Gann's Reasons for Using Just Intonation

Dear tuning list people,

In light of recent controversies, I offer my own, idiosyncratic,
please-don't-try-this-at-home:

REASONS FOR USING JUST INTONATION

For anyone, no matter whom, to make the assumption that composers who
use just intonation do so from a desire to hear pure, beatless
intervals, and *only* for that reason, would be presumptuous and naive.
For instance, I use just intonation with synthesizers, and not very
sophisticated ones at that. I am under no illusion that I am going to
get beatless consonance. I usually can't even get a single beatless
tone. I've written just-intonation pieces on an Akai sampler with only
a 6-cent resolution and been happy as a clam. I think the only composer
who is really looking for perfectly beatless consonances is La Monte
Young, and he has an extreme synthesizer, extreme ears, and extreme
patience. I don't. Since I appear to be taken as one of the main
proponents of just intonation around here (and am complimented to be
considered as such), I thought for the record I'd list a few of the
many reasons I work in just intonation.

1. I have never succeeded in finishing a piece with more than 31
pitches in it. I don't see how Partch kept 43 pitches in his head. To
get the accuracy I need for 7th and 11th harmonics, I need to be able
to have pitches as close as 15 cents apart. To use an equal-tempered
scale that would give me pitches that close would require more than 60
pitches per octave, and I only have 128 pitches in my MIDI controls and
61 keys on my keyboard. Just intonation gives me criteria for choosing
only the pitches I need and leaving out all the rest.

2. I like the harmonic implications of fractions and ratios. I like
knowing, when I use a 21/16 interval, that the upper pitch is an
implied seventh harmonic of the dominant of the lower pitch. I like the
way a 7/6 minor third not only has a different flavor than a 6/5, but
also a different implied set of related pitches, so that it suggests
ways to harmonize it. I like the way the numbers keep every pitch
related to every other one in the system via a series of implied,
interconnected harmonic series'. The higher the numbers, the more
exotic the pitch seems. That's an interesting thing to work with
compositionally. No matter how dissonant and atonal you get, the
gravitational pull of 1/1 keeps you oriented toward a fixed point in
the universe.

3. I have never liked the concept of transposability. I don't like the
way (and noticed this even as a child), in some of Mozart's piano
sonatas, the theme sounds so perfectly placed as to register in the
exposition, and then when it's transposed a fifth in the recap, it
doesn't sound as good. I've always instinctively agreed with Dane
Rudhyar that to transpose a sonority is to diminish its absolute value
as a sonic phenomenon and reduce it to a set of relationships. Even
within the classical tradition, I tend to prefer composers who do not
transpose material (Satie) to those who transpose all over the place
(Schoenberg). I like the way Beethoven carefully limits his
transposition levels in the Hammerklavier Sonata. Therefore, the
universal transposability of sonorities in an equal-tempered scale
holds no charms for me. It is actually a deficit.

4. Relatedly, I like having different-sized intervals available on
different scale steps. It makes the scale have a "natural" feel to me,
like I'm carving a gnarly piece of wood instead of in smooth, mass-
produced plastic. The material gives me feedback: I run up against
things I can't do, keys I can't modulate to, and composing becomes a
dialogue between me and the scale. I enjoy that. Perhaps I would also
enjoy a nonequal, non-just-intonation scale, but I wouldn't know how to
start making one. And why would I try?

5. I do like the sound of my music better when I can minimize beats,
instead of having certain intervals (like thirds on the piano) pop out
randomly with buzzing overtones. I frequently use intervals like 40/27
to get beats, but I'd prefer the beats be deliberate. Nevertheless, the
fact that even my simple consonances are not exactly perfect has never
once bothered me. I live in the real world, where nothing is perfect.

6. I have a tremendous natural talent for fractions and logarithms. It
would be a shame to let it go to waste. I tell my students that if they
don't have a good head for fractions and logarithms they should leave
just intonation alone. It's not for everyone.

7. Not least, I am building on the work of four composers whose music I
deeply love, Harry Partch, Ben Johnston, Terry Riley, and La Monte
Young. I don't know of any other microtonalists working in any other
kinds of scales whose music I love nearly so much. I wouldn't start
working in 72tet or any other -tet unless I first heard some music in
that scale that blew me away, on an emotional as well as technical
level.

All these are not really separate reasons, but multiply interconnected.
Just intonation, for me, represents the ability to use every note with
an intense awareness of its harmonic interconnectedness with every
other note. The theoretical harmonic purity of numerical relationships
is the basis of that interconnectedness, but the ultimate sonic
manifestation does not have to be pure for the composing process to
have the intensity I love about it. As Ben always says, "Better to have
a perfect model and get an imperfect realization of it, than to have an
imperfect model to begin with." Or as Charles Ives asked, "What has
sound got to do with music?"

I have nothing against equal temperaments 19 and over, but I do
consider them inefficient for my purposes. I love the sound of
meantone, but it does pretty much limit you to triads. I prefer that
classical repertoire be played in well temperament. Anything but 12tet.
May a hundred thousand scales flourish.

Respectfully,

Kyle Gann

--- In tuning@y..., "orangedoor190" <orangedoor190@y...> wrote:

> 1. I have never succeeded in finishing a piece with more than 31
> pitches in it. I don't see how Partch kept 43 pitches in his head.

My advice would be to think of them on a lattice.

> I have nothing against equal temperaments 19 and over, but I do
> consider them inefficient for my purposes. I love the sound of
> meantone, but it does pretty much limit you to triads.

Meantone does 7 and 9 quite happily, and 31-et is an excellent 11-limit system.

> Dear Kyle!

I have to greatly agree on these two points. Especially 3. Outside of myself , I had not heard it expressed any other place, even though I had read some of Dane's works, so how missed it. (Granites, and Stars I quite love)
Working with a structure like the Eikosany or even the diamond, where a sonority occurs or can only occur at certain levels becomes a highly expressive element of the music as well as a "structural " element. I like the dialog with the tuning comment also.

In investigating and collecting tunings from around the world, although they might not be JI are almost universally non equal (even when they are described as "equal". But i have elsewhere expressed that this term equal mean different things in different cultures.). Probably like myself you are not interested in "not" doing something so the idea of doing non-anything remains a point lacking in inspiration.

>
> From: "orangedoor190" <orangedoor190@yahoo.com>
> Subject: Gann's Reasons for Using Just Intonation
>
>
>
> 3. I have never liked the concept of transposability. I don't like the
> way (and noticed this even as a child), in some of Mozart's piano
> sonatas, the theme sounds so perfectly placed as to register in the
> exposition, and then when it's transposed a fifth in the recap, it
> doesn't sound as good. I've always instinctively agreed with Dane
> Rudhyar that to transpose a sonority is to diminish its absolute value
> as a sonic phenomenon and reduce it to a set of relationships. Even
> within the classical tradition, I tend to prefer composers who do not
> transpose material (Satie) to those who transpose all over the place
> (Schoenberg). I like the way Beethoven carefully limits his
> transposition levels in the Hammerklavier Sonata. Therefore, the
> universal transposability of sonorities in an equal-tempered scale
> holds no charms for me. It is actually a deficit.
>
> 4. Relatedly, I like having different-sized intervals available on
> different scale steps. It makes the scale have a "natural" feel to me,
> like I'm carving a gnarly piece of wood instead of in smooth, mass-
> produced plastic. The material gives me feedback: I run up against
> things I can't do, keys I can't modulate to, and composing becomes a
> dialogue between me and the scale. I enjoy that. Perhaps I would also
> enjoy a nonequal, non-just-intonation scale, but I wouldn't know how to
> start making one. And why would I try?
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

>
> In investigating and collecting tunings from around the world, although they might not be JI are almost universally non equal (even when they are described as "equal". But i have elsewhere expressed that this term equal mean different things in different cultures.). Probably like myself you are not interested in "not" doing something so the idea of doing non-anything remains a point lacking in inspiration.
>

Hi Kraig,

Thanks for replying. You're right, I've heard some really interesting
non-equal scales in Arabic and Asian musics. I'm a little theoretically
inclined, so I can't quite imagine working with a tuning whose
underlying principle I didn't develop myself to express some idea that
intrigued me. But I get more intuitive as I get older, so maybe someday
I'll pick up one of those scales and figure out how to compose with it.
They sound great. I've got some Native American wooden flutes with some
nice scales that I've always wanted to use.

Henry Cowell reported in New Musical Resources that "Javanese music
divides the scale into five equal parts," for a rather bizarre scale of
240-cent steps. Decades later, of course, some ethnomusicologist
reported that Javenese musicians do indeed insist that their scale
steps are equal, but that they'll also admit, "but we don't really play
it that way."

Kyle

Hello, there, Kyle Gann and everyone.

Please let me thank you for your stimulating comments on some reasons
for using just intonation.

Interestingly some of your reasons, at times modified or reworded a
bit, might explain my penchant both for rational intonation (RI)
systems, and for "gentle temperaments" with the fifths gently wide of
pure.

A humorous touch was that when I first read your reasons, I sometimes
had the reaction, "This statement might reflect Kyle's style or
orientation toward tonal harmony rather than JI in general." Then,
however, reflecting a bit more, I'd realize that I could find
something similar in my own experience, as I discuss more below.

Before getting into my response, Kyle, please let me thank you
especially for octave.txt, an invaluable reference to a large number
of the ratios located within an octave. I often use this either to
seek out the size of a ratio in cents, or the ratio(s) which a given
tempered interval might most closely approximate -- or just to admire
the continuum.

A curious dilemma I've discussed in some recent posts is whether to
call my approach to rational tunings JI and/or RI. Maybe I should
explain a bit of my philosophy and typical neo-medieval style.

To me, a "just" tuning is one where the fifths and fourths, or at
least most of them, have pure ratios of 4:3 and 3:2. This means that
the complete usual stable concord of 2:3:4 is pure, and also such
relative concords as 6:8:9 or 4:6:9.

In contrast, one of the charms of RI is that imperfect concords such
as thirds and sixths may have a plethora of finely shaded ratios. For
regular thirds, in addition to the traditional 81:64 and 32:27, there
are also modern favorites such as 14:11 and 33:28, or 13:11 and 33:26.

In both RI and "gently tempered" systems, a "just" or "near-just"
sound often implies some pure or near-pure ratios of 7 for imperfect
concords: 12:14:18:21 or 14:18:21:24.

These modern sounds illustrate a point made by David Doty: it's
possible for a sonority to be quite "concordant" or blending in
itself, and yet to resolve very effectively to a simpler concord such
as the complete 2:3:4 or a fifth.

Most typically, both my RI and gently tempered systems involve two
12-note chains of fifths placed at some convenient distance to
generate a variety of melodic steps and intervals. The regularity of
each chain makes it easy to find fifths and fourths, the primary
stable concords, while combining notes from the two chains increases
the variety of the system, both vertical and melodic.

Another engaging RI approach is to tune a kind of 17-note
"quasi-circle" or "orbit" with most fifths pure, a few impure by a tad
less than 5 cents, and a few more dramatically impure by up to a full
64:63. Given the beauty and utility of a sonority such as Keenan
Pepper's 16:21:24:28, the 64:63 in such a scheme becomes not merely an
"inevitable flaw" but a valued adornment.

My purpose in writing this is to suggest how approaches to JI or RI
can vary, and also that, at least to me, RI systems and gentle
temperaments have many similar qualities, including both the regular
chains or pure or slightly wide fifths, and the diversity of
intervals.

Now for your reasons, some of which for me can apply to both types of
systems:

> 1. I have never succeeded in finishing a piece with more than 31
> pitches in it....

For me, 24 is a convenient number, and easy to map to two conventional
"regularized" keyboards, each having the same arrangement of steps and
intervals. In an RI scheme, each keyboard is in a standard Pythagorean
tuning (Eb-G#), while in a gentle temperament the fifths are typically
about 2 cents wide.

For me with 24 notes, as for you with up to 31, it's nice to be able
to choose pitches so as to make fine distinctions: for example steps
of 11:12:13:14, or nuances of color like 14:11 and 23:18, or 17:14 and
11:9.

Here maybe I'm experiencing a bit of a "12th-century Renaissance" a la
Gothic Europe with the influence of ancient Greek and medieval Near
Eastern systems favoring lots of different superparticular steps.

Either an RI system or a "near-superparticular" type of gentle
temperament can realize or approximate this ideal, and I find the fine
distinctions very engaging.

> 2. I like the harmonic implications of fractions and ratios. I
> like knowing, when I use a 21/16 interval, that the upper pitch is
> an implied seventh harmonic of the dominant of the lower pitch....

At first blush, my reaction was, "Isn't this more a reflection of
Kyle's chosen tonal style than of JI or RI generally?"

However, as someone oriented to modal or focal harmony (the latter
based on the tendency of the major third to expand to a fifth and the
major sixth to an octave, for example), I can come up with my own
examples.

For example, how about the implications of an 81/56 above a given
vertical center or "local 1/1," the 7-based tritone which forms a 9:7
with the 9/8 inviting very efficient expansion to a stable fifth. For
example, in reference to F as "local 1/1," and using four-voice
harmony in the modern fashion (14:18:21:24 to 2:3:4):

E5 27/14 F5 2/1
D5 27/16 D5 3/2
B4 81/56 C5 3/2
G4 9/8 F3 1/1

There's also the beauty of those 28/27 semitones, and the 9/8
whole-tone steps -- sometimes it's nice to be superparticular.

Also, this is more or less what is approximated in gentle temperaments
in cadences involving "near-7" sonorities.

For me, RI leaves open the musical question of "What's the center" or
the "local 1/1," as I might call it, at a given moment -- sometimes an
artfully ambiguous question in Machaut also, as Sarah Fuller has
eloquently shown. However, either in a basic Pythagorean intonation or
a 7-based version, for example, following the dance of ratios can add
a measure of enjoyment to improvising, listening, and composing.

> 3. I have never liked the concept of transposability. I don't like
> the way (and noticed this even as a child), in some of Mozart's
> piano sonatas, the theme sounds so perfectly placed as to register
> in the exposition, and then when it's transposed a fifth in the
> recap, it doesn't sound as good. I've always instinctively agreed
> with Dane Rudhyar that to transpose a sonority is to diminish its
> absolute value as a sonic phenomenon and reduce it to a set of
> relationships.

Here I might say that for me, _some_ transposibility can be quite
desirable, but typically a basic 12-note gamut of Eb-G# (a typical
14th-century European keyboard tuning) provides enough to get the
regular cadences on the usual steps. Two such 12-note chains combine
to provide intervallic variety, and often many of the benefits of
transposibility such as a great range of available cadences.

For example, with two Eb-G# chains, I might not have a regular Db
available for a cadence like Db4-F4-Bb4 to C4-G4-C5 with descending
semitones, but often something like C*4-F4-Bb4 to C4-G4-C5 is very
nice equivalent with 7-based or near-7-based ratios in a system with
chains something like a 28:27 apart.

Having the tuning system call specifically for 7-based cadences at
these locations is a touch of "modal color" that I like.

> 4. Relatedly, I like having different-sized intervals available on
> different scale steps. It makes the scale have a "natural" feel to
> me, like I'm carving a gnarly piece of wood instead of in smooth,
> mass- produced plastic. The material gives me feedback: I run up
> against things I can't do, keys I can't modulate to, and composing
> becomes a dialogue between me and the scale. I enjoy that. Perhaps
> I would also enjoy a nonequal, non-just-intonation scale, but I
> wouldn't know how to start making one. And why would I try?

This is something my last example also touches upon, and which your
present point suggested to me: the role of creative asymmetry in RI
systems especially, and also to a degree in "near-superparticular"
temperaments and the like.

To take a stronger example: I have a 12-note RI tuning where F-B are
usual Pythagorean, and C# and G# are a bit different but musically
equivalent -- but there's no F#. This means that one can't make a
conventional cadence on G like A4-C#5-F#5 to G4-D5-G5 -- opening for
me a beautiful alternative. I could use F4-G#5-C4 to G4-D5, with the
middle voice descending (G#-G) by a pure 14:13 or "2/3-tone" of about
128.30 cents. Here F4-G#4-C5 has thirds of 63:52 and 26:21, about
332.21 cents and 369.75 cents (values courtesy of your octave.txt).

Here necessity was the mother of invention, and as a result I found
myself looking for this kind of progression in other tuning systems
where a usual cadence on G was also available. Of course, given my
taste for "supraminor/submajor" or "semi-neutral" thirds, I might have
come up with this progression anyway -- and in fact I had used it now
and then before. However, I usually had preferred a version with
ascending 2/3-tones.

That special 12-note tuning had put the form with descending 14:13
steps "on the map," now a landmark I looked for (sometimes with a
range of semi-neutral or neutral steps) in other systems also.

For me, the special charms of RI and near-superparticular systems
don't exclude the beauties of a system like 29-tET, or the 12-note and
24-note subsets I tune.

Using different kinds of intonational systems can be a refreshing
change of pace -- not that I would want to imply than an exquisite
tuning like 29-tET needs justification as merely a "change of pace."
The "Pythagorean-like but just a bit different" appeal, the delights
of the regular and supraminor/submajor thirds, and those striking
intervals in a 24-note tuning near 15:13, 13:10, and 26:15, etc., are
something that gives this world of 29-equal its own integrity and
special place.

At the same time, I'd like to say also that the element of creative
asymmetry comes to the fore in unequal well-temperaments also, for
example George Secor's superb 17-note well-temperament. While 17-equal
is a fine tuning in itself, look what variety George has achieved by
some very artfully unequal temperament.

Here I've replied to only some of your points, but hope that the
sharing of perspectives that you've invited will enrich this list, as
your most eloquent post has already done.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> Interestingly some of your reasons, at times modified or reworded a
> bit, might explain my penchant both for rational intonation (RI)
> systems, and for "gentle temperaments" with the fifths gently wide of
> pure.

I tend to think of those as the bold temperaments, and of meantone as gentle. Anyone else care to vote on this?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> > ...might explain my penchant both for rational intonation (RI)
> > systems, and for "gentle temperaments" with the fifths gently
> > wide of pure.
>
> I tend to think of those as the bold temperaments, and of meantone as gentle. Anyone else care to vote on this?

Sure, count me in: I say let the good woman call it whatever she likes. If "gentle" works for Margo, I say adopt it as a world-wide standard!

Cheers,
Jon

Hi Margo,

> Before getting into my response, Kyle, please let me thank you
> especially for octave.txt, an invaluable reference to a large number
> of the ratios located within an octave. I often use this either to
> seek out the size of a ratio in cents, or the ratio(s) which a given
> tempered interval might most closely approximate -- or just to admire
> the continuum.

Thanks, glad that's a help for you - sometimes I look at it just to
admire the continuum too. (Geez, what does that say about us?) You
probably know, or have, Alain Danielou's Table of Comparative Musical
Intervals (title from memory, accuracy not guaranteed)? La Monte Young
showed me that book and I found an old, beat-up copy of it through
bibliofind.com, a list of thousands of intervals within an octave.
That's what inspired my chart, and I'd consider expanding it or maybe
using Danielou's entire list, but I use the list for teaching a lot,
and thousands of intervals becomes too much information to be useful to
students. Or useful to me, for that matter.

> In contrast, one of the charms of RI is that imperfect concords such
> as thirds and sixths may have a plethora of finely shaded ratios. For
> regular thirds, in addition to the traditional 81:64 and 32:27, there
> are also modern favorites such as 14:11 and 33:28, or 13:11 and 33:26.

This is exactly what several of my pieces use JI for. I've never used
33:28, but you've got me curious about it. For a couple of years I've
been trying to base a piece around 16:13 and 39:32 thirds, but
something about the 13-limit defeats me. I find that I can't compose in
intervals that my ear hasn't internalized.

> Given the beauty and utility of a sonority such as Keenan
> Pepper's 16:21:24:28, the 64:63 in such a scheme becomes not merely an
> "inevitable flaw" but a valued adornment.

16:21:24:28 is a sonority (as you may well know) basic to many of the
Theatre of Eternal Music's improvisations in the '60s. La Monte's way
of using a 9:8 as an ornament to 7:4 made that 63:32 and the 64:63
between that and the octave really palpable - and singable.

> For example, how about the implications of an 81/56 above a given
> vertical center or "local 1/1," the 7-based tritone which forms a 9:7
> with the 9/8 inviting very efficient expansion to a stable fifth. For
> example, in reference to F as "local 1/1," and using four-voice
> harmony in the modern fashion (14:18:21:24 to 2:3:4):
>
> E5 27/14 F5 2/1
> D5 27/16 D5 3/2
> B4 81/56 C5 3/2
> G4 9/8 F3 1/1
>
> There's also the beauty of those 28/27 semitones, and the 9/8
> whole-tone steps -- sometimes it's nice to be superparticular.
>
>
> To take a stronger example: I have a 12-note RI tuning where F-B are
> usual Pythagorean, and C# and G# are a bit different but musically
> equivalent -- but there's no F#. This means that one can't make a
> conventional cadence on G like A4-C#5-F#5 to G4-D5-G5 -- opening for
> me a beautiful alternative. I could use F4-G#5-C4 to G4-D5, with the
> middle voice descending (G#-G) by a pure 14:13 or "2/3-tone" of about
> 128.30 cents. Here F4-G#4-C5 has thirds of 63:52 and 26:21, about
> 332.21 cents and 369.75 cents (values courtesy of your octave.txt).

Nice progression, the first one. (You did mean C5 3/2, I assume?) Do
you have any music using these tunings? Are they meant for the
realization of ancient music, or for new compositions? I'm always
looking for music that exploits the particular advantages of an unusual
tuning - not only because I teach it, but because I love listening to
it. Let me know if you have some music samples I can try out.

Yours,

Kyle

Hello, Kyle, and it's really fun to compare musical notes in this way,
with an opportunity also for me to confess my ignorance of some
important and intriguing history to which you've called my attention,
a whole world of happenings I didn't know was out there.

[on your octave.txt]

> Thanks, glad that's a help for you - sometimes I look at it just to
> admire the continuum too. (Geez, what does that say about us?) You
> probably know, or have, Alain Danielou's Table of Comparative
> Musical Intervals (title from memory, accuracy not guaranteed)? La
> Monte Young showed me that book and I found an old, beat-up copy of
> it through bibliofind.com, a list of thousands of intervals within
> an octave. That's what inspired my chart, and I'd consider
> expanding it or maybe using Danielou's entire list, but I use the
> list for teaching a lot, and thousands of intervals becomes too much
> information to be useful to students. Or useful to me, for that
> matter.

Actually, I learned about Danielou's book through your octave.txt, and
it sound like a very worthwhile thing to find.

Maybe another side of "admiring the continuum" is having some new (to
me) ratio, especially a large or exotic superparticular one, pop up
as the difference between two familiar ratios. Yesterday, for example,
I came across 442:441 (~3.921 cents), the difference between 17:14
(~336.13 cents) and 63:52 (~332.21 cents).

[In response to a comment I made on the variety of finely shaded
ratios:]

> This is exactly what several of my pieces use JI for. I've never
> used 33:28, but you've got me curious about it. For a couple of
> years I've been trying to base a piece around 16:13 and 39:32
> thirds, but something about the 13-limit defeats me. I find that I
> can't compose in intervals that my ear hasn't internalized.

Here are a few ideas which just might get you moving in some
interesting direction.

First, I find 13:8 a sweet sixth which, like others, I've found at
least sometimes seems to lean curiously a bit in a "major" direction,
and 13:7 a neat kind of "submajor seventh." These intervals could be
one place to start, especially if you tend to branch out from simpler
ratios.

Two sonorities you might like are 13:16:22 and 13:16:22:28 -- tastes
can vary, and experimenting a bit could prove catalytic in finding
your own musical approach to these or related ratios. I could make
MIDI files in Scala for a couple of progressions.

By the way, at least for me, having an actual 33:28 (the minor third
which together with 14:11 forms a 3:2 fifth) is one of things that
tends to distinguish between a just tuning an a "near approximation."
In a temperament, I find that often 14:11 is at or very close to pure,
with the fifth about two cents wide, and a minor third somewhere
between 33:28 (~284.45 cents) and the larger 13:11 (~289.21 cents).
To get both 14:11 and 33:28 pure, the fifth has to be pure as well.

>> Given the beauty and utility of a sonority such as Keenan
>> Pepper's 16:21:24:28, the 64:63 in such a scheme becomes not merely
>> an "inevitable flaw" but a valued adornment.

> 16:21:24:28 is a sonority (as you may well know) basic to many of the
> Theatre of Eternal Music's improvisations in the '60s. La Monte's
> way of using a 9:8 as an ornament to 7:4 made that 63:32 and the
> 64:63 between that and the octave really palpable - and singable.

Please let me thank you for at once informing my ignorance, and
calling my attention to a whole world of music and history that I'd
like to learn more about, now that you've opened the door. The idea of
people using the 16:21:24:28 in the 1960's is something that really
draws me into learning more about LaMonte Young.

I've read a bit about his use of ratios of 3 and 7, and also of some
very complex JI sonorities with amazingly precise tunings, so that
what you say fits very nicely into the picture -- which I realize I've
been admiring from a distance when I should take a closer look, and
listen.

As it happened, the place where I learned about the 16:21:24:28 was
Keenan Pepper's famous post about "crunchy chords" which has now
become something of a legend, so I called this a "Pepperian"
sonority. He described it as a "3-7 square."

/tuning/topicId_11922.html#11922

Learning now that LaMonte Young and his colleagues were using this
sonority in the 1960's not only enlightens me but makes Pepper's
famous words about "crunchiness" part of a richer history. Thank you
for opening this door of knowledge, which also might help to explain
one comment I recently saw that Young can "bring out the justness of a
64:63."

> Nice progression, the first one. (You did mean C5 3/2, I assume?)
> Do you have any music using these tunings? Are they meant for the
> realization of ancient music, or for new compositions? I'm always
> looking for music that exploits the particular advantages of an
> unusual tuning - not only because I teach it, but because I love
> listening to it. Let me know if you have some music samples I can
> try out.

First, why don't I give the progression with the correct octave
numbers, and with F4-C5 indeed a 3:2 fifth:

C5 -- +9:8 -- D5
G#4 -- -14:13 -- G4
F4 -- +9:8 -- G4

Generally my tunings are meant mainly for new music in "neo-medieval"
styles, typically based mainly on Gothic Europe in the 13th-14th
centuries, but with some novel xenharmonic elements, and also nowadays
some influence from the medieval Near Eastern traditions.

Of course, with a 24-note Pythagorean-based system, each keyboard is a
standard 12-note tuning such as one might have found on a 14th-century
instrument, and the mixture of new and old is one of the attractions
of this approach.

There are delicate questions about "historically informed"
performances of medieval music, especially when some of the
"information" could suggest flexible intonation practices that might
or might not be approximated by one's favorite "neo-medieval" JI/RI
system.

Maybe I should get into this more in another post, where I'll also
include some links to MIDI files.

In the meantime, thanks for educating me about LaMonte Young, and
sharing your experience and enthusiasm in such a friendly way.

Most appreciatively,

Margo Schulter
mschulter@value.net

----- Original Message -----
From: "M. Schulter" <MSCHULTER@VALUE.NET>

> In the meantime, thanks for educating me about LaMonte Young, and
> sharing your experience and enthusiasm in such a friendly way.

I highly recomend Kyle's article "The Outer Edge of
Consonance Snapshots from the Evolution of La Monte
Young's Tuning Installations" in the book "Sound and Light:
La Monte Young Marian Zazeela". It's a detailed
look at La Monte's tunings.

* David Beardsley
* http://biink.com
* http://mp3.com/davidbeardsley

Hello, there, Gene and Jon and everyone.

In some recent posts, I have used the term "gentle temperaments"
without giving a full definition and more of an explanation about the
stylistic context. This article, hopefully, may make both the
definition and the implicit stylistic setting and assumptions more
clear.

Here I would like especially to thank Gene for a most helpful query
suggesting that a bit of cultural and stylistic context could indeed
make "gentle temperaments" more understandable; and also Jon for some
general encouragement also playing a catalytic role in the writing of
this article.

First of all, I would like to emphasize the point that "terms of art"
such as "gentle temperaments" are descriptions in relation to specific
styles. What sounds "gentle" in one cultural or stylistic setting
could sound quite otherwise in a different style where different
expectations apply.

For me, the term "gentle temperaments" refers specifically to the
world of what I call "common music" (or _musica communis_), a kind of
offshoot of the 13th-14th century styles of Gothic Europe influenced
also by medieval Near Eastern musics and tuning systems.

In a different setting, for example a European Renaissance one calling
for meantone as a norm, these same temperaments would likely seem
anything but "gentle" -- they would represent, at least in many usual
harmonic timbres, the right tuning system in the wrong place.

The main characteristics of "gentle temperaments," with fifths
tempered in the wide direction by about 1.49-2.65 cents, are about as
follows:

(1) Regular major and minor thirds are at or in the general
vicinity of 14:11 (~417.51 cents) and 13:11 (~289.21
cents).

(2) Augmanted seconds and diminished fourths have sizes not
too far from 17:14 (~336.13 cents) and 21:17 (~365.83
cents).

(3) Additionally, 24-note systems based on two 12-note chains
of fifths or (at the upper end of the range) a single
24-note chain often feature "7-based" intervals at or near
such ratios as 7:6 (~266.87 cents), 9:7 (~435.08 cents),
and 7:4 (~968.83 cents).

The favored ratios of 14:11, 13:11, 17:14, and 21:17 -- poetically
known as the "Four Convivial Ratios" -- define the general region of
gentle temperaments.

Here it is interesting that while the term "gentle temperaments"
refers to the relatively mild compromising of fifths and fourths, it
also implies a _minimum_ amount of tempering so as reasonably to
approximate the Four Convivial Ratios -- about as in 29-EDO, with
fifths around 703.45 cents (~1.49 cents wide).

The upper end of the region at around 704.61 cents (~2.65 cents wide)
represents a point at which a single 24-note chain of fifths can
provide a reasonably accurate approximation of 7-based ratios. The
near-7:4 is virtually pure, while the near-7:6 is about 2.37 cents
narrow, and the near-9:7 more imprecisely about 5.03 cents wide.

Another term for the region of gentle temperaments is the "704-cent
neighborhood," since this fifth size marks about the middle of the
region, although the low end is actually slightly closer to 703 cents,
and the high end to 705 cents.

In the lower and middle portions of the region, 7-based intervals are
typically achieved by a 24-note system featuring two 12-note chains
placed at a distance equal to the difference between a regular major
second and a pure 7:6 minor third. This arrangement produces pure
ratios of 7:6 and 12:7, with 9:7 and 7:4 (and likewise 14:9 and 8:7)
impure by the same amount as the fifth.

A basic stylistic assumption of these gentle temperaments is that
thirds and sixths are active intervals often resolving to stable
sonorities involving primary stable concords such as fifths and
fourths. Progressions featuring resolutions such as minor third to
unison, major third to fifth, and major sixth to octave often define
an efficiently directed verticality, as in the style of 14th-century
Europe.

Also, timbres are chosen so as generally to fit a musical role for
thirds and sixths as _relatively_ concordant and at the same time
active and unstable. This aspect of "gentleness" can call for a bit of
discretion.

What I would like to commnicate is that the world of _musica communis_
has its own expectations and patterns, and that what is "gentle"
within this world could be quite otherwise in another musical setting
such as that of 16th-century or 18th-century European counterpoint,
where fifths are customarily tempered in the _narrow_ direction. In
each case, intonational form follows, or at least accords with,
musical function.

A more technical discussion might approach gentle temperaments in
terms of certain characteristic "unison vectors" such as 896:891,
352:351, 364:363, and 10648:10647.

In more nontechnical terms, we might say that in these temperaments
regular thirds represent ratios of 14:11 and 33:28 (~284.45 cents),
and likewise 13:11 and 33:26 (~412.75 cents). Further, the same
regular major third represents both 14:11 and 33:26, while the regular
minor third represents both 13:11 and 33:28 -- ratios distinguished in
just or rational systems.

From a practical standpoint, these equivalences distinguish a gentle
temperament of _musica communis_ from a system such as 72-EDO, where
there are comparably accurate representations of 3:2 and 14:11, for
example, but where a regular major third at 400 cents is _not_
identical to the best approximation of 14:11 at 416-2/3 cents.

Also, gentle temperaments contrast with just or rational systems in
_musica communis_ providing ratios of 14:11, 33:28, 13:11, and 33:26,
where some fifths are pure, but others impure by 352:351 (~4.93 cents)
or 364:363 (~4.76 cents). The following diatonic scale illustrates
such a system:

---------------- Scala file begins on next line of text --------------

! msdiat7.scl
!
Diatonic scale, symmetrical tetrachords based on 14:11 and 13:11 thirds
7
!
44/39
14/11
4/3
3/2
22/13
21/11
2/1

----------------- Scala file ended on blank line after "2/1" ---------

This scale, by the way, includes some intervals differing by the small
superparticular ratio of 10648:10647, e.g. 13:11 (~289.210 cents) and
968:819 (~289.372 cents), or about 0.163 cents. This ratio is equal to
the difference between 352:351 and 364:363.

In such a rational system, some fifths are pure, while others are
impure by almost 5 cents; in a gentle temperament, all regular fifths
are equally and mildly impure by about 1.49-2.65 cents.

Most appreciatively,

Margo Schulter
mschulter@value.net