F*4
G3 A3 D*4 C*4
D*3 D3 C4 C*4
C3 D3 or G*3 F*3

In the first example, a near-12:14:18:21 (with the 7:6 pure and other
intervals impure by ~3.93 cents) contracts to a stable fifth. In the
second, a near-14:18:21:24 (with the 12:7 pure) expands to a
near-2:3:4 trine

In this arrangement, the supraminor third has a "conventional"
keyboard arrangement as minor-third-plus-diesis (e.g. E3-G*3), and the
submajor third also as major-third-less-diesis (e.g. E*3-G#3).

In a version of this tuning with two 12-of-17-tET subsets, an
advantage of the metachromatic arrangement is that we get the 2-3-7-9
types of sonorities in more locations.

For example, we get a near-9:7 as fourth-less-diesis, in 11 locations;
with a spacing of 15.48 cents, we'd get it in 8 locations as a regular
major third plus the "comma" between the keyboards.

With the 55.11-cent spacing, we get a pure 7:6 as tone-plus-diesis in
10 locations; with the other spacing, we'd get it in 9 locations as
minor-third-less-comma.

However, the "metachromatic" diesis arrangement gives a near-7:4 as
major-sixth-plus-diesis in 9 locations, while we get 10 with the
15.48-cent spacing, where it is played as minor-seventh-less-comma.

With a full tuning of two complete 17-tET sets, the difference would
seem purely one of taste, I guess.

Most appreciatively,

Margo Schulter
mschulter@value.net