Re: 17-tET "set-consistency" (for Joseph Pehrson)

Hello, there, Joseph Pehrson and everyone.

Thank you for raising a very important point both about 17-tone equal
temperament (17-tET) and about Paul Erlich's famous charts of equal
temperaments in his 22-tET article, which invites some further
discussions. What follows is in addition to Paul's remarks in recent
threads on the intentions of those charts, warmly inviting any
corrections or clarifications from Paul.

The critical point I would immediately make about Paul's charts is
that they are based on a Partch-like "n-odd-limit" concept which
assumes that each limit includes the intervals of any lower limits.
If any of these lower limits is found to be inconsistent for a given
n-tET, then the charts don't look at higher-limit ratios for which
what I term "set-consistency" (and Carl Lumma may have termed
something else when he delved into this concept) may indeed obtain.

To translate this into what may be plainer language: Paul assumes, for
example, that a kind of system on which he focuses, if it seeks ratios
of 7, 9, or 11 odd, will also seek ratios of 5 odd. The tetradic
system he describes, neatly realized in 22-tET, is an example: it
calls for a stable sonority of 4:5:6:7.

Thus from Paul's viewpoint (in this article), if an n-tET is
inconsistent for ratios of 5, this is a fundamental flaw from the
viewpoint of building his tetrads -- so rather than proceeding to
survey all of its other resources, he simply shows the highest n-limit
at which it is consistent.

If I wanted to pick an n-tET for which this procedure does _not_
suggest the total range of resources under other assumptions, an ideal
example might be 17-tET.

Very simply, for a 7-limit tetradic (4:5:6;7) music, the most relevant
fact is that 17-tET is neither consistent nor accurate for the
5-limit; in fact, 17-tET may be an ideal temperament for avoiding
rather than approximating ratios of 5. From the viewpoint of Paul's
article, and his charts, that about concludes the inquiry as far as an
ideal 7-limit tetradic tuning goes; the charts do show that 17-tET is
both consistent and rather highly accurate at the 3-limit (fifths at
~705.88 cents, ~3.93 cents wide).

Now let's consider a radically different sort of problem: how about
17-tET as a neo-Gothic tuning, where 3-limit ratios form the basis of
stable sonorities, but unstable intervals such as thirds and sixths
can take on all kinds of sizes -- e.g. "thirds" just about anywhere in
the range from around 15:13 (~248 cents) to 13:10 (~454 cents)?

In contrast to Paul's 4:5:6:7 tetrad, our stable trine at 2:3:4 has an
excellent representation in 17-tET of around 0-706-1200 cents; less
accurate than some other favorite neo-Gothic tunings, but more
accurate than a typical Renaissance meantone, for example. From Paul's
charts alone, which show the accurate 3-limit realization for this
tuning, we might find this happy situation an expected one.

From a neo-Gothic point of view, once we know that a tuning has a
reasonable approximation of 2:3:4 -- or sometimes a "not-so-pure" one,
for example 20-tET at 0-720-1200 cents (fifth ~18.04 cents wide) -- we
can look into the various other resources from a quite different
perspective than would be involved in a music based on 4:5:6 triads or
4:5:6:7 tetrads, for example.

Rather than following an "n-limit" model which may not so closely
apply, I'll consider intervals in certain musical groupings which are
intuitive to me (not necessarily to others on this List <grin>).

First, let's consider what I'd call the "basic Pythagorean 9-based
intervals," 9:8 and 16:9, equal respectively to the difference between
a fifth and a fourth, and the sum of two fourths.

Here the 17-tET counterparts are ~7.85 cents wide of 9:8 (3/17 octave)
and narrow of 16:9 (14/17 octave) -- in each case, a variance equal to
twice the temperament of the fifth or fourth, since these intervals
are formed from chains of two fifths or fourths.

---------------------------------------------------
Ratio Cents 17-tET Cents Variance
---------------------------------------------------
9:8 ~203.91 3/17 octave ~211.76 +~7.85
16:9 ~996.09 14/17 octave ~988.24 -~7.85
---------------------------------------------------

Thus we have quite nice although by no means "pure" representations of
such important Gothic/neo-Gothic sonorities as 6:8:9 or 4:6:9,
considerably more accurate than in 22-tET let alone 20-tET -- where,
interestingly, in appropriate timbres these sonorities can remain
recognizable and pleasing.

How about ratios of 2-3-7? On Paul's charts, based on a "7-limit"
concept or ratios of 2-3-5-7, the inconsistency of 17-tET for ratios
of 5 makes an inquiry into 2-3-7 somewhat beside the point; but for
neo-Gothic, 2-3-7 (without any need for 5) represents _one_ possible
intonational ideal.

As it happens, although I haven't yet confirmed this in a rigorous
manner, 17-tET appears consistent for ratios of 2-3-7, but not
especially accurate (albeit somewhat more so than 12-tET). For
neo-Gothic purposes, the ratios of interest are often, in order of
interval size, 8:7, 7:6, 9:7, 12:7, and 7:4 -- with 14:9 also pleasant
as a minor sixth.

Accuracy (or rather its inverse, variance) ranges from around 20 cents
for 8:7 and 7:4 to around 16 cents for 7:6 and 12:7, and around 12
cents for 9:7 and 14:9.

---------------------------------------------------
Ratio Cents 17-tET Cents Variance
---------------------------------------------------
8:7 ~231.17 3/17 octave ~211.76 -~19.41
7:4 ~968.83 14/17 octave ~988.24 +~19.41
7:6 ~266.87 4/17 octave ~282.35 +~15.48
12:7 ~933.13 13/17 octave ~917.65 -~15.48
9:7 ~435.08 6/17 octave ~423.53 -~11.55
14:9 ~764.92 11/17 octave ~776.47 +~11.55
---------------------------------------------------

The longer the chain of fifths gets, the greater the accuracy -- in
fact, 17-tET might be described as very close to a 1/7-septimal-comma
temperament. Seven fifths up, curiously, gives us a chromatic semitone
of 2/17 octave or ~141.18 cents -- very close to a "septimal chromatic
semitone" of 243:224, the difference between a 9:8 whole-tone and a
28:27 semitone (~140.95 cents).

As far as I've been able to tell by my crude inductive methods, these
intervals "add up" without any special surprises: we can use the best
approximation for each interval of something like 12:14:18:21 or
14:18:21:24 (the two favorite neo-Gothic quads, or unstable four-voice
sonorities). It may not be especially accurate, but it seems
consistent -- here, unlike on Paul's charts, factors of 5 aren't
involved.

How about ratios of 2-3-7-9-11 -- again without the complication of
Paul's ratios of 5 (a basic musical ingredient for his purposes in the
article)?

As it turns out, 17-tET here seems to me both consistent and often
rather accurate.

Here are some ratios of this 11-based variety, which can have a rather
"mysterious" quality in neo-Gothic, typically resolving by way of
progressions involving the melodic interval of 2/17 octave, the
chromatic semitone:

---------------------------------------------------
Ratio Cents 17-tET Cents Variance
---------------------------------------------------
11:9 ~347.41 5/17 octave ~352.94 +~5.53
18:11 ~852.59 12/17 octave ~847.06 -~5.53
11:8 ~551.32 8/17 octave ~564.71 +~13.39
16:11 ~648.68 9/17 octave ~635.29 -~13.39
12:11 ~150.63 2/17 octave ~141.18 -~9.46
11:6 ~1049.37 15/17 octave ~1058.82 +~9.46
---------------------------------------------------

In addition to these "special effects" intervals, we might also
consider 14:11 and 11:7, routinely used (or their close
approximations) as regular major thirds and minor sixths in various
neo-Gothic tunings.

---------------------------------------------------
Ratio Cents 17-tET Cents Variance
---------------------------------------------------
14:11 ~417.51 6/17 octave ~423.53 +~6.02
11:7 ~782.49 11/17 octave ~776.47 -~6.02
---------------------------------------------------

In conclusion, if we take Paul's criterion of what I might call
minimal "ballpark" accuracy -- a variance of around 20 cents[1] --
then 17-tET seems to permit consistent expression for a large and
possibly complete set of ratios of 2-3-7-9-11.

From a neo-Gothic viewpoint, however, if the question is what 17-tET
really does best, I might prefer a more "flavor-oriented" approach:
not necessarily looking at simpler ratios alone, but also at more
complex rational ratios, and at the Noble Mediant of Complexity which
Dave Keenan has introduced to discussions of harmonic entropy.

In neo-Gothic parlance, 17-tET presents a model "23-flavor" tuning,
because it has a major third very close to 23:18 (~424.36 cents), and
a minor third not too far from 27:23 (~277.59 cents), with
corresponding sizes for major and minor sixths.

If we apply Dave Keenan's Noble Mediant, we find that the major third
is also very close to the theoretically estimated point of maximum
complexity or "entropy" between 5:4 and 9:7, ~422.48 cents. Paul
Erlich's entropy model likewise suggests a maximum around 423 cents.

More generally, as Dave and I have agreed, 17-tET might be taken as a
kind of overall optimization of complexity for major and minor thirds
and major sixths: these intervals tend toward maximal complexity, and
in many harmonic timbres toward maximal tension.

From a neo-Gothic perspective, this fine tuning may call for a bit of
discretion with timbres: we want thirds to be active and unstable, but
at the same time to have a _degree_ of blend or "partial concord." To
borrow a typical 13th-14th century description, these intervals are
"imperfect concords," and a bit of diplomacy with fifth partials and
the like can bring out this desired effect.

Of course, from other perspectives, bringing out the extra tension in
those thirds and sixths could facilitate various kinds of effects like
thirds resolving to seconds (a kind of reversal of the Renaissance
suspension), as described by theorists such as Ivor Darreg or John
Chalmers. Neo-Gothic is only one musical approach.

Curiously, from my neo-Gothic standpoint, 17-tET seems a bit analogous
to 12-tET for someone involved in tertian music of the tonal variety
of which Paul writes. Its symmetry gives it a special status and place
on the spectrum; it appears to be consistent for lots of ratios,
although not necessarily especially accurate in comparison with other
tunings; and it serves as a kind of starting point for unequal 17-note
well-temperaments of the kind which I described here some months ago.

Anyway, thanks to both you and Paul for an opportunity to touch on
some interesting matters -- and emphasize the point that one article,
by Paul or anyone else, can only look at tunings from a small subset
of the possible viewpoints. Similarly, my description of a tuning from
a neo-Gothic perspective might omit all kinds of things that might be
both obvious and essential for someone involved in a different kind of
music.

Incidentally, I'd welcome either confirmation or artful contradiction
of my tentative impression that 17-tET is "set-consistent" for ratios
of 2-3-7-9-11.

----
Note
----

1. 1. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament,"
_Xenharmonikon_ 17:12-40 (Spring 1998), at 27: "Table 1 shows that all
ratios involving 9, 11, 15, and 17 -- in addition to 1, 3, 5, and 7 --
can be expressed consistently (and with a maximum error of 20.1 cents)
in 22-equal" (see also Table 1, at 25).

Most appreciatively,

Margo Schulter
mschulter@value.net

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

/tuning/topicId_18806.html#18806

> Hello, there, Joseph Pehrson and everyone.
>
> Thank you for raising a very important point both about 17-tone
equal temperament (17-tET) and about Paul Erlich's famous charts of
equal temperaments in his 22-tET article, which invites some further
> discussions. What follows is in addition to Paul's remarks in recent
> threads on the intentions of those charts, warmly inviting any
> corrections or clarifications from Paul.
>
> The critical point I would immediately make about Paul's charts is
> that they are based on a Partch-like "n-odd-limit" concept which
> assumes that each limit includes the intervals of any lower limits.
> If any of these lower limits is found to be inconsistent for a given
> n-tET, then the charts don't look at higher-limit ratios for which
> what I term "set-consistency" (and Carl Lumma may have termed
> something else when he delved into this concept) may indeed obtain.
>

Thank you very much, Margo Schulter, for your comments on what I see
now as a wonderful tuning, 17-tET. I guess I should not be faulted,
then, if I initially saw it as rather "deficient" after looking at
Paul's "sonorities" chart. After all, it's mostly just a blank line
with virtually no equivalencies on it... well, except the 3-limit
one, which IS indicated as having a very high correspondence.

However, the initial impression, looking at this chart is that 17-tET
is "poor." I find this particularly fascinating, since it is a VERY
clear illustration of "editorial content," actually pretty similar to
"points of view" that are expressed in newspaper articles and
photographs, without the editor actually WRITING anything at all!
It's just what is included or excluded that makes the statement.

In this case, I *did* notice that every limit that was approximated
for each n-tET had all the other odd whole number limits under it as
well. I just always assumed that was the way it has to be.

Now, according to your description, it could be possible to evaluate
such tunings in an ENTIRELY different way... say just omitting the 5
limit, for example, and finding them quite wonderful and
set-consistent otherwise.

As someone who is a great lover of Medieval and Renaissance music (I
listen to it in the morning EVERY DAY... well OK, occasionally a jazz
record slips in, but that is the exception, rather than the rule...)
I always study your posts carefully and look forward to hearing Paul
Erlich's response on this "editorializing...!"

Thank you again...

________ ______ _____ _
Joseph Pehrson

Margo wrote,

>The critical point I would immediately make about Paul's charts is
>that they are based on a Partch-like "n-odd-limit" concept which
>assumes that each limit includes the intervals of any lower limits.

Yes, that is correct -- the paper is basically a synthesis of Partch and
Yasser, since those were the only two microtonal books I had read when I
came up with the whole idea (in 1991).

>If any of these lower limits is found to be inconsistent for a given
>n-tET, then the charts don't look at higher-limit ratios for which
>what I term "set-consistency" (and Carl Lumma may have termed
>something else when he delved into this concept) may indeed obtain.

The justification for this might be that musicians will naturally want to
incorporate any intervals they could possibly experience as "consonant" into
the particular music at hand. This makes the assumption that "consonance" is
defined by being below a particular level of discordance; which as we know,
when octave-equivalence is assumed, corresponds to a particular odd-limit
(provided that's not too high). Anyway, you can look at Partch's book to see
how he justifies it.

>Now let's consider a radically different sort of problem: how about
>17-tET as a neo-Gothic tuning, where 3-limit ratios form the basis of
>stable sonorities, but unstable intervals such as thirds and sixths
>can take on all kinds of sizes -- e.g. "thirds" just about anywhere in
>the range from around 15:13 (~248 cents) to 13:10 (~454 cents)?

I would say that most of these thirds and sixths are too complex to
reasonably be considered as identifiable ratios, so the characterization of
some neo-gothic tunings as 3-limit is quite correct -- with the slight
complication that being exceptionally good in the 3-limit implies that 9:4
and equivalents may be concordant as well.

>As it happens, although I haven't yet confirmed this in a rigorous
>manner, 17-tET appears consistent for ratios of 2-3-7, but not
>especially accurate (albeit somewhat more so than 12-tET)

You're right.

>Accuracy (or rather its inverse, variance) ranges from around 20 cents
>for 8:7 and 7:4 to around 16 cents for 7:6 and 12:7, and around 12
>cents for 9:7 and 14:9.

>---------------------------------------------------
>Ratio Cents 17-tET Cents Variance
>---------------------------------------------------
> 8:7 ~231.17 3/17 octave ~211.76 -~19.41
> 7:4 ~968.83 14/17 octave ~988.24 +~19.41
> 7:6 ~266.87 4/17 octave ~282.35 +~15.48
>12:7 ~933.13 13/17 octave ~917.65 -~15.48
> 9:7 ~435.08 6/17 octave ~423.53 -~11.55
>14:9 ~764.92 11/17 octave ~776.47 +~11.55
>---------------------------------------------------

I would say deviation rather than variance, since variance often implies
squared deviation.

>In conclusion, if we take Paul's criterion of what I might call
>minimal "ballpark" accuracy -- a variance of around 20 cents[1] --
>then 17-tET seems to permit consistent expression for a large and
>possibly complete set of ratios of 2-3-7-9-11.

Excellent! I don't think I ever thought of including ratios of 11 in 17-tET!

>More generally, as Dave and I have agreed, 17-tET might be taken as a
>kind of overall optimization of complexity for major and minor thirds
>and major sixths: these intervals tend toward maximal complexity, and
>in many harmonic timbres toward maximal tension.

Another cool feature of 17-tET. Yet the minor triads of 17-tET, despite
minor thirds "maximally tense" between 7:6 and 6:5 and major thirds
"maximally tense" between 5:4 and 9:7, are rather pleasant, while the major
triads are far more unpleasant. Looks like a job for triadic harmonic
entropy!!!

>Incidentally, I'd welcome either confirmation or artful contradiction
>of my tentative impression that 17-tET is "set-consistent" for ratios
>of 2-3-7-9-11.

Paul Hahn's method is simplest:
In 17-tET,
Deviation of 3 = 0.0556
Deviation of 7 = 0.2750
Deviation of 9 = 0.1113
Deviation of 11 = 0.1897
Since the difference between the highest deviation and the lowest deviation
is less than .5 (and indeed the deviations are all positive, automatically
consistent), 17-tET is consistent with respect to the set (1,3,7,9,11).

Joseph wrote,

>Now, according to your description, it could be possible to evaluate
>such tunings in an ENTIRELY different way... say just omitting the 5
>limit, for example, and finding them quite wonderful and
>set-consistent otherwise.

Absolutely . . . though it would be more correct to say "omitting the ratios
involving 5" since "omitting the 5-limit" would omit ratios of 3 as well as
ratios of 5, and also would fail to omit ratios involving 5 and a higher odd
number.

>As someone who is a great lover of Medieval and Renaissance music (I
>listen to it in the morning EVERY DAY... well OK, occasionally a jazz
>record slips in, but that is the exception, rather than the rule...)
>I always study your posts carefully and look forward to hearing Paul
>Erlich's response on this "editorializing...!"

Well, I definitely think Medieval music is 3-limit and Renaissance music is
5-limit in the Partch sense.

But I think set-consistency is a great idea.

Even common practice 12-tET music might be described as exploiting 12-tET's
set-consistency with respect to (1,3,5,7,17) or even (1,3,5,7,17,19).

And of course 22-tET is consistent (and fairly accurate) with respect to the
set (1,3,5,7,9,11,15,17) -- numerologists out there, note that "unlucky 13"
is omitted!

Paul Erlich wrote,

<<Excellent! I don't think I ever thought of including ratios of 11 in
17-tET!>>

Wow, I always assumed "set-consistency" was just part of the whole
deal... ? (So with 17-tET for example you could add the 13 as well so
long as you ditch the 5, etc.)

<<Since the difference between the highest deviation and the lowest
deviation is less than .5 (and indeed the deviations are all positive,
automatically consistent), 17-tET is consistent with respect to the
set (1,3,7,9,11).>>

What Paul means by .5 here is the absolute value of the differences
between

((LOG(u)-LOG(1))*(t/LOG(2)))

and

ROUND ((LOG(u)-LOG(1))*(t/LOG(2)))

Where "u" represents the sequential odd numbers (i.e., the odd-limit),
and "t" an equal temperament.

So the first absolute value difference you come to between any two
odd-limit ratios that is > .5 breaks the consistent chain. So Margo's
"set-consistency" would be those cases where it could be beneficial to
chuck the break in the chain and group some still consistent set --
such as the 1,3,7,9,11 or 1,3,7,9,11,13 set in 17-tET -- together.

--Dan Stearns

On Fri, 16 Feb 2001 17:58:58 -0500, "Paul H. Erlich"
<PERLICH@ACADIAN-ASSET.COM> wrote:

>>Incidentally, I'd welcome either confirmation or artful contradiction
>>of my tentative impression that 17-tET is "set-consistent" for ratios
>>of 2-3-7-9-11.
>
>Paul Hahn's method is simplest:
>In 17-tET,
>Deviation of 3 = 0.0556
>Deviation of 7 = 0.2750
>Deviation of 9 = 0.1113
>Deviation of 11 = 0.1897
>Since the difference between the highest deviation and the lowest deviation
>is less than .5 (and indeed the deviations are all positive, automatically
>consistent), 17-tET is consistent with respect to the set (1,3,7,9,11).

Hmm, let's see how this works out for 15-TET..... (skipping 9, since 15-TET
doesn't have a good approximation of 9/8.)

Deviation of 3 = 0.2256
Deviation of 5 = 0.1711
Deviation of 7 = -0.1103
Deviation of 11 = 0.1085

They're not all positive, but they're within 0.3359 between the highest and
lowest deviation. This result is certainly consistent with my experience
with 15-TET, and confirms that it's a good scale for 11-limit harmony if
you don't care about factors of 9.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Herman wrote,

>Hmm, let's see how this works out for 15-TET..... (skipping 9, since 15-TET
>doesn't have a good approximation of 9/8.)

>Deviation of 3 = 0.2256
>Deviation of 5 = 0.1711
>Deviation of 7 = -0.1103
>Deviation of 11 = 0.1085

>They're not all positive, but they're within 0.3359 between the highest and
>lowest deviation.

Right -- hence 15-tET is consistent with respect to the set (1,3,5,7,11).

>This result is certainly consistent with my experience
>with 15-TET, and confirms that it's a good scale for 11-limit harmony if
>you don't care about factors of 9.

Right -- as long as you don't use 9:8, 9:7, 10:9, or 11:9, 15-tET is great
for 11-limit.

P.S. Herman, I think you should be included in Joe Monzo's ET listing under
15-tET, if not others. Could you provide Joe with an appropriate year, etc.?

On Fri, 16 Feb 2001 22:19:24 -0500, "Paul H. Erlich"
<PERLICH@ACADIAN-ASSET.COM> wrote:

>P.S. Herman, I think you should be included in Joe Monzo's ET listing under
>15-tET, if not others. Could you provide Joe with an appropriate year, etc.?

My eventual goal is to explore all the equal scales at least up to 31-TET.
So far I've written in:

5-TET (2001)
13-TET (1998)
14-TET (2000)
15-TET (1996)
16-TET (1998)
17-TET (1997)
19-TET (1987)
20-TET (1999)
26-TET (1998)

I've also experimented and improvised in 9-TET, 11-TET, 18-TET, 22-TET, and
31-TET.

"Rriladeni Tharnien" (1987) was originally in a subset of 31-TET, although
I was essentially using it as a meantone tuning.

From 1989, "Cinq et Sept" (my piece in Owen Jorgensen's Five and Seven
Temperament) is divided into separate sections featuring 5-TET, 7-TET, and
the complete 12-note scale.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Hello, there, and thanks to Paul Erlich and others for their responses
to my post on "17-tET set consistency," including the mathematical
test which I need to understand more fully.

As people have mentioned, in addition to the ratios of 2-3-7-9-11
considered in my post, 17-tET has consistent approximations for
ratios of 2-3-7-9-11-13. To fill out the set of tables in my original
post, here's one for these ratios of 13:

-----------------------------------------------------
Ratio Cents 17-tET Cents Difference
-----------------------------------------------------
13:7 ~1071.70 15/17 octave ~1058.82 -~12.88
14:13 ~128.30 2/17 octave ~141.18 +~12.88
13:8 ~840.53 12/17 octave ~847.06 +~6.53
16:13 ~359.47 5/17 octave ~352.94 -~6.53
13:9 ~636.62 9/17 octave ~635.29 -~1.32
18:13 ~563.38 8/17 octave ~564.71 +~1.32
13:11 ~289.21 4/17 octave ~282.35 -~6.86
22:13 ~910.79 13/17 octave ~917.65 +~6.86
13:12 ~138.57 2/17 octave ~141.18 +~2.60
24:13 ~1061.43 15/17 octave ~1058.82 -~2.60
-----------------------------------------------------

Curiously, in a kind of "numerical pun" which another recent thread on
19-tET and 7-limit JI tunings may suggest, it turns out that these
17-tET approximations of 13-based ratios (2-3-7-9-11-13) are all
accurate to within about 13 cents.

(As for the roughly 7-cent differences between 19-tET and a 7-limit JI
tuning, it occurs to me that this is, at least in part, a predictable
consequence of the tempering of the fifth or fourth in 19-tET by
around 7.22 cents. Thus 19-tET major thirds will be smaller than 5:4
by this amount, etc. This raises the possibility of a 38-note
"adaptive quasi-JI" system with two 19-tET manuals at ~7.22 cents
apart. I'm not sure if some previous posts may have covered these
points, and my first-blush reading may not have caught everything.)

People might well observe at this point that all of the above 13-based
ratios with the exception of 13:7 and 13:8, stated in the form a:b,
have a*b > 105, so that it's debatable whether they would be
"recognizable" as Keenanesque "just" ratios even if purely tuned.
However, the same comment might be made about many of the ratios of 13
and 17 approximated by other tunings such as 22-tET, so I'm presenting
these tables while leaving the "recognizability" issue open.

By the way, Paul, I much agree that neo-Gothic thirds are really
flexible and don't need to match or approximate any "recognizable
ratio," so that we might speak of a "complex 3-limit system." My use
of "flavors" rather than limits to map the spectrum may in part
reflect this distinction: whether a major third is tuned at or around
81:64, 14:11, 23:18, or 9:7 is a question of "shading" rather than
categorical recognition.

Just how best to describe this kind of system, and what the
theoretical implications might be, is the kind of issue inviting lots
of dialogue from lots of viewpoints. However, avoiding "n-limit"
terminology in describing unstable neo-Gothic intervals may at least
avoid some opportunities for confusion and misunderstanding, and I'd
like to thank you, Dave Keenan, and others for some exchanges which
have educated me on this not inconsequential point.

One point on which different musical practices may lead to different
groupings of intervals: like medieval European theorists, I regard the
ratios 9:8 and 16:9 (and also 9:4) as "basic." The Pythagorean
"quadrichord" with string lengths of 12:9:8:6 includes 9:8 as one of
the four principal nonunisonal intervals of the system (2:1, 3:2, 4:3,
9:8). Thus I tend to group 9:8 -- and its octave complements or
extensions 9:4 and 16:9 -- with the 3-based 3:2 and 4:3.

In a 3-prime-limit system, 9:8, along with 3:2 and 4:3, is a
superparticular ratio, giving it a certain special status in medieval
theory along with the stable concords, all founded on multiplex (n:1)
or superparticular (n+1:n) ratios.

There's lots more to discuss here, Paul, thanks to your creative
comments and those of others, but for now I hope that maybe the list
of 13-based ratios in 17-tET might fill out this point.

Most appreciatively,

Margo Schulter
mschulter@value.net

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

> People might well observe at this point that all of the above 13-
based
> ratios with the exception of 13:7 and 13:8, stated in the form a:b,
> have a*b > 105, so that it's debatable whether they would be
> "recognizable" as Keenanesque "just" ratios even if purely tuned.
> However, the same comment might be made about many of the ratios of
13
> and 17 approximated by other tunings such as 22-tET, so I'm
presenting
> these tables while leaving the "recognizability" issue open.

Please remember, Margo, that the a*b limit of 105 was derived in a
_dyadic_ context, while even the consistency question itself concerns
triads and larger chords. I know from personal experience that 13 can
occur in "recognizable" triads and 17 in "recognizable" tetrads.

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> People might well observe at this point that all of the above
13-based
> ratios with the exception of 13:7 and 13:8, stated in the form a:b,
> have a*b > 105, so that it's debatable whether they would be
> "recognizable" as Keenanesque "just" ratios even if purely tuned.

I fear I am in danger of being misrepresented as implying that only
dyads matter (in another thread as well). I believe my recent
definitional attempts made it very clear that I consider context to be
all-important.

So don't forget that this sort of a*b limit is only intended to be
applied to bare dyads. Although 9:13 (for example) on its own is very
unlikely to be perceived as just, even if tuned within 0.5 c, it might
well be perceived as such in the context of a larger otonal chord,
with a suitable timbre etc etc.

However, as you point out, in the case of 17-tET, the deviations of
the other approximate ratios of 13 make this extremely unlikely.

Regards,
-- Dave Keenan