Yahoo Tuning Groups Ultimate Backup tuning For Mike, on 7:9:11, counterpoint rules

Hi Margo, and sorry for taking so long to respond to this; I'll try to
resurrect this conversation now...

On Thu, Nov 8, 2012 at 3:24 AM, Margo Schulter <mschulter@...> wrote:
>
> You're right that this is almost exactly 3/5 of the way into the
> piece: around 0:36 out of 1:03 in all. And your comment is really
> fascinating, because to me this is a fairly standard idiom, and
> analogous to a transition in Monteverdi or Gesualdo, say, from a
> basically diatonic section of a madrigal to a rather melancholy
> chromatic section with various slower and more dramatic passages
> like suspensions. It's a rhetorical gesture to set the words _Ex
> dumeto illaquante bello_, "Out of the entangling thicket of
> war..." But clearly the impact for you want beyond that!

It's that I wasn't able to cognize the sound I was hearing in terms of
notes in a scale. For instance, imagine you're playing a song on your
car radio and the person behind you honks - it's likely you'll hear
the honk as just a sound, not something tonally related to what you're
listening to. That's what happened to me until I listened to it again
and managed to figure out the "functions" of each note in the chord
(for lack of a better term).

There's a big difference in my mind between hearing a half step above
the root (sounds like noise) and hearing a half step above the root
(sounds like phrygian). The latter implies some additional depth of
musical cognition that's not present in the former; the sound's been
"activated" in a sense. So for most of these past , I went on a quest
to push my brain to "activate" as many sounds in this way as possible,
including various scales that are typically heard as "weird," like the
modes of melodic and harmonic minor and such.

It's the difference between hearing lydian augmented #2 and thinking
"that's weird" and hearing it and imagining a chord like
C-G#-E-F#-B-D# moving to C-A-E-G#-B-E (notes in ascending order)

Now I'm trying to similarly push my brain to "activate" various
microtonal scales but it's difficult because I don't know what I'm
doing. :)

> Since we're listening to the same realization of this piece,
> factors like timbre are controlled -- so the difference in
> impressions signals a distinction in musical perceptions!
> But one possible distinction, especially if you have really good
> high frequency hearing, is that as an older listener, I may not
> be so aware of beating between higher partials. It's sometimes
> theorized that Mersenne may have become a bit more favorable
> toward 12-EDO as a harpsichord tuning in his later years because
> high-frequency hearing loss may have made the beating less
> obvious!

It could be, though I suspect it's some deeper sort of music-cognitive
thing that has to do with assigning learned interval "functions" to
sounds. It sounded completely different once I managed to place it in
terms of the scale, it's hard for me to describe.

The first time it sounded like a nice sound effect of some sort. For
instance, you can imagine some early 60s sunshine pop song where they
have a recording of the outdoors with birds chirping and such. The
first time I heard it, it was like birds chirping - the individual
notes didn't seem like they were tonally related to the rest of what's
going on.

The second time I managed to fit the sound into the scale I was
hearing and hear the dyads in it as having some sort of
diatonic/chromatic function, like I finally managed to cram the
slightly-misshapen jigsaw puzzle piece into place.

> A final and very important lesson from your observation is that
> much the same difference of perception might have happened in a
> Parapyth tuning such as MET-24 or the POTE, as well as
> Peppermint: fine points of the generator size over a range of
> about 0.4 cents or whatever shouldn't distract us from more basic
> and dramatic things like the effect of the 7:9:11 on different
> listeners.

Another reason why I think there's more going on than just 7:9:11 is:
here's a little ditty I wrote a while ago in 11-EDO that features
7:9:11 as a sonority

http://soundcloud.com/mikebattagliamusic/tonal-study-in-orgone-temperament-11-edo

This is using orgone[7] in 11-EDO, which is LLsLsLs. Maybe I'll let
you listen and make an impression first before I describe what's going
on...

> A humorous note here, before I address your question where it's
> relevant, is that parallel fifths as well as fourths are quite
> routine in 13th-14th century and earlier medieval progressions
> and cadences! Some "modern" 14th-century treatises give a rule
> against parallel fifths or octaves in two-voice, note-against-note,
> writing -- but it doesn't apply even in theory with three or more
> voices until around the mid-15th century.

Yes, that's true, and rock music uses them pretty commonly too.

So why aren't they allowed in Bach-era counterpoint? What's the effect
we're trying to get away from, exactly?

> Since the 3/2 is the generator in medieval Pythagorean tuning,
> that obviously isn't a problem. Intuitively, I would say that a
> 3/2 has a "higher-complexity" status where 2:3:4 is a saturated
> and complete harmony than where 4:5:6 is starting to fill that
> role, and that's likely one of the relevant variables.

Hmm, what do you mean by higher-complexity here?

> And even in the 16th century, it's not the _sound_ of parallel
> fifths in textures for three or more voices that's the issue, but
> the relative loss of independence of parts where 3/2 is no longer
> the standard of rich stability, but less ideally euphonious than
> a third, sixth, or tenth. If the two voices cross which form the
> fifths, there's no violation of the rule. Some anachronistic
> criticism of 16th-century compositions treats this as a flaw; but
> it's a flaw in the eyes of the beholder, not of 16th-century
> practice or theory.

Not sure I understand; do you mean that they're treating 3/2 as
dissonant in general? As in, it's not as "stable"?

> And in 20th-century counterpoint, as in the 13th century, it's no
> problem at all. The norm is that in serious writing we do want a
> variety of intervals and motions, which implies not overdoing
> parallelism of any kind -- but why not fifths, as long we keep
> some balance and moderation? In a medieval style, they're the
> most euphonious intervals of all, just as parallel thirds or
> sixths would be tolerated in Renaissance styles (although using
> them all the time between the same voices wouldn't be considered
> serious counterpoint). And in a 20th-century style,
> quartal-quintal harmony likewise is often very much to taste.

Well, sure, I agree with that. But I do note there's a certain sound
that happens if you avoid parallel fifths which is nice, just like
there's a certain sound you get if you use them often...

> And I should add that the acceptability of parallel fifths in a
> medieval or 20th-century style holds just as much in 12-EDO (not
> too far from Pythagorean) as in Porcupine or Parapyth or any of
> the other nifty systems we're discussing here. And parallel
> fifths between two voices (noncrossing) would be just as
> incorrect in a 16th-century style played in 12-EDO as in meantone
> or 5-limit JI.

Well, consider porcupine. Porcupine is generated by a slightly sharp
11/6, or by an interval which is about halfway between 11/6 and 9/5.
If I want to emulate 16th century counterpoint in porcupine
temperament, do I avoid parallel 3/2's, parallel 11/6's (aka
generators), or both?

I think it's both - the more you play the same dyad in a row, the more
it tends to fuse and sound like a single timbre (aka what Ravel does
with 1:3:5 chords in Bolero), and this happens more if the dyad is
simpler, like 3/2. That's my guess, anyway.

Thanks Margo and sorry for taking so long to reply!

-Mike

🔗Margo Schulter <mschulter@...>

> Hi Margo, and sorry for taking so long to respond to this; I'll try to
> resurrect this conversation now...

Dear Mike,

Thank you for a great reply, where the timing was fine because I
was away a lot of the time; in fact, I apologize for taking a few
extra days to reply to your wonderful new post because of matters
involving my Mom's health.

This is a conversation very much worthwhile, because your remarks
very clearly explain your perceptions of my piece while raising
some perennial as well as newer questions -- and also a matter
which might interest Dave Keenan about a feature of the Orgone
tuning in your piece.

I'll feel free to trim my own old quotes, so that I can focus on
yours. Also, I'll give links to a MIDI file and a PDF score (in
Sagittal notation) of my piece we're discussing, _O Europae_:

<http://www.bestII.com/~mschulter/O_Europae.mid>
<http://www.bestII.com/~mschulter/O_Europae.pdf>

We are discussing the point at about 0:36 (m. 25 in the score)
where a ~7:9:11 sonority occurs that seemed to you at first
"so acoustically strong that it destroyed my context of the
tonality and sounded just like a raw sonic object (like a car
horn or something, and I had to go back and listen over it again
to be able to `frame' the scale in the proper context." I'm
quoting from your earlier post, upon which you comment further
here.

> It's that I wasn't able to cognize the sound I was hearing in
> terms of notes in a scale. For instance, imagine you're
> playing a song on your car radio and the person behind you
> honks - it's likely you'll hear the honk as just a sound, not
> something tonally related to what you're listening to. That's
> what happened to me until I listened to it again and managed
> to figure out the "functions" of each note in the chord (for
> lack of a better term).

That's a vivid description, and a helpful one.

> There's a big difference in my mind between hearing a half
> step above the root (sounds like noise) and hearing a half
> step above the root (sounds like phrygian). The latter implies
> some additional depth of musical cognition that's not present
> in the former; the sound's been "activated" in a sense. So for
> most of these past , I went on a quest to push my brain to
> "activate" as many sounds in this way as possible, including
> various scales that are typically heard as "weird," like the
> modes of melodic and harmonic minor and such. It's the
> difference between hearing lydian augmented #2 and thinking
> "that's weird" and hearing it and imagining a chord like
> C-G#-E-F#-B-D# moving to C-A-E-G#-B-E (notes in ascending
> order) Now I'm trying to similarly push my brain to "activate"
> various microtonal scales but it's difficult because I don't
> know what I'm doing. :)

Thanks for calling my attention to some of these modes like
lydian augmented #2! And I get your point about "activated"
patterns: those one can parse and recognize "on the fly" as
indeed meaningful patterns. How different people learn ways to
parse something like lydian augmented #2 could be a discussion in
itself, of course. Is this F-G#-A-B-C-D-E-F, for example?

As you also noted in the earlier post, previous familiarity with
a tuning like Peppermint or MET-24 and the neomedieval idioms
used in a piece like _O Europae_ can make the parsing easier: in
effect, you gained this familiarity through experience.

>> Since we're listening to the same realization of this piece,
>> factors like timbre are controlled -- so the difference in
>> impressions signals a distinction in musical perceptions!
>> But one possible distinction, especially if you have really
>> good high frequency hearing, is that as an older listener, I
>> may not be so aware of beating between higher partials. It's
>> sometimes theorized that Mersenne may have become a bit more
>> favorable toward 12-EDO as a harpsichord tuning in his later
>> years because high-frequency hearing loss may have made the
>> beating less obvious!

> It could be, though I suspect it's some deeper sort of
> music-cognitive thing that has to do with assigning learned
> interval "functions" to sounds. It sounded completely
> different once I managed to place it in terms of the scale,
> it's hard for me to describe.

Could this be a bit like optical illusions where we might see
either two faces or a vase against a background, for example?

But I get the point that the issue here likely isn't "dissonance"
or beats in the usual sense. Maybe it's more like what is
sometimes termed "melodic dissonance," but a bit different from
that either. Not so much, "Is this a smooth melody?" as "Is this
a single texture, however smooth or jagged or whatever?"

> The first time it sounded like a nice sound effect of some
> sort. For instance, you can imagine some early 60s sunshine
> pop song where they have a recording of the outdoors with
> birds chirping and such. The first time I heard it, it was
> like birds chirping - the individual notes didn't seem like
> they were tonally related to the rest of what's going on.

A very helpful description, and I seem to recall effects like
this a bit later on from the Beatles.

> The second time I managed to fit the sound into the scale I
> was hearing and hear the dyads in it as having some sort of
> diatonic/chromatic function, like I finally managed to cram
> the slightly-misshapen jigsaw puzzle piece into place.

This is very important, and something I hadn't considered: not
only can greater familiarity make a given progression sound more
"meaningful," but sometimes it can make what originally seemed to
be a "raw sonic object" (quoting your earlier post) emerge as
part of a connected progression or a single texture.

>> A final and very important lesson from your observation is
>> that much the same difference of perception might have
>> happened in a Parapyth tuning such as MET-24 or the POTE, as
>> well as Peppermint: fine points of the generator size over a
>> range of about 0.4 cents or whatever shouldn't distract us
>> from more basic and dramatic things like the effect of the
>> 7:9:11 on different listeners.

> Another reason why I think there's more going on than just
> 7:9:11 is: here's a little ditty I wrote a while ago in 11-EDO
> that features 7:9:11 as a sonority
> <http://soundcloud.com/mikebattagliamusic/tonal-study-in-orgone-temperament-11-edo>

> This is using orgone[7] in 11-EDO, which is LLsLsLs. Maybe
> I'll let you listen and make an impression first before I
> describe what's going on...

Getting my impressions first, and then describing more about the
piece and what's going on, is an approach I like. And I'm very
curious to get your more detailed perspective.

My first impression: a bit like a Bartok prelude from
_Mikrokosmos_. Most relevantly here, I here no discontinuity or
sense of sounds arising separate from the previous texture. We
might add, of course, that 11-EDO is a lot further from a
traditional European diatonic/chromatic style than _O Europae,
which sort of mixes late medieval and weirdly neo-Renaissance
idioms.

The piece has lots of rhythmic or motor energy, a trait shared by
lots of Bach and Bartok, and very winningly communicated in your
piece. Whatever you did, and however you did it, I like the
result!

So your orgone mode is 0-218-436-545-764-873-1091-1200. And this
raises the point as to triadic harmonic entropy -- or
recognizability -- that might interest Dave Keenan.

Here I take it that the 7:9:11 representation must be 0-436-764
cents, close to a just 1/1-9/7-14/9 or 0-435-765 cents or
63:81:98, in contrast to a just 7:9:11 at 0-435-782 cents.

Now in a simple dyadic context, I might guess that 764 cents or a
just 14/9 would be unlikely to represent 11/7 -- but with 7:9:11,
the isoharmonic triad might have a greater attractive power so
that 0-436-764 cents in orgone _can_ be perceived as 7:9:11!

Back around 2000-2001, Dave would often point out that dyads not
so easy to recognize or tune by ear could nevertheless be quite
recognizable in a triadic or more complex sonority like an
isoharmonic one such as 7:9:11. And Paul Erlich was much
fascinated with the related question of triadic harmonic
entropy.

Another theme was that if some intervals were close to just in
such a sonority, others might be a bit off without interference
with recognizability. On this point, in orgone 0-435-764, the
upper interval of 327 cents, just a tad smaller than 98:81,
represents 11:9 or 347 cents in a 7:9:11 perception. That's a
difference of 20 cents, but the isoharmonic triad may have a
"pull" that overcomes this difference.

Of course, I'm interested not only in how you see -- and maybe
Dave, if you're reading this -- the 7:9:11 question, but also the
other details of what's going on.

Our parallel fifths discussion I'm treating as a separate post,
to keep the focus on _O Europae_, your orgone piece, and the
question of when and how textures are perceived as unified or as
an interrupted texture plus "raw sonic object" or the like.
Also, this will give me a chance to pare down my remarks a bit on
the parallel fifths question, while thanking you in advance for
your excellent questions that helped me write more clearly!

Best,

Margo

🔗Margo Schulter <mschulter@...>

Hi, Mike.

This is picking up on the question of parallel fifths we were
discussing, and I'll try to clarify in some places where you
asked excellent questions.

Also, it's my intuition that the generator for a given scale or
tuning set may not necessarily have a connection to parallelism
restraints, although this may invite lots of exploration. Your
example of porcupine and counterpoint rules which might be
analogous to those of the 16th century is a great exercise, and
could prompt some compositions!

>> Intuitively, I would say that a 3/2 has a
>> "higher-complexity" status where 2:3:4 is a saturated and
>> complete harmony than where 4:5:6 is starting to fill that
>> role, and that's likely one of the relevant variables.

> Hmm, what do you mean by higher-complexity here?

Quickly, I'm thinking of relative or "contextual" complexity in a
given composition or style. A couple of examples might help.

A performance of a three-part piece by Perotin around 1200 that I
first heard about 30 years ago starts out with a touch not in
Perotin's score. First we hear the lowest voice alone, a unison;
then the second voice at a 3/2, showing how rich a simple fifth
can be; and finally the third voice at a 2/1, for a complete
2:3:4 trine rich and resonant enough to encompass the universe.
That's what I call a "saturated" sonority, as complex as we can
get and still be restful and stable.

By the later 15th and 16th centuries, however, 4:5:6 or 10:12:15
is taking over the role of a complete or saturated stable
sonority, and Vicentino (1555) and Zarlino (1558) make it
official. In this context, 2:3:4 is no longer so rich and
satisfying from a perceptual point of view -- the "richness of
harmony," as Vicentino terms it, calls for something like 2:3:4:5
or 2:4:5:6, etc.

So various people, including Joseph Yasser if I recall, have
suggested that parallel fifths in a medieval setting are like
parallel thirds or sixths in a 16th-century setting. As the
favorite stable concords, they are acceptable as parallels -- but
limited in serious counterpoint by the general preference for a
variety of motions (especially contrary) and intervals.

>> And even in the 16th century, it's not the _sound_ of parallel
>> fifths in textures for three or more voices that's the issue, but
>> the relative loss of independence of parts where 3/2 is no longer
>> the standard of rich stability, but less ideally euphonious than
>> a third, sixth, or tenth. If the two voices cross which form the
>> fifths, there's no violation of the rule. Some anachronistic
>> criticism of 16th-century compositions treats this as a flaw; but
>> it's a flaw in the eyes of the beholder, not of 16th-century
>> practice or theory.

> Not sure I understand; do you mean that they're treating 3/2 as
> dissonant in general? As in, it's not as "stable"?

A good question that helps me communicate more clearly! There's
no question that 3/2 or 2:3:4 is still stable in the 16th
century, but it's increasingly felt as "stable but incomplete."
By the middle of the century, an ending like the following with
1:2:3:4, which occurs in some later pieces also, might sound a
bit "conservative":

... A5 G#5 A5
E5 E5
B4 A4
E4 A3

The alternative is to have the second voice cadence on C#4, for
example, giving us a closing sonority of 2:5:6:8.

... A5 G#5 A5
E5 E5
B4 C#4
E4 A3

So cadences of the first kind, common around 1500, were still
used late in the century; but the second type, specifically with
a major third or tenth in the final sonority, was standard by
around the 1520's, and typified Vicentino's "richness of harmony"
or Zarlino's "perfect harmony" -- the same role as 2:3:4 in a
medieval setting.

> Well, sure, I agree with that. But I do note there's a certain
> sound that happens if you avoid parallel fifths which is nice,
> just like there's a certain sound you get if you use them
> often...

True!

And Don Randel has suggested that the prohibition against
parallel fifths, as it became more consistently observed by
around 1450, played a large role in shaping 16th-century and
later progressions by a process of exclusion. As someone most
often following a medieval style, with countless cadences and the
like routinely involving parallel fifths, I can appreciate his
point.

> Well, consider porcupine. Porcupine is generated by a slightly
> sharp 11/6, or by an interval which is about halfway between
> 11/6 and 9/5. If I want to emulate 16th century counterpoint
> in porcupine temperament, do I avoid parallel 3/2's, parallel
> 11/6's (aka generators), or both?

> I think it's both - the more you play the same dyad in a row,
> the more it tends to fuse and sound like a single timbre (aka
> what Ravel does with 1:3:5 chords in Bolero), and this happens
> more if the dyad is simpler, like 3/2. That's my guess,
> anyway.

Here I agree that both factors would be relevant: any kind of
parallelism tends to reduce independence of voices, and more so
with simple intervals like 3/2 (especially if intervals more
complex than 3/2 or 4/3 are deemed fully stable and restful).

The interesting thing is that, if one assumes our 20/11 or
whatever is stable and ideally euphonious, analogous to 5/4 in a
16th-century style, then applying both factors might lead to
tentative rules we could try out like these:

(1) Avoid parallel 3/2's in "serious" 16th-century-like
writing;

(2) Use parallel 20/11's or 51/28's or whatever in a
restrained way, especially in two-part writing, because
they're ideally rich and euphonious, but a row of more
than two or three creates something more like a textural
effect than the development of two independent voices.

(3) With three or more voices, we might loosen up a bit on
rule (2), since two voices moving in parallel 20/11's or
whatever can be balanced by other voices in oblique or
better yet contrary motion.

As Richard Crocker humorously commented 50 years ago, rule (3) is
not just an artistic liberty but a necessity, since while we may
have three or more voices, there are only two directions that
voices can move -- up or down! So what I might term "ideally
euphonious parallels" are not only acceptable and pleasant, but
almost unavoidable. The presence of passages in fauxbourdon even
in the best 16th-century writing -- despite the distaste of both
Vicentino and Zarlino, who find the parallelism unpleasant rather
than sweet -- suggests that a bit of leeway might be interesting
to experiment with in porcupine to see how things balance out.

> Thanks Margo and sorry for taking so long to reply!

No problem, and taking some time led you to some really helpful
questions which I saw shortly after getting back online. And
thanks for your patience in waiting for my reply!

> -Mike

Best,

Margo

🔗Mike Battaglia <battaglia01@...>

Hi Margo, and sorry to hear about your Mom's health. Thanks for the
thoughtful reply; my points below:

-Mike

On Thu, Dec 6, 2012 at 4:03 AM, Margo Schulter <mschulter@...> wrote:
>
> Thanks for calling my attention to some of these modes like
> lydian augmented #2! And I get your point about "activated"
> patterns: those one can parse and recognize "on the fly" as
> indeed meaningful patterns. How different people learn ways to
> parse something like lydian augmented #2 could be a discussion in
> itself, of course. Is this F-G#-A-B-C-D-E-F, for example?

Right, exactly. I note that it's specifically a sort of harmonic
parsing that I'm talking about here, by the way. I wonder if a similar
11- or 13-limit perspective exists behind the various maqams, and if
it's just waiting for someone to activate the sound in the right way
for me.

> As you also noted in the earlier post, previous familiarity with
> a tuning like Peppermint or MET-24 and the neomedieval idioms
> used in a piece like _O Europae_ can make the parsing easier: in
> effect, you gained this familiarity through experience.

Right. I just wish I had some sort of idea about which specific
phenomenon is involved when I throw these hand-wavy terms like parsing
around. Since you seem to be familiar with the phenomenon I suggest,
do you have any idea what exactly it means? Unlike when we talk about
intonation, where there's lots of science to spare in terms of how
things like the harmonic series works, I'm completely lost here. I
don't even know where to start reading if I wanted to.

> > It could be, though I suspect it's some deeper sort of
> > music-cognitive thing that has to do with assigning learned
> > interval "functions" to sounds. It sounded completely
> > different once I managed to place it in terms of the scale,
> > it's hard for me to describe.
>
> Could this be a bit like optical illusions where we might see
> either two faces or a vase against a background, for example?

I think it may be related to that, although a more apt analogy for the
phenomenon you describe would be a 350 cent interval flip-flopping
between a major and minor third to my ear.

However, as a side note, I have the feeling that every time I hear a
tuning like 10-EDO, for instance, where I hear notes that are "between
12-EDO notes" and my brain tries to continually make sound like
altered versions of their nearest 12-EDO equivalents, the phenomenon
you describe is taking place. That is, I'm hearing broken pieces of
the order in 12-EDO that are being reflected onto the tuning system,
because it's just the best guess that my brain can come up with as to
what's going on. It's like this picture:

http://images1.wikia.nocookie.net/__cb20060330234506/psychology/images/3/37/Emergence.jpg

If the tuning system is this picture, I'm only seeing the "spots." I
have a very strong feeling that if I knew how to play the system
"correctly," or if some Mozart came along and wrote the perfect 10-EDO
concerto, it would somehow communicate the correct "perspective" from
which to view the tuning system and I would then instead see the big
picture of the 10-EDO "dalmatian." I'm using fluffy and hand-wavy
terms to describe my perceptual experience here because again, I don't
have any idea what sort of correlate in the literature exists to
describe what I'm talking about. (Sorry if this is too much of a
random tangent.)

> But I get the point that the issue here likely isn't "dissonance"
> or beats in the usual sense. Maybe it's more like what is
> sometimes termed "melodic dissonance," but a bit different from
> that either. Not so much, "Is this a smooth melody?" as "Is this
> a single texture, however smooth or jagged or whatever?"

It could be. I honestly think that it's something totally learned,
like I managed to take the sound I was hearing and "relate it" to
everything else I was hearing. The only problem is, I don't know what
exactly the "relation" is that I've come up with! I think it might be
something like: I heard the chord the second time and was able to plug
it into the circle of fifths, or something like that. I don't know if
that's actually it, but it's something like it.

I'm really fascinated by this aspect of music cognition because I feel
like if I could identify and really nail down this sort of "relation,"
and figure out what it is, then it would become self-evident how the
rest of music works. It might help to answer the question: when people
experience the mess of sound that is music - why they don't always
just hear it as raw sound, like a train horn? What is the experience
that's being added on top of that?

I think the answer is that they hear the notes all being related to
one another in a certain way. Keenan Pepper has added to this the
Krumhansl-esque realization that "tonal" music, specifically, allows
us not only to perceive these relations, but to predict somewhat which
notes are most likely to occur next. These are very simple cognitive
things we're talking about, it's easy to confirm with a bit of
introspection that these "features" of our musical paradigm are indeed
present, and it's easy to see how a much richer musical structure
could emerge from such simple building blocks. However, I still feel
like there's something else to add to that.

> > was hearing and hear the dyads in it as having some sort of
> > diatonic/chromatic function, like I finally managed to cram
> > the slightly-misshapen jigsaw puzzle piece into place.
>
> This is very important, and something I hadn't considered: not
> only can greater familiarity make a given progression sound more
> "meaningful," but sometimes it can make what originally seemed to
> be a "raw sonic object" (quoting your earlier post) emerge as
> part of a connected progression or a single texture.

Yes, and from my perspective even more meaningfully, it can change the
perception of a chord from "a raw sound" to "a collection of notes
which occupy a certain place in a scale, and a relation to other notes
in the scale."

> Here I take it that the 7:9:11 representation must be 0-436-764
> cents, close to a just 1/1-9/7-14/9 or 0-435-765 cents or
> 63:81:98, in contrast to a just 7:9:11 at 0-435-782 cents.
>
> Now in a simple dyadic context, I might guess that 764 cents or a
> just 14/9 would be unlikely to represent 11/7 -- but with 7:9:11,
> the isoharmonic triad might have a greater attractive power so
> that 0-436-764 cents in orgone _can_ be perceived as 7:9:11!

I think so, though I'm not sure what you mean by perceiving something
as 7:9:11 - I hear the intonation is reasonably concordant, if that's
what you mean. However, the key thing to my ear is that the 7:9:11 is
part of the 2 2 1 2 pentachord in 11-EDO. When you play it, it sounds
like you're playing C D E F G, however the whole thing is stretched so
that C-E-G is 7:9:11. In that context, this version of 7:9:11 is
completely different than yours; to my ears, mine sounds like a really
stretched major chord (though I didn't always use it that way),
whereas yours has a very complex functional relationship that took me
a second to grab a hold of.

I may need to make some more examples of 7:9:11 in 11-EDO to make my
point a little bit better on that, but to sum it up, I guess I'm
suggesting that a separation of function and intonation exists.

I think that many people start writing about how things like 7:9:11
sound or how they work musically, but they don't realize that in their
minds, the thing they're calling 7:9:11 has a lot of associated
"functional baggage" with it - because they're still force-fitting it
to the functional structure that they know, mentally. Then, when they
describe "here's how you use 7:9:11," they actually mean "here's how
you use [combination of functions they're remembering which happen to
be intoned as 7:9:11]." And one way to tease that apart is to show
that a chord like 7:9:11 can take on completely different functions in
different scales that cause you to force-fit it differently!
Underneath it all is still that raw sound - the 7:9:11 train horn -
but I think that it's the stuff we're projecting on top of it that
gives it its real musical flavor. Well, that's my thought, anyway.

> Another theme was that if some intervals were close to just in
> such a sonority, others might be a bit off without interference
> with recognizability. On this point, in orgone 0-435-764, the
> upper interval of 327 cents, just a tad smaller than 98:81,
> represents 11:9 or 347 cents in a 7:9:11 perception. That's a
> difference of 20 cents, but the isoharmonic triad may have a
> "pull" that overcomes this difference.

Yes, I agree with this, and furthermore I think that if you play a
4:7:9:11 chord and take the 4 and 7 away, that 327 cent interval
remaining might sound a lot different if you're still remembering
those notes "lingering" than it would if you just played it cold.

> Our parallel fifths discussion I'm treating as a separate post,
> to keep the focus on _O Europae_, your orgone piece, and the
> question of when and how textures are perceived as unified or as
> an interrupted texture plus "raw sonic object" or the like.
> Also, this will give me a chance to pare down my remarks a bit on
> the parallel fifths question, while thanking you in advance for
> your excellent questions that helped me write more clearly!

Well, I hope I haven't added too much with the extra stuff above! But
thanks for taking the time to respond to my post. The reason I'm
curious to ask you about this stuff is that you've studied the time
period when these sort of functional relationships shifted
dramatically, e.g. major thirds stopped being "avoid intervals" and
started being obvious consonances. I have no idea how someone could
simply adjust their brain to make that happen, but I wonder if it's
the same sort of "activation" process I discussed above with something
like lydian augmented #2. Dunstable simply got the major chord
"activated" for him or something.

-Mike

🔗dkeenanuqnetau <d.keenan@...>

🔗Margo Schulter <mschulter@...>

Dear Mike,

Thank you for a wonderful post that led me to do two experiments
I describe below on our 7:9:11 perceptions and idioms in 22-EDO
and 11-EDO. Your ideas were very much supported by these
experiments.

While you express concern about going off on tangents, or going
into "extra stuff," really I'm the one who should be concerned:
would that I had your gift of concisely and very readably
expressing my ideas in your relaxed and conversational way!

> Right, exactly. I note that it's specifically a sort of
> harmonic parsing that I'm talking about here, by the way. I
> wonder if a similar 11- or 13-limit perspective exists behind
> the various maqams, and if it's just waiting for someone to
> activate the sound in the right way for me.

An interesting question: in a given situation, is this parsing
mainly melodic (relating to melodic intervals) or vertical
("harmonic" in the sense of relating to simultaneous intervals or
sonorities)? With our 7:9:11 idioms, the vertical is obviously a
factor. With something like Lydian augmented #2, it could be
either: as pure melody as in traditional maqam, or as a mode for
polyphony of some kind.

> Right. I just wish I had some sort of idea about which specific
> phenomenon is involved when I throw these hand-wavy terms like
> parsing around. Since you seem to be familiar with the
> phenomenon I suggest, do you have any idea what exactly it
> means? Unlike when we talk about intonation, where there's lots
> of science to spare in terms of how things like the harmonic
> series works, I'm completely lost here. I don't even know where
> to start reading if I wanted to.

It's a term that I throw around with a very sketchy sense of
exactly what I'm talking about <grin>, but by analogy with
grammar and theories of language, e.g. parsing a sentence. Maybe
"pattern recognition" is the general idea.

> I think it may be related to that, although a more apt analogy
> for the phenomenon you describe would be a 350 cent interval
> flip-flopping between a major and minor third to my ear.
> However, as a side note, I have the feeling that every time I
> hear a tuning like 10-EDO, for instance, where I hear notes
> that are "between 12-EDO notes" and my brain tries to
> continually make sound like altered versions of their nearest
> 12-EDO equivalents, the phenomenon you describe is taking
> place. That is, I'm hearing broken pieces of the order in
> 12-EDO that are being reflected onto the tuning system, because
> it's just the best guess that my brain can come up with as to
> what's going on. It's like this picture:

> <http://images1.wikia.nocookie.net/__cb20060330234506/psychology/images/3/37/Emergence.jpg>

An open question: when you speak of "between 12-EDO notes," is it
specifically 12-EDO that is your standard? Or is more generally
certain diatonic patterns we might also find in 17-EDO, 31-EDO,
or 22-EDO -- but not in 10-EDO? This is sort of an Easley
Blackwood distinctions between "regular diatonic tunings" (or
reasonably close irregular variations) and others.

To take an extreme case I learned about from Robert Walker:

1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1

This parses easily for me as Dorian, a less usual tuning of a
familiar and recognizable pattern. So a 49/48 step at 36 cents
can be a very small semitone. And Paul Erlich found that he could
play the Aeolian mode in 22-EDO with near-6:7:9 chords and have
people enjoy it evidently much as with 12-EDO -- presumably
including some regular diatonic semitones at 55 cents.

For me, the thing with 10-EDO (or a 20-EDO superset) is that lots
of the interval sizes are familiar, but the nondiatonic structure
(by European or maqam standards) isn't. For example, a "septimal
Rast," as I might call it, at 16:14:13:12, is 0-231-359-498
cents -- as compared to 0-240-360-480 cents in 10-EDO. Yet the
latter sounds a bit curious. A difference is that 14:13:12 or
128-139 cents in JI gets compressed to 120-120 cents, and 7:6
thus to 240 cents. Harmonically, 240 cents can make a wonderful
"xeno-third" contracting to a unison, for example, but the
melodic compression of 14:13:12 to this degree might be what
seemed "curious" to me.

The ambiguity of an 11:9, say, is something that for me comes out
especially in a harmonic context, when it is "something else
again." Basically that's okay, because in a neomedieval setting
it's another kind of relatively complex third tending to some
interesting resolution. But add 13:8 at a tempered fourth higher,
and things get much more clear and defined. This feels a lot to
me like starting with a bare 9:7 expanding to a fifth, with the
third a bit blurry in a lower or middle register. Either moving
to a higher register, or adding the fourth above so we have
7:9:12 expanding to a full 2:3:4, clarifies things.

Melodically, however, something like 1/1-44/39-11/9-4/3 or
39:44:48:52 sounds just fine: "Ah, a good neutral step." And I'd
look at any maqam-based polyphony with directed progressions
involving thirds (major, minor, or neutral) as a kind of hybrid
phenomenon. So I suspect my parsing here is primarily melodic,
and might or might not be much like that of a Near Eastern
performer.

> If the tuning system is this picture, I'm only seeing the
> "spots." I have a very strong feeling that if I knew how to
> play the system "correctly," or if some Mozart came along and
> wrote the perfect 10-EDO concerto, it would somehow
> communicate the correct "perspective" from which to view the
> tuning system and I would then instead see the big picture of
> the 10-EDO "dalmatian." I'm using fluffy and hand-wavy terms
> to describe my perceptual experience here because again, I
> don't have any idea what sort of correlate in the literature
> exists to describe what I'm talking about. (Sorry if this is
> too much of a random tangent.)

It's not a random tangent at all, something I may be at a lot
more risk of indulging in. And the Mozart scenario is intriguing!
Fluffy and hand-wavy is the way I tend to talk and think also, so
we're on the same page there.

As I understand it, a lot of the RMP outlook -- and this isn't
unique to RMP, but one of the themes it tends to emphasize, if
I'm right -- is finding some kind of mode or tuning that will be
radically nondiatonic, and yet comparably compelling or readily
parsed, so to speak.

What I'm curious about: do you have a favorite or a number of
favorite "radically nondiatonic" systems -- say miracle or
porcupine -- where you have come to feel at home? This is what
Brian McLaren said about EDO's (I recall that he composed in
every one of them from 5-EDO to 53-EDO): Give yourself a bit of
time to get acclimated, and you'll find each has a logical and
increasingly familiar structure, whether you're listening or
yourself composing.

With something like 13-EDO, "the scale of 13 limmas," I found it
wasn't too hard to feel at home: I started by designing a few
timbres where 738 cents sounded like a tempered 3/2 (Bill
Sethares would have done it much more expertly!). From there, it
was mostly, "The intervals don't add up in the usual way, but the
more-or-less standard 13th-14th century progressions are neat!"
I'll spare you the stylistic mini-essay (another post), but I
felt quite at home.

In contrast, 14-EDO was beautiful but more challenging, because
individual cadences were great, and especially that 943-cent
interval, but the structure seemed further from "quasi-diatonic"
than 13-EDO. Maybe 14-EDO is my equivalent of a miracle subset:
lovely intervals, and a radically nondiatonic structure.

> It could be. I honestly think that it's something totally
> learned, like I managed to take the sound I was hearing and
> "relate it" to everything else I was hearing. The only problem
> is, I don't know what exactly the "relation" is that I've come
> up with! I think it might be something like: I heard the chord
> the second time and was able to plug it into the circle of
> fifths, or something like that. I don't know if that's
> actually it, but it's something like it.

Maybe this is a bit like learning a foreign language, where not
only the new sounds and structures but the element of "language
interference" or confusion caused by attempts to parse in
familiar patterns make the process challenging for an adult.

> I'm really fascinated by this aspect of music cognition
> because I feel like if I could identify and really nail down
> this sort of "relation," and figure out what it is, then it
> would become self-evident how the rest of music works. It
> might help to answer the question: when people experience the
> mess of sound that is music - why they don't always just hear
> it as raw sound, like a train horn? What is the experience
> that's being added on top of that?

Here it might be helpful to do studies of individual styles, and
then try to generalize. I can articulate some 14th-century
European relations, but other people know 18th-century European
relations better, or maqam, or gamelan, and so forth.

> I think the answer is that they hear the notes all being
> related to one another in a certain way. Keenan Pepper has
> added to this the Krumhansl-esque realization that "tonal"
> music, specifically, allows us not only to perceive these
> relations, but to predict somewhat which notes are most likely
> to occur next. These are very simple cognitive things we're
> talking about, it's easy to confirm with a bit of
> introspection that these "features" of our musical paradigm
> are indeed present, and it's easy to see how a much richer
> musical structure could emerge from such simple building
> blocks. However, I still feel like there's something else to
> add to that.

What I might get from Krumhansl (who I need to read) or Paul
Erlich's summary -- rightly or wrongly! -- is the idea that in a
system of polyphonic progressions, it helps to draw the
sonorities and progressions from the same material as the melodic
steps. For example, a piece of maqam-based polyphony in Rast may
sound more coherent if the sonorities use the regular steps of
Maqam Rast, or inflections already present for melodic reasons.

The open question is with something like the D-mode in Machaut.
Here the basic melodic material is an eight-note collection: the
six stable diatonic steps plus the fluid B/Bb. But some of the
most powerful expectations of "which notes are most likely to
occur next" come, for example, from hearing E-G#-C# and expecting
D-A-D, indeed the arch-typical 14th-century cadence.

It's like I'm Machaut's dog, and for the last 45 years he's had
me very well clicker-trained, a clicker being a device to
_quickly_ reward a dog for desirable behavior by signalling
something good that often (although not always) happens after it
sounds, like a treat. So when that E-G#-C# clicker sounds, I
anticipate the satisfying treat of a cadence to D-A-D. Or
likewise A-C#-F# to G-D-G, etc. And the inflections are a big
part of that "click."

But not every click means an actual treat, at least not
immediately. For example, we might have a section ending with an
interrupted cadence: E-G#-C#, and then a pause. Maybe the next
sonority is D-A-D; but maybe things stay up in the air for a bit
before we have a clear cadence. And relishing these pregnant
pauses and moments of uncertainty is part of the training also.

So maybe it's a question, for a given style, of going to the
right puppy classes.

[On a 7:9:11 perception of 0-436-764 cents in 11-EDO porcupine]

> I think so, though I'm not sure what you mean by perceiving
> something as 7:9:11 - I hear the intonation is reasonably
> concordant, if that's what you mean. However, the key thing to
> my ear is that the 7:9:11 is part of the 2 2 1 2 pentachord in
> 11-EDO. When you play it, it sounds like you're playing C D E
> F G, however the whole thing is stretched so that C-E-G is
> 7:9:11. In that context, this version of 7:9:11 is completely
> different than yours; to my ears, mine sounds like a really
> stretched major chord (though I didn't always use it that
> way), whereas yours has a very complex functional relationship
> that took me a second to grab a hold of.

Your post moved me to try two experiments.

The first was to try my 7:9:11 idiom in 22-EDO (usual harmonic
timbre), where the tuning of 0-436-764 cents is identical to
11-EDO, thus controlling that variable. I found that this shade
was charming, a stimulating variant on the _O Europae_ version in
Peppermint (0-437-784 cents), and maybe more accessible for
people getting acquainted with this progression both because
22-EDO is a more familiar system, and because comma niceties are
tempered out. It's a bit like getting acquainted with
16th-century keyboard counterpoint in meantone rather than
5-limit JI.

The second was to move to timbres meant to maximize the
consonance of 6/11 octave (655 cents) as a "quasi-fifth," and
explore the same 0-436-764 or 7:9:11 sonority. In the right
timbre and in a higher range, your "stretched major third and
effect" was very convincing! In the regular timbre during the
earlier 22-EDO experiment, I _could_ hear it that way to a
degree; but in this tweaked timbre and the right range, it was
immediate, obvious, and natural. almost like meantone!

Part of the range thing may be something I hear in a temperament
like MET-24 and a usual timbre also: 9/7 gets a lot mellower and
"aurally just (or near-just)" as we go higher, also a reason why
something like 7:12:18, e.g. in a medieval-style cadence to
1:2:3, can be breathtaking. The higher frequencies may reduce the
critical band kind of dissonance; one experiment, I recall, found
that when people heard the two tones of a 9/7 with one tone
presented in each earphone to avoid beats, it was hard to
distinguish from 5/4.

So maybe 0-436-764 cents in 11-EDO under the right circumstances
is a 14:18:21, with 9/7 shorn of its sensory dissonance, and 764
cents heard in context as a stretched fifth. That's just a guess,
but whatever it is, it's amazingly impressive!

In other 11-EDO contexts, again with these tweaked timbres, 764
cents as a dyad did sound like a small minor sixth, which could
contrast with and move to 655 cents as a weird kind of fifth.
For example, in a pelog of 0-109-436-655-764-1200 or 1-3-2-1-4,
the 655-cent step was clearly _kempyung_ (the "third" step,
equivalent to a European fifth), and 764 a "minor sixth" often
leading up by a "large major third" (436 cents) to the 2/1.

So the sensory consonance of 9/7 might be a dyadic effect (range,
and also timbre), while the 7:9:11 "stretched fifth" effect could
be triadic (i.e. 764 cents as part of a "major third and fifth").

In MET-24, I can sometimes hear a similar effect in the upper
register with C*-F-G# (0-438-772 cents) or the like -- which is
fine for a version of the progression I'm about to discuss from
_O Europae_. The experience of vagueness I get now and then in a
usual timbre, "This is like an active major third and fifth,"
seems connected with some kind of beating or difference tone
effect between the upper voices, maybe interacting with the 9/7.
When I do the resolution we're about to consider, this resolves
the intriguing ambiguity: "Well, it turns out to have been a
large major third plus a small minor sixth, with the third rising
to the fourth and the sixth falling to the fifth of a 6:8:9
suspension, etc."

Now an explanation of this neomedieval resolution and its 22-EDO
version that was the focus of my other experiment, starting with
links (especially for people who may be joining this thread) to
the original version in _O Europae_ we've been discussing in
Peppermint with 7:9:11 at 0-437-784 cents. The relevant passage
occurs at around 0:36, or measures 25-30 of the pdf score:

<http://www.bestII.com/~mschulter/O_Europae.mid>
<http://www.bestII.com/~mschulter/O_Europae.pdf>

Here's my 22-EDO version, with spellings based on a regular chain
of 709-cent fifths, intervals in cents, and some ratios the
sonorities might represent.

F E F
C# D C B C Bb
A Bb

764 709 764
436 491 273 218 273 55
0 55

11 9 9 12 9 3
9 8 7 9 7 2
7 6 6 8 6 2

When I tried this out, it was fine, a stimulating variation on my
usual tuning in something like Peppermint or MET-24. In those
temperaments, the opening minor sixth is very close to a just
11/7 (782 cents), and actually a tad larger at 784 or 785 cents.
Here it's 764 cents or a near-just 14/9 with a quality I find a
bit more languid and quite charming.

From this colorful 7:9:11, the small minor sixth descends to the
fifth while the 9/7 major third rises to the fourth, arriving at
a 6:8:9 suspension, here 0-491-709 cents. This is actually rather
concordant, but by historical association invites for me a
"neoclassical" resolution of the middle voice downward from a 4/3
to the 7/6 minor third of 6:7:9. Then, after a bit of
ornamentation for a momentary 8:9:12 sonority (also rather
consonant), the 6:7:9 resolves in a classical 14th-century
fashion to a fifth, with the minor third contracting to a unison
and the major third expanding to a fifth.

So for me, 22-EDO was another and quite successful shading of a
familiar idiom. Two intervals are especially important here: the
709-cent fifth and 55-cent diatonic semitone. The 709-cent fifth
clearly represents 3/2 and provides a stable goal for the whole
progression: directly for the cadential 6:7:9, and ultimately for
the 7:9:11 and everything else leading up to it. And the 55-cent
semitone is at once the defining difference and the vital melodic
bridge between the opening minor sixth at 764 cents and the fifth
to which it resolves at the beginning of the progression, as well
as a vital ingredient in the closing cadence from 6:7:9 to 3:2.

More generally, the idea of a minor sixth resolving by oblique
motion downward to a fifth is a basic classical idiom for me,
whether as a cadential gesture complete in itself as in some
13th-century pieces, or as part of a larger chain of cadential
events.

So a central context here is "small minor sixth descending to
fifth via a small semitone" -- in 22-EDO, 764 cents to 709 cents
via a 55-cent semitone. I came to 22-EDO with a scenario, and the
tuning system very nicely realized that scenario.

However, listening from a different angle, I also noticed how
0-436-764 cents might well be heard as a kind of stretched
sonority with major third and fifth. But the "minor sixth
inviting a resolution to stable fifth" scenario was so familiar
and attractive to me, based on my musical experience and
conditioning, that it took priority. And, as I was to learn in
the 11-EDO experiment described above, timbre and register are
both vital! With a usual timbre, the "stretched fifth" effect was
"a possible alternative way of hearing." With a tweaked timbre
and higher register, it was overwhelming!

> I may need to make some more examples of 7:9:11 in 11-EDO to
> make my point a little bit better on that, but to sum it up, I
> guess I'm suggesting that a separation of function and
> intonation exists. I think that many people start writing
> about how things like 7:9:11 sound or how they work musically,
> but they don't realize that in their minds, the thing they're
> calling 7:9:11 has a lot of associated "functional baggage"
> with it - because they're still force-fitting it to the
> functional structure that they know, mentally. Then, when they
> describe "here's how you use 7:9:11," they actually mean
> "here's how you use [combination of functions they're
> remembering which happen to be intoned as 7:9:11]." And one
> way to tease that apart is to show that a chord like 7:9:11
> can take on completely different functions in different scales
> that cause you to force-fit it differently!

Agreed, as our two examples may demonstrate quite effectively!
It's curious how "activating" the stretched-fifth thing took a
bit of focused listening in 22-EDO with a usual timbre, but
happened immediately in 11/22-EDO with the tweaked timbre and a
high register.

With my 7:9:11 progression, the historical "baggage" could at
least partly explain why you likewise needed a bit of listening
to tune into the musical grammar based on 14th-16th century
idioms in a curious mixture.

As to the "force-fitting thing," I might sum up the wisdom of
your last paraphrase by saying that "Here's how you use 7:9:11"
often means "Here's how I use 7:9:11 (in tunings with some room
to accommodate my particular baggage)."

For example, my neomedieval idiom gets accommodated just fine in
22-EDO, with its 709-cent fifth to which the 764-cent "minor
sixth" can resolve by way of a 55-cent semitone -- and likewise
the 491-cent fourth to which the 436-cent fourth can rise at
the opening of the progression to form a 6:8:9 suspension. But in
11-EDO, these bets are off: the "functional baggage" won't fit!

With your stretched-fifth idiom, I'd guess it could be there in
any tuning system with intervals around 7:9:11, and the right
timbre and register making the effect really unmistakeable.
Its recognition might be especially likely in 11-EDO because
there's no obvious "perfect fifth" to take priority.

> Underneath it all is still that raw sound - the 7:9:11 train
> horn - but I think that it's the stuff we're projecting on top
> of it that gives it its real musical flavor. Well, that's my
> thought, anyway.

My experiments very much confirm your view!

[On 327 cents in 11-EDO 0-436-764 cents representing 9:11]

> Yes, I agree with this, and furthermore I think that if you play a
> 4:7:9:11 chord and take the 4 and 7 away, that 327 cent interval
> remaining might sound a lot different if you're still remembering
> those notes "lingering" than it would if you just played it cold.

When I tried the 7:9:11 in 22-EDO, I did find that somehow that
upper interval of 327 cents had a different quality than as a
simple dyad. Alone, it seemed fine -- but somehow more "neutral"
or something in the three-voice sonority. And that might get back
to the attractive or cohesive power of isoharmonic sonorities,
whether treated as unstable or stable.

One question of this kind is whether 0-415-761-1037 cents could
approximate 11:14:17:20 (0-418-754-1035 cents). Here I'm using 346
cents, close to 11:9, as a 17:14 (336 cents) -- a bit analogous,
although the difference is 10 cents rather than 20 cents! In
other words: if 11:14:20 are all reasonably close, can 17 be off
by 10 cents and still have a 11:14:17:20 synergy?

> Well, I hope I haven't added too much with the extra stuff
> above! But thanks for taking the time to respond to my
> post. The reason I'm curious to ask you about this stuff is
> that you've studied the time period when these sort of
> functional relationships shifted dramatically, e.g. major
> thirds stopped being "avoid intervals" and started being
> obvious consonances. I have no idea how someone could simply
> adjust their brain to make that happen, but I wonder if it's
> the same sort of "activation" process I discussed above with
> something like lydian augmented #2. Dunstable simply got the
> major chord "activated" for him or something.

Your "extra stuff" is invaluable and absolutely fine: it's mine
I'm worried about, wondering if I've gone into too much detail
that might not communicate well without musical examples (a good
next step).

To cut to the chase on one fine point, I'd say that major thirds
in the 13th-14th centuries, or minor sevenths for that matter for
a composer like Perotin or Machaut who evidently likes them a
great deal, are analogous to seventh chords in Bach:
"avoid-in-closing intervals" that signal pleasant and creative
tension and motion rather than arrival and repose.

But with Dunstable for major thirds, or some of the 19th-century
Romantic composers for sevenths, the sense of motion and tension
is getting more and more "floating" and laid-back, so to speak.
And intonations may change in the process: the stable 4:5:6 of
the Renaissance, or Paul Erlich's stable 4:5:6:7 tetrad.

And new patterns do get "activated." A quick example: the
movement by Dunstable and Dufay and others toward restful thirds
leads to a situation where a third can be a relatively and
eventually quite stable resolution of a more tense interval, and
specifically of a diminished fifth, e.g. B-F to C-E. Zarlino
tells us that this is the one dissonance -- unlike seconds or
sevenths -- that can be written "in the same percussion" without
benefit of a suspension.

So while the increasingly "laid-back" nature of major thirds in
the 15th-century is at first partly a matter of color, and indeed
may be seen as a _blurring_ of 14th-century directedness where
these thirds are vital in defining cadential tension (Richard
Crocker's insight in the 1960's), it leads to a new kind of
directedness in 16th-century polyphony, defining one type of
frequent cadence. And in later 17th-19th century tonality, this
theme of "tritone tension" becomes a central organizing
principle.

With many thanks,

Margo

🔗Margo Schulter <mschulter@...>

🔗Mike Battaglia <battaglia01@...>

Hello Margo, and sorry for the delayed response; due to a death in the
family I've been busier than usual and family is coming in all this
week. I'll respond as quickly as I can...

On Wed, Dec 12, 2012 at 5:07 AM, Margo Schulter <mschulter@...>
wrote:
>
> > Right, exactly. I note that it's specifically a sort of
> > harmonic parsing that I'm talking about here, by the way. I
> > wonder if a similar 11- or 13-limit perspective exists behind
> > the various maqams, and if it's just waiting for someone to
> > activate the sound in the right way for me.
>
> An interesting question: in a given situation, is this parsing
> mainly melodic (relating to melodic intervals) or vertical
> ("harmonic" in the sense of relating to simultaneous intervals or
> sonorities)? With our 7:9:11 idioms, the vertical is obviously a
> factor. With something like Lydian augmented #2, it could be
> either: as pure melody as in traditional maqam, or as a mode for
> polyphony of some kind.

The best way for me to describe it is that it's "scalar." I have very
clear scale degrees in my head which get activated when I hear musical
intervals. You might call these scale degrees "functions" instead, or
"categories" or something. Perhaps this is "melodic," I dunno - I
don't know how to use it to write catchy melodies, that's for sure :)

But it's not intonational, and that's what I'm sure of, unless there's
some secret way in which it is that I'm not aware of. Minor chords are
"minor" to me whether or not they're 10:12:15 or 6:7:9 or etc; the
minor "feeling" is preserved regardless, though there are some clear
subtle differences in mood. And all of my musical emotions seem to be
dependent on these interval functions first and foremost, and the
intonations of them secondly.

Also, I'll go out a limb and say that this phenomenon accounts for
about 90% of my musical experience, whereas ratios and intonation
account for 10% of it. So it seems to me that if I want to experience
new and fundamentally different musical emotions, which is what got me
into this field to begin with, I need to learn how to perceive a
radically new set of interval functions, however I choose to intone
them. I've spent the past few years learning all of this theory which
is built around ratios and temperaments and intonation and such, but
now it's clear that in order for me to get to the point I want to, I
need to learn to hear music relative to a completely different
foundation of functions and scales and dyads.

> It's a term that I throw around with a very sketchy sense of
> exactly what I'm talking about <grin>, but by analogy with
> grammar and theories of language, e.g. parsing a sentence. Maybe
> "pattern recognition" is the general idea.

I agree with this, but I wonder what's being recognized...

> An open question: when you speak of "between 12-EDO notes," is it
> specifically 12-EDO that is your standard? Or is more generally
> certain diatonic patterns we might also find in 17-EDO, 31-EDO,
> or 22-EDO -- but not in 10-EDO? This is sort of an Easley
> Blackwood distinctions between "regular diatonic tunings" (or
> reasonably close irregular variations) and others.

Half of it seems to be 12-EDO specifically, and half of it seems to be
diatonic/chromatic/chain of fifths stuff specifically. When I first
started out with different tunings, it was the former - 7/4 sounded
"crunchy but wrong" to me at first, because I knew that to split it
into two fourths, those fourths would also have to be flat, and then
the LLs tetrachord would have to be compressed flat, but that would
make the major thirds so flat that three wouldn't equal an octave...
etc. I just intuitively knew that it would make everything break.

Sooner or later I got over that, by learning to allow the notes in my
head to deform a bit to accomodate the new intervals, So I started
learning to hear things like 7/4 as a type of minor seventh, and 11/8
as a type of aug 4, etc.

These days, it still seems to be diatonic based though, or maybe I
should say "chromatic-based." If I play in something like 16-EDO, I
generally just snap it into my head in terms of some 12 note chromatic
scale. But, if I play in 19-EDO, something strange happens. I can now
hear a very clear difference between sharps and flats. E-A# is lydian,
and E-Bb is diminished. Also, the diesis is now the same size as the
chromatic semitone, so I can play resolutions that exploit that and
move from things like A#-Bb and hear the whole structure of the
tonality swing waaaay across the circle of fifths to the other side.

I think it's pretty safe at this point to say that I have a set of
19-EDO "categories" which are "perfect" in my head just like 12-EDO's
are; they developed naturally as an outgrowth of the extended meantone
structure. So maybe we can say that it used to be 12-EDO specifically,
but now it's something more general, like diatonic-based or
meantone-based or chain-of-fifths based or something. I really don't
know.

> To take an extreme case I learned about from Robert Walker:
>
> 1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
>
> This parses easily for me as Dorian, a less usual tuning of a
> familiar and recognizable pattern. So a 49/48 step at 36 cents
> can be a very small semitone. And Paul Erlich found that he could
> play the Aeolian mode in 22-EDO with near-6:7:9 chords and have
> people enjoy it evidently much as with 12-EDO -- presumably
> including some regular diatonic semitones at 55 cents.

Right, well, how about 1/1 12/11 8/7 21/16 3/2 12/7 7/4 2/1? Because
that's clearly Dorian for me too, except that now 8/7 is a subsubminor
third instead of a major second of some sort.

If you can also hear 1/1-8/7-3/2 as minor in the context of the above
scale, instead of sus2, then all I can say is that I think that's a
really, really significant thing, not just a trivial sort of
curiosity. In my mind, that completely makes the concept of "learning
how different JI chords" work meaningless. How does 1/1-8/7-3/2 work?
Is it a sus2 chord, as in your scale, or a minor chord, as in my
scale? When I "detwelvulate" my ears to hear things in some supposedly
more natural way, what's the "correct" way to hear the chord? It's now
obvious that the answer to every single one of these questions is
"mu," to quote a famous Zen koan.

> For me, the thing with 10-EDO (or a 20-EDO superset) is that lots
> of the interval sizes are familiar, but the nondiatonic structure
> (by European or maqam standards) isn't. For example, a "septimal
> Rast," as I might call it, at 16:14:13:12, is 0-231-359-498
> cents -- as compared to 0-240-360-480 cents in 10-EDO. Yet the
> latter sounds a bit curious. A difference is that 14:13:12 or
> 128-139 cents in JI gets compressed to 120-120 cents, and 7:6
> thus to 240 cents. Harmonically, 240 cents can make a wonderful
> "xeno-third" contracting to a unison, for example, but the
> melodic compression of 14:13:12 to this degree might be what
> seemed "curious" to me.

Well, if you already know how to hear 240 cents as a type of third, I
guess the thing I wrote above will be old hat to you :)

> The ambiguity of an 11:9, say, is something that for me comes out
> especially in a harmonic context, when it is "something else
> again." Basically that's okay, because in a neomedieval setting
> it's another kind of relatively complex third tending to some
> interesting resolution. But add 13:8 at a tempered fourth higher,
> and things get much more clear and defined.

I totally agree with this too. 16:24:26:39 is magic. But, I still
don't know if the ratios I just laid out have anything to do with this
magicness.

> This feels a lot to
> me like starting with a bare 9:7 expanding to a fifth, with the
> third a bit blurry in a lower or middle register. Either moving
> to a higher register, or adding the fourth above so we have
> 7:9:12 expanding to a full 2:3:4, clarifies things.

Is the root staying the same in both cases?

> As I understand it, a lot of the RMP outlook -- and this isn't
> unique to RMP, but one of the themes it tends to emphasize, if
> I'm right -- is finding some kind of mode or tuning that will be
> radically nondiatonic, and yet comparably compelling or readily
> parsed, so to speak.

The RMP outlook doesn't really address these questions. It just
assumes that we want to find temperaments with chords that are close
to small-integer ratios. It has nothing at all to say about the
phenomenon with 8/7 that I presented above. It doesn't know what
parsing is. It just tells you how to find tunings that have nice
tempered approximations to simple JI ratios, and lots of them.

Quite frankly, I think that the existing RMP completely fails to
explain this side of music, which to my ears comprises about 90% of
what I hear, and that we're in need of some new basic realization to
begin a new point of departure, just like Parncutt's work (among
others) served as a point of departure for the existing RMP.

> What I'm curious about: do you have a favorite or a number of
> favorite "radically nondiatonic" systems -- say miracle or
> porcupine -- where you have come to feel at home?

A few. All of them are MOS's with more large than small steps, and
where the ratio of large to small step is close to 2/1.

Machine in 16-EDO is fairly amazing; I can get into it pretty easily.
That would be the scale 3 3 1 3 3 3.

Also, I've found that I'll have these moments in mavila temperament in
16-EDO sometimes, where I'll get a glimmer of this new sort of "order"
lurking behind things, and it all makes sense - but then it goes away
and I start imposing the 12-EDO order on it instead. It's very weak.
Mavila[7] in 16-EDO is 2 3 2 2 3 2 2, and mavila[9] is 2 2 2 1 2 2 2 2
1. It's sometimes hard for me to tell the 2's apart from the 3's in
mavila[7], and the 2's apart from the 1's in mavila[9]; this seems to
interfere with making a very "clear" and "stable" set of scale degrees
for me to be able to comprehend. Strangely, mavila[9] in 25-EDO is
much clearer; it's now 3 3 3 2 3 3 3 3 2, though I don't know what the
source of this stability is.

Sensi temperament doesn't have any great scales that I like, but the
basic idea is that two 9/7's makes a 5/3 - huge chains of tempered
9/7's like that sound amazing to me, like deep space or wind or
something. I have no idea why it sounds that way, but it does; it's a
completely new, "xenharmonic" sound to me. In contrast, Orwell has two
7/6's tempered to make an 11/8, and then three 7/6's tempered to make
a 8/5. Orwell just sounds weird and diminished to me; it's very dark
and just sounds diminished and messed up. I doubt that there's some
magic pattern in the ratios which makes it work that way.

Blackwood temperament in 15-EDO has some very very very crazy "tonal"
effects going on because the circle of fifths is five notes wide. That
really makes everything sound very "related" and this creates some
sort of "intense" emotional effect. However, the scale 2 1 2 1 2 1 2 1
2 1, which is blackwood[10], just sounds weird and noisy to me; I can
never tell the 2's from the 1's. In 20-EDO, it becomes 3 1 3 1 3 1 3 1
3 1, and now everything is much clearer, despite the chords not being
major and minor anymore.

In terms of chromatic scales, 15-EDO seems to be the most "stable"
nondiatonic chromatic scale out there that I've found other than
12-EDO. In 15-EDO, I need to start grouping 7/6 and 8/7 as different
versions of the same "thing," if you will, whereas 6/5 and 7/6 are now
completely and obviously different. I "sort of" get the hang of this.

One of the craziest scales I've ever heard is Bohlen-Pierce. I've
spent some time in BP and while I still don't have the hang of tritave
equivalence, I'm now fully aware that something about BP is really
crazy. If I spend enough time in it, I'll start confusing 7/3 for 3/1.
This is a mistake I would never make in real life, but after prolonged
exposure to BP, my brain seems to start adjusting in very strange
ways. I'm not sure what it means.

All of those are sort of "weak." Machine is probably the strongest of
them. At any rate, that's all I can think of so far.

> This is what
> Brian McLaren said about EDO's (I recall that he composed in
> every one of them from 5-EDO to 53-EDO): Give yourself a bit of
> time to get acclimated, and you'll find each has a logical and
> increasingly familiar structure, whether you're listening or
> yourself composing.

I play in either 19-EDO, 22-EDO, and 16-EDO almost every day, and I
have for a year or two now. Maybe I'll regret saying this, but I
expect that I would perform very well, comparatively speaking, on any
sort of ear training test for all of these, demonstrating my
familiarity with those tunings in a measurable way. Of course, it
could well be the case that I perform comparatively poorly, at which
point I'd be happy to eat those words and humbly ask what sorcery
you're all partaking in which allows you to hear completely novel
22-EDO interval categories, where 7/6 and 6/5 are as different as 6/5
and 5/4, which I have yet to discover.

The more time I spend in these tunings, the more I realize that
there's a deeper theory that there is to stumble upon, which we don't
currently know. I think that if we administered an ear training test
to a lot of people, the people who do best on the ear training test
will be the people who also have had the same realization. I have come
to conclude that in situations where 6:7:9 and 10:12:15 sound the
same, they're 'supposed' to sound the same. Then, once that's
accepted, one can then start looking for times when they're 'supposed'
to sound different. For instance, hedgehog temperament in 22-EDO is a
scale which suddenly makes it very easy to distinguish between 7/6 and
6/5, and it's given by 3 3 2 3 3 3 2 3. I have no idea how or why this
works; I feel like I've made some progress towards understanding this,
but that I only know half of the picture.

> With something like 13-EDO, "the scale of 13 limmas," I found it
> wasn't too hard to feel at home: I started by designing a few
> timbres where 738 cents sounded like a tempered 3/2 (Bill
> Sethares would have done it much more expertly!). From there, it
> was mostly, "The intervals don't add up in the usual way, but the
> more-or-less standard 13th-14th century progressions are neat!"
> I'll spare you the stylistic mini-essay (another post), but I
> felt quite at home.

I would be very interested to hear what you came up with in this tuning!

> In contrast, 14-EDO was beautiful but more challenging, because
> individual cadences were great, and especially that 943-cent
> interval, but the structure seemed further from "quasi-diatonic"
> than 13-EDO. Maybe 14-EDO is my equivalent of a miracle subset:
> lovely intervals, and a radically nondiatonic structure.

Yes, I absolutely agree with this. I didn't mention 14-EDO above
because I'm still new to it, but I think there's magic to it. In
14-EDO, 5/4 and 6/5 become two different intonations of the same
thing, and 7/6 and 8/7 become two different intonations of the same
thing. Also, 14-EDO magically contains all of the touch tone intervals
on your phone. There is definitely something interesting about 14-EDO.
However, I have trouble hearing 5/4 and 6/5 as just two different
versions of the 14-EDO neutral third.

> Maybe this is a bit like learning a foreign language, where not
> only the new sounds and structures but the element of "language
> interference" or confusion caused by attempts to parse in
> familiar patterns make the process challenging for an adult.

Yes, I think so; specifically I think this happens when I try to play
in really radically nondiatonic tuning systems. I just keep hearing
12-EDO patterns and I'm not sure what sorts of things to play which
convey to my brain a really strong, and most importantly "stable",
sort of novel pattern in the new tuning system, so that my brain is
likely to gravitate to this new pattern and not the old broken 12-EDO
patterns it ends up hearing instead, corresponding to my brain trying
its best and giving up.

> > I'm really fascinated by this aspect of music cognition
> > because I feel like if I could identify and really nail down
> > this sort of "relation," and figure out what it is, then it
> > would become self-evident how the rest of music works. It
> > might help to answer the question: when people experience the
> > mess of sound that is music - why they don't always just hear
> > it as raw sound, like a train horn? What is the experience
> > that's being added on top of that?
>
> Here it might be helpful to do studies of individual styles, and
> then try to generalize. I can articulate some 14th-century
> European relations, but other people know 18th-century European
> relations better, or maqam, or gamelan, and so forth.

What sorts of relations do people hear?

> What I might get from Krumhansl (who I need to read) or Paul
> Erlich's summary -- rightly or wrongly! -- is the idea that in a
> system of polyphonic progressions, it helps to draw the
> sonorities and progressions from the same material as the melodic
> steps. For example, a piece of maqam-based polyphony in Rast may
> sound more coherent if the sonorities use the regular steps of
> Maqam Rast, or inflections already present for melodic reasons.

Hm, I'm not sure I understand this - what do you mean to draw the
sonorities from the same material as the melodic steps, exactly?

> It's like I'm Machaut's dog, and for the last 45 years he's had
> me very well clicker-trained, a clicker being a device to
> _quickly_ reward a dog for desirable behavior by signalling
> something good that often (although not always) happens after it
> sounds, like a treat. So when that E-G#-C# clicker sounds, I
> anticipate the satisfying treat of a cadence to D-A-D. Or
> likewise A-C#-F# to G-D-G, etc. And the inflections are a big
> part of that "click."

Yes, this is exactly how I hear it too! But I wonder, is classical
conditioning all that there is to resolutions, or is there some
scale-theoretic reason which makes the tritone resolution more
"forceful" than the double leading tone cadence? In other words, if we
imagine that the listener is constantly maintaining a copy of the
scale in his/her head at all times, is there some scale-theoretic
reason why the tritone resolution might "refresh" more of this scale
than the double leading tone resolution, or is it just classical
conditioning?

> The second was to move to timbres meant to maximize the
> consonance of 6/11 octave (655 cents) as a "quasi-fifth," and
> explore the same 0-436-764 or 7:9:11 sonority. In the right
> timbre and in a higher range, your "stretched major third and
> effect" was very convincing! In the regular timbre during the
> earlier 22-EDO experiment, I _could_ hear it that way to a
> degree; but in this tweaked timbre and the right range, it was
> immediate, obvious, and natural. almost like meantone!

So if I understand correctly, you flattened the third harmonic of the
timbre to 655 cents, and then 764 cents sounded like a fifth...? I
would have expected that if anything, sharpening the third harmonic to
764 cents would have made this happen more...

While you're in 22-EDO, you may want to play in hedgehog, which is 3 3
2 3 3 3 2 3. If you're like me, that 3 3 2 is very clearly some sort
of do re mi fa, except the "mi" is 6/5, and the "fa" is 9/7, which is
apparently a shade of perfect fourth today...

> Part of the range thing may be something I hear in a temperament
> like MET-24 and a usual timbre also: 9/7 gets a lot mellower and
> "aurally just (or near-just)" as we go higher, also a reason why
> something like 7:12:18, e.g. in a medieval-style cadence to
> 1:2:3, can be breathtaking. The higher frequencies may reduce the
> critical band kind of dissonance; one experiment, I recall, found
> that when people heard the two tones of a 9/7 with one tone
> presented in each earphone to avoid beats, it was hard to
> distinguish from 5/4.

This is probably part of it; it's also been shown that the tendency
towards virtual pitch integration increases as the VF gets higher in
pitch. So as the 9/7 gets higher, there may be more of a tendency to
hear it as (1):7:9, where the (1) is a virtual fundamental which you
can hear if you focus on it. For me, I need only to imagine the 9/7 as
part of a rootless (4):7:9 to totally transform the sound of it to
something much more musically consonant.

> So maybe 0-436-764 cents in 11-EDO under the right circumstances
> is a 14:18:21, with 9/7 shorn of its sensory dissonance, and 764
> cents heard in context as a stretched fifth. That's just a guess,
> but whatever it is, it's amazingly impressive!

Perhaps... I would suspect, however, that this sort of analysis makes
ratios into a procrustean bed we're fitting something to, whereas the
fundamental thing we're hearing is something much deeper and related
to the properties of scales. I think that 14:18:21 sounds like
root-major third-fifth, not the other way around, with the latter
being the thing which is truly "fundamental" - at least to a 12-EDO
listener. That being said, I can't claim that I really know what's
going on.

> In other 11-EDO contexts, again with these tweaked timbres, 764
> cents as a dyad did sound like a small minor sixth, which could
> contrast with and move to 655 cents as a weird kind of fifth.

In the context of the scale 2 2 1 2 2 2, if I go up to the octave and
then go two whole steps down, it sounds like a small minor sixth,
whereas if I go up from the tonic via 2 2 1 2, it sounds like a large
perfect fifth - to my ears, anyway.

> In MET-24, I can sometimes hear a similar effect in the upper
> register with C*-F-G# (0-438-772 cents) or the like -- which is
> fine for a version of the progression I'm about to discuss from
> _O Europae_. The experience of vagueness I get now and then in a
> usual timbre, "This is like an active major third and fifth,"
> seems connected with some kind of beating or difference tone
> effect between the upper voices, maybe interacting with the 9/7.
> When I do the resolution we're about to consider, this resolves
> the intriguing ambiguity: "Well, it turns out to have been a
> large major third plus a small minor sixth, with the third rising
> to the fourth and the sixth falling to the fifth of a 6:8:9
> suspension, etc."

So you're saying that at first, it seems like some sort of stretched
major chord, right? But then, once you resolve it in the manner you
describe, your perception of the chord changes -retroactively-, right?
I hadn't thought about this before, but I totally agree that this
happens, though not for me this time - it required repeated listens
for me to "decipher" what was happening.

Ironically, after this discussion and a year or so's worth of parallel
offlist discussions with Keenan Pepper, I'm now starting to feel like
we have -too many- different leads! But the question that I have which
lurks underneath all of this is: how are all of these different
concepts - classical conditioning of resolutions, scale-based and
non-ratio-based "function" assignment of intervals, retroactive
reidentification of functions, ease of distinguishing intervals in a
scale if they're all as far apart from one another as possible
(haven't mentioned this yet but it's true), formation of a "map" of
said functions (haven't talked about this yet either but it's also
true) in a novel tuning system - what is the underlying theme that
unifies all of these things? What are we even talking about? I've got
a "eureka" ready to go if anyone can figure out what part of reality
we're even discussing here...

> From this colorful 7:9:11, the small minor sixth descends to the
> fifth while the 9/7 major third rises to the fourth, arriving at
> a 6:8:9 suspension, here 0-491-709 cents.

This is how I heard it after four or five listens. What happened with
me when I listened to the segment that you describe is - at first, I
clicked right to the chord, and heard it as a completely foreign and
atonal sort of thing; couldn't figure it out at all. Then I heard it
resolve to the A-D-E and A-C-E, and knew I was at the right place.
Then I tried listening again, a few bars earlier, and still it sounded
very strange when that chord came up. Then after a few more listens, I
started hearing the chord as a sort of augmented chord, which then
resolved to A-D-E and then to A-C-E. Now that I know what it is, I can
immediately snap it into that sort of augmented framework. (You may be
labeling the note names differently; I'm just describing what I hear.)

> However, listening from a different angle, I also noticed how
> 0-436-764 cents might well be heard as a kind of stretched
> sonority with major third and fifth. But the "minor sixth
> inviting a resolution to stable fifth" scenario was so familiar
> and attractive to me, based on my musical experience and
> conditioning, that it took priority.

I am personally very familiar with this phenomenon. I think that my
problem is that when I play in radically nondiatonic scales, there are
a lot of 12-EDO things which constantly "take priority" in my head. I
suspect that there's some way to play things in a tuning system like
10-EDO, for instance, which would be so "strong" due to unknown
factors that almost any listener hearing would have to interpret those
in this novel "strong" way instead of some old 12-EDO "best I could
do" sort of way. If only I knew how...

> And, as I was to learn in
> the 11-EDO experiment described above, timbre and register are
> both vital! With a usual timbre, the "stretched fifth" effect was
> "a possible alternative way of hearing." With a tweaked timbre
> and higher register, it was overwhelming!

Well, I note that I hear the stretched fifth even with normal timbres
- in the scale 2 2 1 2 2 2, it's fairly hard to avoid for me. What
happens if you use a very relaxed timbre, like an ocarina sound?

I know that for me, even with normal timbres, if I load up 2 2 1 2 2
2, play a root-third-fifth triad and transpose it up and down the
scale, I hear something like C-E-G -> D-F-A -> E-G-C -> F-A-D -> etc,
sort of a gospel-ish hexatonic thing. I'm aware that it's not really
the 12-EDO version of that, but that's the resemblance that I sort of
hear.

Did you hear the tonic chord in "tonal study in orgone" as being some
sort of root-major third-fifth? That was also 7:9:11.

> With my 7:9:11 progression, the historical "baggage" could at
> least partly explain why you likewise needed a bit of listening
> to tune into the musical grammar based on 14th-16th century
> idioms in a curious mixture.

Agreed. I think that about answers that question as best as it can be
answered; it's definitely true that your past experiences biased you
towards parsing the chord in a certain way, whereas it took me a
second to catch on. That being said, I wish I knew how to model this
"parsing" process, or even knew what it was parsing, or what I'm
recognizing things as - what the building blocks of something like
"musical grammar" are.

> As to the "force-fitting thing," I might sum up the wisdom of
> your last paraphrase by saying that "Here's how you use 7:9:11"
> often means "Here's how I use 7:9:11 (in tunings with some room
> to accommodate my particular baggage)."

Yes, I absolutely agree with this. I strongly suspect that
microtonalists are secretly manipulating 12-EDO/western/common
practice/etc functions and categories in their heads when they play in
extended JI, but then ascribe those powers to something like a set of
ratios. This is just a hunch, though; there may be some among us who
truly do hear a completely novel set of interval functions which are
radically unlike anything we're used to in 12-EDO, and where 7/6 is
its own novel interval class, not just a sort of minor third. However,
if these sorts of "enlightened" listeners are among us, they seem to
be the silent lurker type...

> For example, my neomedieval idiom gets accommodated just fine in
> 22-EDO, with its 709-cent fifth to which the 764-cent "minor
> sixth" can resolve by way of a 55-cent semitone -- and likewise
> the 491-cent fourth to which the 436-cent fourth can rise at
> the opening of the progression to form a 6:8:9 suspension. But in
> 11-EDO, these bets are off: the "functional baggage" won't fit!

Right. Not only won't it fit, but you can come up with new things that
do fit which effectively supercede the old things, which is what I
tried to do in my tonal study for orgone. Now, to figure out how to
supercede 12-EDO when I play in 15-EDO...

> > Yes, I agree with this, and furthermore I think that if you play a
> > 4:7:9:11 chord and take the 4 and 7 away, that 327 cent interval
> > remaining might sound a lot different if you're still remembering
> > those notes "lingering" than it would if you just played it cold.
>
> When I tried the 7:9:11 in 22-EDO, I did find that somehow that
> upper interval of 327 cents had a different quality than as a
> simple dyad. Alone, it seemed fine -- but somehow more "neutral"
> or something in the three-voice sonority. And that might get back
> to the attractive or cohesive power of isoharmonic sonorities,
> whether treated as unstable or stable.

This is 100% in line with my experience, though I experience the same
thing with pretty much any sort of chord containing that dyad as
though it were 11/9. For instance, 8:9:10:11:12 is an example of such
a chord, as is 8:9:11:12 or 8:9:10:11. This comes up a lot in
porcupine. If I play 8:9:10:11:12 and then take everything but the
11/9 away, it definitely starts sounding "neutral" all of a sudden.
I'm completely certain that I have no idea how that works, though I do
know it also works with non-isoharmonic chords, so long as I perceive
the "root" of the 327 cent interval as being a 9/8 below the lowest
note...

> One question of this kind is whether 0-415-761-1037 cents could
> approximate 11:14:17:20 (0-418-754-1035 cents). Here I'm using 346
> cents, close to 11:9, as a 17:14 (336 cents) -- a bit analogous,
> although the difference is 10 cents rather than 20 cents! In
> other words: if 11:14:20 are all reasonably close, can 17 be off
> by 10 cents and still have a 11:14:17:20 synergy?

In terms of the actual effect caused by isoharmonic chords, which is
what we were calling "periodicity buzz," it'd have some sort of
effect, but the 17 cents being off would definitely make the whole
thing not seem to buzz in sync with itself. Though, after everything
above, it now seems to me to be rather unlikely that the fundamental
qualia of our musical experience are ratios, or that we really
fundamentally hear things "as" ratios at all...

> Your "extra stuff" is invaluable and absolutely fine: it's mine
> I'm worried about, wondering if I've gone into too much detail
> that might not communicate well without musical examples (a good
> next step).

I feel the same way. For me, my head has been a chaotic swirl of these
ideas for the past year; I'm desperately trying to find a way to
organize it all, but I'm not sure where to start. Maybe it would be a
good idea to make a list of these various phenomena or something.

One thing I can definitely say is that I'm very appreciative of your
ability to introspectively identify exactly what you're hearing in
such an unconfused manner, and to relate it to all of these other
things you've heard before, such as pelog and maqam scales, and then
to state all of those connections clearly enough that I can understand
them from halfway across the country and over a mailing list. I think
I understand your thought process very well when you constantly
identify these intervals which you hear as common across tuning
systems, such as a certain type of pelog fifth which is also sort of
close to 16/11 which also makes an appearance in some mode of rast, or
something like that.

I've been writing the same sorts of descriptions of the things I hear
on here and on other fora, except instead of relating the stuff I'm
hearing to something like maqams or pelog, it's to MODMOS's of
porcupine, or to 19-EDO, or to 16-EDO, or to some sort of extended
12-EDO jazz harmony, which are the musical systems I've immersed
myself in daily for the past few years. However, I've at times been
frustrated, because I rarely seem to find anyone who can understand a
thing I'm saying. Some people on these lists claim to not hear a
network of very clear interval functions, but rather they "just hear
sound"; whether that's true, or whether they're just not sure how to
express the functions they do here, I don't know, but I do know that
I've found the process of communication on this sort of level to be
difficult at times. It occurred to me in this discussion that I should
perhaps not take for granted that I am now talking to someone who
seems to think in exactly the same way that I do and who understands
what I'm saying and can respond in kind, and I am grateful for that. I
also think it's very interesting to see you going through the same
sort of inter-tuning comparison process that I've also come to, but
with a totally different set of tunings as a base than those which I
have.

> To cut to the chase on one fine point, I'd say that major thirds
> in the 13th-14th centuries, or minor sevenths for that matter for
> a composer like Perotin or Machaut who evidently likes them a
> great deal, are analogous to seventh chords in Bach:
> "avoid-in-closing intervals" that signal pleasant and creative
> tension and motion rather than arrival and repose.

I think I can sort of see what you mean; I'm thinking of Machaut's
Kyrie here, from the Messre de Nostre Dame, and I guess it's true that
he never really ends a phrase on a major chord, does he? Maybe that's
a key aspect of the signaling. Major chords are always sort of the
comma in the middle of a sentence for him, but never a period and
full-stop. But I have to say that the notion that major thirds weren't
just resolved consonances didn't really click for me until I heard
Marcel Peres' version of Machaut's Credo; he sings it in a much lower
register than usual, so that the major thirds start to fall within a
critical band and sound somewhat rough, and adds lots of vibrato to
the thirds. In other words, he does every single thing that you could
possibly do to make a major chord not just sound like a static, calm,
blending 4:5:6 chord. And indeed this works very well: just like
sensory concordance can enhance grammatical musical consonance, Peres
uses sensory discordance to enhance grammatical musical dissonance.
That really blew my mind the first time I heard it; to hear the thirds
suddenly becoming dissonant in a very real, visceral sense.

> And new patterns do get "activated." A quick example: the
> movement by Dunstable and Dufay and others toward restful thirds
> leads to a situation where a third can be a relatively and
> eventually quite stable resolution of a more tense interval, and
> specifically of a diminished fifth, e.g. B-F to C-E. Zarlino
> tells us that this is the one dissonance -- unlike seconds or
> sevenths -- that can be written "in the same percussion" without
> benefit of a suspension.

What does this mean exactly - that it doesn't have to go B-F -> C-F ->
C-E, or something like that?

> So while the increasingly "laid-back" nature of major thirds in
> the 15th-century is at first partly a matter of color, and indeed
> may be seen as a _blurring_ of 14th-century directedness where
> these thirds are vital in defining cadential tension (Richard
> Crocker's insight in the 1960's), it leads to a new kind of
> directedness in 16th-century polyphony, defining one type of
> frequent cadence. And in later 17th-19th century tonality, this
> theme of "tritone tension" becomes a central organizing
> principle.

But why, I wonder, are tritones tense?

I've experimented with diatonic scales from 7-EDO to 5-EDO. One really
interesting one is in 52-EDO, 10 10 1 10 10 10 1. In this scale, it
takes me a second to hear the tiny 26 cent half steps as being half
steps, but I inevitably catch on and the whole thing snaps into place.
Once it does, the diminished fifth is dissonant and enjoys resolution
to the major third, as it usually does; if I play I ii iii IV V vi
vii° I, that vii° is definitely recognizable as a diminished chord to
me. However, the diminished fifth is 508 cents; it's now close to 4/3
and is in fact a better approximation to 4/3 than the actual perfect
fourth of the scale.

Thanks,
Mike

🔗dkeenanuqnetau <d.keenan@...>

🔗Margo Schulter <mschulter@...>

> Hello Margo, and sorry for the delayed response; due to a
> death in the family I've been busier than usual and family is
> coming in all this week. I'll respond as quickly as I can...

Please let me begin, of course, by wishing you my most sincere
condolences for your family tragedy, and by thanking for a most
generous reply while noting that I'm the one who took longer to
reply in turn.

> The best way for me to describe it is that it's "scalar." I
> have very clear scale degrees in my head which get activated
> when I hear musical intervals. You might call these scale
> degrees "functions" instead, or "categories" or something.
> Perhaps this is "melodic," I dunno - I don't know how to use it
> to write catchy melodies, that's for sure :)

I've often read that really good melody is hard to teach; rules
of counterpoint or maqam modulation might be a bit easier, at
least as a starting point. But it is true that there are lots of
categorical or, as you say, "functional" perceptions.

> But it's not intonational, and that's what I'm sure of, unless
> there's some secret way in which it is that I'm not aware
> of. Minor chords are "minor" to me whether or not they're
> 10:12:15 or 6:7:9 or etc; the minor "feeling" is preserved
> regardless, though there are some clear subtle differences in
> mood. And all of my musical emotions seem to be dependent on
> these interval functions first and foremost, and the
> intonations of them secondly.

While we may be discussing different styles, I find likewise that
a 14th-century cadence in Machaut style like C#-E-G# to D-A might
tune the minor third sonority at 6:7:9, 22:26:33, or indeed the
supraminor 52:63:78 (minor third in MET-24 at 265, 289, or 333
cents), and with a semitone or supraminor step at 57, 81, or 127
cents. The 52:63:78 and 14:13 step are a good example of
"paraschismic" tuning (e.g. Eb-F#-Bb to E-B). But they all seem
like shadings of the same 14th-century "functionality."

As George Secor and I found in 2001-2002, though, when we get to
something like 18:22:27 or 22:27:33 with cadential steps like 139
or 150 cents, it's something else again -- neither major nor
minor, but neutral or Zalzalian. And hearing in terms of the
basic categories "minor - neutral - major" is a bit different,
maybe like recognizing a distinctive vowel in a new language as
something other than a blurring of two familiar English
categories or phonemes. And, of course, traditional Near Eastern
musicians take this for granted as part of their "mothertongue."

> Also, I'll go out a limb and say that this phenomenon accounts
> for about 90% of my musical experience, whereas ratios and
> intonation account for 10% of it. So it seems to me that if I
> want to experience new and fundamentally different musical
> emotions, which is what got me into this field to begin with,
> I need to learn how to perceive a radically new set of
> interval functions, however I choose to intone them. I've
> spent the past few years learning all of this theory which is
> built around ratios and temperaments and intonation and such,
> but now it's clear that in order for me to get to the point I
> want to, I need to learn to hear music relative to a
> completely different foundation of functions and scales and
> dyads.

What I find is that Near Eastern maqamat are at once a bit the
same (e.g. mostly diatonic tetrachords), and a bit different
(neutral intervals as an everyday, primary reality). But medieval
European rules of polyphony can adapt quite well -- although I'd
emphasize that pure maqam melody is a goal in itself, with any
polyphony making it a different and hybrid art. But the rules
just get expanded a bit, for the most part: "A minor, neutral, or
major seventh often contracts effectively to a fifth, etc."

But there are those "radically different" moments, like with
this in MET-24:

Xeno-Najdi Xeno-Hijaz
|----------------------|--------------|
F* G* Bb B C# D E* F*
0 207 438 565 772 853 1118 1200
207 231 127 207 81 265 82

Is that 772-cent interval (264/169, or 18/13 plus 44/39) a small
minor sixth, or some kind of "quasi-fifth"? This is the Battaglia
effect, as in the lower pentachord of 11-EDO machine, 2 2 1 2.
So that "Xeno-Najdi" pentachord really spans "a small minor
sixth"; and that "Xeno-Hijaz," or maybe "Hijaz al-Tammany"
tetrachord <grin>, a large major third! Yet they can sound like a
pentachord and tetrachord!

Now change that C# back to the usual C, and we're back in Kansas,
or maybe Cairo, at least from my perspective of what a "normal"
maqam sounds like:

Najdi Xeno-Hijaz
|----------------------|----------------|
F* G* Bb B C* D E* F*
0 207 438 565 703 853 1118 1200
207 231 127 138 150 265 82

Nadji, 3-3-2-2 in simple 17-step notation, is sort of a Zalzalian
or neutral variation on European Lydian; the upper Hijaz would be
2-4-1. Now tastes might vary: some Near Eastern musicians might
say "We really aren't into the 8/7 thing here" or "A Hijaz with a
12/11 step sounds more like Persian Chahargah than our local
style," etc.

But those are "normal" differences on fine-tuning, etc., as
opposed to a scenario where usual interval perceptions are
getting bent in striking ways!

> Half of it seems to be 12-EDO specifically, and half of it
> seems to be diatonic/chromatic/chain of fifths stuff
> specifically. When I first started out with different tunings,
> it was the former - 7/4 sounded "crunchy but wrong" to me at
> first, because I knew that to split it into two fourths, those
> fourths would also have to be flat, and then the LLs
> tetrachord would have to be compressed flat, but that would
> make the major thirds so flat that three wouldn't equal an
> octave... etc. I just intuitively knew that it would make
> everything break. Sooner or later I got over that, by
> learning to allow the notes in my head to deform a bit to
> accomodate the new intervals, So I started learning to hear
> things like 7/4 as a type of minor seventh, and 11/8 as a type
> of aug 4, etc.

What strikes me is that you and the RMP approach generally have
taken a much more strenuous and risky artistic path than mine:
you have dared to run the risk of "making everything break."
For me, starting in 1998 after 30 years of medieval and
Renaissance music played in 12-EDO by default and heard on record
and CD in who knows what intonations, it wasn't a question of
"breaking" anything, just using the "historically correct"
Pythagorean and meantone tunings.

Oh, yes, I _was_ interested in 1998 in trying a "what-if": how
about 24-note Pythagorean on two MIDI keyboards? That kept the
12-note LLs structure securely in place, with 9:12:16 on each
keyboard, and 7/4 also available in lots of places as an
"alternative minor seventh." By 2001, I was delightedly using
12:14:18:21 as the new fauxbourdon -- and I must record this! --
but, again, with the regular diatonic structure of each 12-MOS
untouched.

Neutral intervals were an acquired test for me: first mostly
things I played by mistake in 17-EDO, for example; then
"charmingly impressionistic"; then great in supraminor and
submajor flavors as a variation for 14th-century cadences, or in
the guise of an 11/8 wolf fourth in B-Eb-G# to A-E-A, etc; and
finally, during my collaboration with George Secor in 2001-2002,

So we both went through a process of getting acquainted with 7/4
and 11/8 -- but I never felt the chain-of-fifths framework was at
risk, because it was there on each manual. And this is where I
find it a bit surrealistic to think of Peppermint or MET-24 as
anything but another tuning of the "regular diatonic," or more
specifically a tweak of Pythagorean. For playing Machaut or an
improvisation in a similar style, the basic "mapping" is
identical to that of Pythagorean -- or 12-EDO, for that matter!
Something like miracle or father or porcupine really is a
different "abstract temperament" in that very concrete musical
sense.

> These days, it still seems to be diatonic based though, or
> maybe I should say "chromatic-based." If I play in something
> like 16-EDO, I generally just snap it into my head in terms of
> some 12 note chromatic scale. But, if I play in 19-EDO,
> something strange happens. I can now hear a very clear
> difference between sharps and flats. E-A# is lydian, and E-Bb
> is diminished. Also, the diesis is now the same size as the
> chromatic semitone, so I can play resolutions that exploit
> that and move from things like A#-Bb and hear the whole
> structure of the tonality swing waaaay across the circle of
> fifths to the other side. I think it's pretty safe at this
> point to say that I have a set of 19-EDO "categories" which
> are "perfect" in my head just like 12-EDO's are; they
> developed naturally as an outgrowth of the extended meantone
> structure.

The A#-Bb thing is really interesting in 19-EDO. In something
like MET-24, there's a small thirdtone at 57 cents, a bit smaller
than 1 step of 19-EDO, that likewise can be either clearly a
"diesis" (e.g. 4/3 vs. 11/8) or a small semitone. And there's a
middle thirdtone at 68 or 69 cents that your Lydian example
reminds me of: thus F G A B C or 1/1-44/39-14/11-63/44-3/2 is
Lydian, but F* G* A* B C* or 1/1-44/39-14/11-18/13-3/2 is Najdi.
As in 19-EDO, it's the distinction between 1/3 and 2/3 of a
tone. But the difference is that in 19-EDO, the usual semitone is
126 cents -- almost identical to my apotome (e.g. C-C#)! And my
normal semitone is 81 or 82 cents, so 57 or 69 cents may be a
less dramatic difference from the diatonic than 63 cents where
126 cents is the norm!

> So maybe we can say that it used to be 12-EDO specifically,
> but now it's something more general, like diatonic-based or
> meantone-based or chain-of-fifths based or something. I really
> don't know.

The humorous thing here is that you have gone _far_ beyond me in
exploring things which are definitely not "chain-of-fifths based"
in any usual sense, e.g. 11-EDO or 16-EDO or porcupine!

> Right, well, how about 1/1 12/11 8/7 21/16 3/2 12/7 7/4 2/1?
> Because that's clearly Dorian for me too, except that now 8/7
> is a subsubminor third instead of a major second of some
> sort.

A good question, and a great holiday gift, because you prompted
me to try an experiment! With 240 cents, say in 20-EDO. I've been
familiar with the "quasi-third" effect; and Jacques Dudon has
noted this property of 147/128 or the like. But 231 cents?

What I learned, thanks to you, is that _timbre is critical_!
In a usual harmonic timbre, 8/7 doesn't seem quite enough to pull
off the quasi-third effect, although this is a game of horseshoes
where "almost" does count: it's a wonderful hypermodern sound,
and quite distinct from either 8:9:12 or 6:7:9.

But now for the critical step you prodded me to take last night:
"Try a softer timbre, for example one designed for 14-EDO." And I
found that 1/1-8/7-3/2 _could_ be like 6:7:9, or 1/1-8/7-3/2-12/7
like 12:14:18:12. In this timbre, also, 32/21 made a satisfying
fifth for a cadential resolution! So we might speak of "timbreing
out the 49/48." Maybe it's mostly toning down the critical band
stuff and beating, leaving more room for a "thirdlike"
impression.

> If you can also hear 1/1-8/7-3/2 as minor in the context of
> the above scale, instead of sus2, then all I can say is that I
> think that's a really, really significant thing, not just a
> trivial sort of curiosity. In my mind, that completely makes
> the concept of "learning how different JI chords" work
> meaningless. How does 1/1-8/7-3/2 work? Is it a sus2 chord,
> as in your scale, or a minor chord, as in my scale? When I
> "detwelvulate" my ears to hear things in some supposedly more
> natural way, what's the "correct" way to hear the chord? It's
> now obvious that the answer to every single one of these
> questions is "mu," to quote a famous Zen koan.

Indeed the Zen "mu" might be the best answer to lots of these
questions if not all. Basically, as I understand it in crude
terms, "mu" is the illuminating realization that the reality is
beyond the reach of the question, indescribable in those terms.

But for me, whether 1/1-8/7-3/2 can be like 6:7:9 depends mainly
on timbre: MET-24 (230.9 or 232.0 cents) can do it in my 14-EDO
timbre. And 1/1-8/7-3/2-12/7 is even stronger, if anything, as
12:14:18:21 -- not to mention 14:18:21:24. In a sense, it's as
much a verb as a noun: how it resolves helps determine which way
I hear it (although someone else deciding the resolution would be
a better "scientific test" of that!).

But scale structure may very well make a difference in terms of
whether I hear 1/1-12/11-8/7-4/3 as a diatonic rather than
chromatic tetrachord. Your modification to 1/1-12/11-8/7-21/16,
which I can't do in MET-24 because I don't have both 12/11 and
21/16 above the same note, is likely crucial. It's much easier to
hear 240 cents as a "whole-tone" step -- although it can also be
a harmonic quasi-third -- than 265 cents! But 1/1-12/11-8/7-9/7
in MET-24 can be a sort of diatonic tetrachord -- is this a bit
like the 2-2-1-2 "pentachord" in 11-EDO machine?

> Well, if you already know how to hear 240 cents as a type of
> third, I guess the thing I wrote above will be old hat to you
> :)

Not exactly, although I appreciate the humor, because 240 cents
is one thing in a usual timbre, and 231 cents another. But with
the right timbre -- maybe more subtracting beats than anything
else -- 231 will do, also. And that _is_ news for me! Thanks for
prompting me to try it out. One sign of the timbre is that
G*-Bb-F at 0-231-935 cents not only has the sweetness of
12:14:21, but can resolve to G#*-E at 728 cents!

> I totally agree with this too. 16:24:26:39 is magic. But, I
> still don't know if the ratios I just laid out have anything
> to do with this magicness.

Exploring different shadings of the neutral intervals could be
very interesting.

>> This feels a lot to me like starting with a bare 9:7 expanding
>> to a fifth, with the third a bit blurry in a lower or middle
>> register. Either moving to a higher register, or > adding the
>> fourth above so we have 7:9:12 expanding to a > full 2:3:4,
>> clarifies things.

> Is the root staying the same in both cases?

A good question, and I'm not sure! My training is mainly to count
intervals from the lowest voice, or think of an outer interval
divided into adjacent ones, like 7:9:12 as 7:12 divided into 7:9
plus 3:4. But maybe the audible root does change -- at any rate,
going from 7:9 to 7:9:12 seems to consolidate things a bit, while
14:18:21:24 has the "anchoring" effect of that 3/2 fifth also.

Maybe this is a good place to conclude this portion of my reply.

Best,

Margo

🔗Margo Schulter <mschulter@...>

[This continues our dialogue from my last reply, with your
following comment directed to the idea that RMP could be focused
on the idea of finding tunings with "nondiatonic but melodically
compelling" modes, etc.]

> The RMP outlook doesn't really address these questions. It
> just assumes that we want to find temperaments with chords
> that are close to small-integer ratios. It has nothing at all
> to say about the phenomenon with 8/7 that I presented
> above. It doesn't know what parsing is. It just tells you how
> to find tunings that have nice tempered approximations to
> simple JI ratios, and lots of them. Quite frankly, I think
> that the existing RMP completely fails to explain this side of
> music, which to my ears comprises about 90% of what I hear,
> and that we're in need of some new basic realization to begin
> a new point of departure, just like Parncutt's work (among
> others) served as a point of departure for the existing RMP.

But maybe RMP has pointed to tunings or MOS sets or whatever
where these kind of psychoacoustical phenomena are likely to
arise -- some of them of interest in more "conventional" systems
also.

Of course, there are systems like 11-EDO or 13-EDO or 14-EDO
where the intervals simply don't "add up" as expected. But at
least equally interesting, in a way, are things like a
"pentachord" of 0-207-438-565-772 cents in a system where the
usual 0-207-438-565-703 cents is also available -- and yet we get
the same kind of "aural phenomenon" where 14/9 or a bit larger is
heard as a "fifth."

But the applied RMP stuff -- I mean machine and so forth -- can
port to various types of systems where appropriate subsets happen
to turn up. And it doesn't have to be a port strictly in MOS
form. Thus 0-218-436-545-764 is a MOS, while 0-207-438-565-772
isn't, but the Battaglia effect of "small minor sixth as
perceived fifth" is there in either.

[On some of the MOS systems you've tried]

> A few. All of them are MOS's with more large than small steps,
> and where the ratio of large to small step is close to 2/1.
> Machine in 16-EDO is fairly amazing; I can get into it pretty
> easily. That would be the scale 3 3 1 3 3 3.

Here I'm just quoting a few of the systems you mention where
you've given me some ideas.

The machine thing is pretty amazing, and also the portability of
the Battaglia effect. Was the "small sixth as fifth" effect one
of the purposes of this temperament? Here's the MET-24 version
I've been discussing:

! met24-xenonajdi6_Fup.scl
!
Curious hexatonic like Najdi (17: 3 3 2 2 3 2 2), 3 3 2 3 3 3
6
!
207.42187
438.28125
564.84375
772.26562
992.57812
2/1

> Sensi temperament doesn't have any great scales that I like,
> but the basic idea is that two 9/7's makes a 5/3 - huge chains
> of tempered 9/7's like that sound amazing to me, like deep
> space or wind or something. I have no idea why it sounds that
> way, but it does; it's a completely new, "xenharmonic" sound
> to me.

While MET-24 doesn't have enough 9/7 and 169/132 thirds to
generate a whole subset this way, it does have a large neutral
sixth at 866 or 867 cents (say 33/20 or 104/63) that can divide
into these two large major thirds, something like 439-428 or
428-438. And it is a fascinating "diminished seventh" sonority.

One resolution has that large neutral sixth expanding to an
octave, The other has it "jazzily" acting like the diminished
seventh it technically is, and contracting to a fifth with a
22/21 step in each outer voice!

Eb* E Eb* D*
B B* B D*
F#* E F#* G*

In the first version, the three voices have unequal motions of
three thirdtones (lowest part), one small thirdtone (middle part)
or two thirdtone (highest part) -- 207, 57, or 125 cents. In the
second, we have usual 82-cent semitones in each outer voice, plus
the leap of a 346-cent third (near 11:9) in the middle voice. So
this is one great chromatic progression -- or maybe enharmonic,
with that 57-cent step in the middle voice where the neutral
context is quite different than 1/1-8/7-9/7-4/3 or the like,
where it would just be a small diatonic 28/27!

So thanks for a great present in keeping with the spirit of the
season! And I realize that it could also occur in a neat mode
where the resolution is more conventionally balanced:

Eb* F*
B C*
F#* F*

Here the outer voices move from 104/63 or whatever to 2/1 with the
lower voice moving down by a 14/13 and the upper voice ascending
by a 9/8 or 44/39 (125 and 207 cents). The lower two voices go
from 9/7 to 3/2 with the lower again descending by 14/13, and the
middle ascending by 13/12. So we have all steps this time either
a tone or a neutral second -- in contrast to those semitones or
thirdtones in the other two resolutions.

> In contrast, Orwell has two 7/6's tempered to make an 11/8, and
> then three 7/6's tempered to make a 8/5. Orwell just sounds
> weird and diminished to me; it's very dark and just sounds
> diminished and messed up. I doubt that there's some magic
> pattern in the ratios which makes it work that way.

Well, that gave me an idea: how about a pelog pentatonic taking
the "two 7/6's" idea and running with it?

! met24-oceania_C.scl
!
"Orwellian" peloglike pentatonic
5
!
264.84375
554.29688
704.29688
829.68750
2/1

For slendro, one can just add 21/16 and 7/4; but the pelog
pentatonic is very pleasing to me.

> In terms of chromatic scales, 15-EDO seems to be the most
> "stable" nondiatonic chromatic scale out there that I've found
> other than 12-EDO. In 15-EDO, I need to start grouping 7/6 and
> 8/7 as different versions of the same "thing," if you will,
> whereas 6/5 and 7/6 are now completely and obviously
> different. I "sort of" get the hang of this.

This (240 or 320 cents) is interesting to compare with 20-EDO,
where it's 240 or 300 cents, with either as a minor third,
although 240 can also be a whole tone, or 960 a "quasi-sixth"!

Maybe the situation is something like 15-EDO with 265 and 332
cents -- juxtapositing these above a drone, they seem clearly two
different classes of intervals (minor and small neutral),
although the supraminor qualities of 63/52 could in a different
setting make it more "another kind of minor third, larger than
6/5." But in 20-EDO, I often regarded 240 and 300 as the two
types of "minor thirds."

> I play in either 19-EDO, 22-EDO, and 16-EDO almost every day,
> and I have for a year or two now. Maybe I'll regret saying
> this, but I expect that I would perform very well,
> comparatively speaking, on any sort of ear training test for
> all of these, demonstrating my familiarity with those tunings
> in a measurable way. Of course, it could well be the case that
> I perform comparatively poorly, at which point I'd be happy to
> eat those words and humbly ask what sorcery you're all
> partaking in which allows you to hear completely novel 22-EDO
> interval categories, where 7/6 and 6/5 are as different as 6/5
> and 5/4, which I have yet to discover.

Maybe, just to be safe, I should confirm that I don't claim any
sort of special expertise with 22-EDO. What I know is mostly
parapyth and some meantone or modified meantone for Renaissance
uses. On any of these tunings you play regularly, certainly from
my standpoint, you're the expert! And your skill, patience, and
creative humor call for much admiration.

There's lots more in your reply, but why don't I leave off and
continue my reply in another post.

Happy holidays,

Margo

🔗Margo Schulter <mschulter@...>

> The more time I spend in these tunings, the more I realize that
> there's a deeper theory that there is to stumble upon, which we
> don't currently know. I think that if we administered an ear
> training test to a lot of people, the people who do best on the ear
> training test will be the people who also have had the same
> realization. I have come to conclude that in situations where 6:7:9
> and 10:12:15 sound the same, they're 'supposed' to sound the
> same. Then, once that's accepted, one can then start looking for
> times when they're 'supposed' to sound different. For instance,
> hedgehog temperament in 22-EDO is a scale which suddenly makes it
> very easy to distinguish between 7/6 and 6/5, and it's given by 3 3
> 2 3 3 3 2 3. I have no idea how or why this works; I feel like I've
> made some progress towards understanding this, but that I only know
> half of the picture.

A similar question: is there a "difference in kind" between 332 and
357 cents -- which might both be neutral, but the first more
supraminor and the second in the higher central range? One Turkish
view is that while 357 cents would be a very low Rast (by Turkish
standards where "low" means around 16/13), 332 could actually be
Nahawand, a maqam with a minor third often in practice around 32/27
but sometimes around 6/5 or even higher.

[On 14-EDO as having good cadences, but outside diatonic structure]

> Yes, I absolutely agree with this. I didn't mention 14-EDO above
> because I'm still new to it, but I think there's magic to it. In
> 14-EDO, 5/4 and 6/5 become two different intonations of the same
> thing, and 7/6 and 8/7 become two different intonations of the same
> thing. Also, 14-EDO magically contains all of the touch tone
> intervals on your phone. There is definitely something interesting
> about 14-EDO. However, I have trouble hearing 5/4 and 6/5 as just
> two different versions of the 14-EDO neutral third.

The touch tone thing is interesting! The 943-cent interval is great in
place of 12/7 or 7/4, although I'm not sure if it exactly represents
either -- maybe more substitutes for either. And likewise with 5/4 and
6/5 -- or maybe 14/11 and 13/11 or whatever other regular thirds one
is used to. I tend to guess I might be perceiving things something in
this way: "Well, it expands to a fifth in a familiar way by stepwise
motion in each voice -- so it's some kind of third!" Or, similarly,
"It contracts to a unison by stepwise contrary motion, and doesn't
have enough of a critical band effect to make it a whole-tone, so it's
a third!" That last might apply to the 8/7 effect we've been
discussing.

[On radically nondiatonic systems as like a new natural language]

> Yes, I think so; specifically I think this happens when I try to
> play in really radically nondiatonic tuning systems. I just keep
> hearing 12-EDO patterns and I'm not sure what sorts of things to
> play which convey to my brain a really strong, and most importantly
> "stable", sort of novel pattern in the new tuning system, so that
> my brain is likely to gravitate to this new pattern and not the old
> broken 12-EDO patterns it ends up hearing instead, corresponding to
> my brain trying

And with some of these patterns, they might be more general than
12-EDO itself: e.g. "a major sixth from three generators fairly near
3/2 expands to a 2/1 octave." That could nicely apply to 19-EDO or
1/3-comma meantone in the 16th century, or 22-EDO today -- with the
fifth at around 695 or 709 cents. But it doesn't apply in the usual
way to the systems we're discussing.

>> Here it might be helpful to do studies of individual styles, and
>> then try to generalize. I can articulate some 14th-century
>> European relations, but other people know 18th-century European
>> relations better, or maqam, or gamelan, and so forth.

> What sorts of relations do people hear?

Well, probably most familiarly for many people here, the kinds of
harmonic progressions in late 17th-19th century European music often
represented by Roman numeral analysis, and so forth. Or the kinds of
13th-14th century European structures discussed by writers of the time
and by a modern scholar such as Sarah Fuller. Or the relationship
between Rast and Bayyati in a scheme of Near Eastern maqam modulation.

> Yes, this is exactly how I hear it too! But I wonder, is classical
> conditioning all that there is to resolutions, or is there some
> scale-theoretic reason which makes the tritone resolution more
> "forceful" than the double leading tone cadence? In other words, if
> we imagine that the listener is constantly maintaining a copy of
> the scale in his/her head at all times, is there some
> scale-theoretic reason why the tritone resolution might "refresh"
> more of this scale than the double leading tone resolution, or is
> it just classical conditioning?

Since I actually find 14th-century cadences more compelling than
18th-century cadences -- although each is compelling in the context of
its own style, and tastes vary -- maybe I might offer a friendly
amendment and ask: "Why do tastes vary so much on this?"

Since for 30 years I preferred 14th-century cadences while playing or
improvising in 12-EDO, that may control one variable, at least, and
support your view that often intonation isn't the main issue!

What I've found is that "scale-theoretic" approaches often can
identify ways that the modal languages of the 13th to early 17th
centuries differs from 18th-19th century tonality -- and that the
"anomalies" from a tonal perspective that they identify are often the
reason I prefer the earlier styles!

I remember sharing one of my favorite Spanish pieces around 1500 with
one of these tonal theorists -- a quite expert scholar! -- and getting
the comment, "Of course, having C and C# juxtaposed like that is fine
in modality, but would be impossible in tonal music!" I wouldn't
necessarily have known that such a progression would be positively
excluded in tonality, but I knew that I preferred the 16th-century
approach to accidentals: more fluid and varied, it felt to me.
But that's also a biased description, and doesn't stop me from
enjoying Bach, any more than people who find tonality more coherent or
compelling aren't prevented from enjoying Machaut.

Sometimes intonation can simply accentuate the fact that very
different kinds of "parsing" are going on for people mainly oriented
to different styles. For example, the following approximates pretty
closely a monochord-based interpretation by Jay Rahn of the cadential
intonations advocated by Marchettus de Padua in 1318:

<http://www.bestII.com/~mschulter/PythEnharImprov01.mp3>

One listener who's a very skilled musician remarked that since these
thirds and sixths aren't so harmonious, I really shouldn't have
"landed" on them. The answer, of course, is that those aren't
"landings," they're cliffhangers -- interrupted cadences with an extra
intonational edge. But the very fact I had to explain this suggests
that different patterns of conditioning are involved.

> So if I understand correctly, you flattened the third harmonic of
> the timbre to 655 cents, and then 764 cents sounded like a
> fifth...? I would have expected that if anything, sharpening the
> third harmonic to 764 cents would have made this happen more...

The short story is that likely I mostly detuned the partials to the
point where a "sine wave" situation resulted where 655 or 764 cents or
almost anything larger than a tone could be quite "concordant," with
lots of room for creatively blurred impressions.

> While you're in 22-EDO, you may want to play in hedgehog, which is
> 3 3 2 3 3 3 2 3. If you're like me, that 3 3 2 is very clearly some
> sort of do re mi fa, except the "mi" is 6/5, and the "fa" is 9/7,
> which is apparently a shade of perfect fourth today...

Here 3 3 2 is for me more like re-mi-fa -- Dorian, say -- plus some
kind of chromatic step where I might expect sol. But in MET-24,
playing pure melody without a drone to give any harmonic clues, I find
that Bb*-C-C#*-Eb at 0-150-334-439 cents can suggest do-re-mi-fa. The
fourth might be low by 57 cents from the tempered 4/3, but it sounds
like a slightly inflected "fourth" rather than something really
"nondiatonic." So 150-184-105 cents can create the effect you're
describing for me. But Bb-B*-C# at 0-184-334 cents, or 184-150 cents,
is more re-mi-fa. So the order of steps can make difference. With the
two identical steps of 22-EDO, 164-164 cents, I may lean to re-mi-fa.

> This is probably part of it; it's also been shown that the tendency
> towards virtual pitch integration increases as the VF gets higher
> in pitch. So as the 9/7 gets higher, there may be more of a
> tendency to hear it as (1):7:9, where the (1) is a virtual
> fundamental which you can hear if you focus on it. For me, I need
> only to imagine the 9/7 as part of a rootless (4):7:9 to totally
> transform the sound of it to something much more musically
> consonant.

Maybe the root is the mechanism; one way I've heard it put is that
thirds get less "muddy" in higher registers. By the way, Paul Erlich
would have dialogues where he was surprised that I found 12:14:18:21
smoother and less tense than 4:6:7:9, with the later still a very
intense kind of "concord," but not so easy to "bask" in by comparison
with 12:14:18:21. Maybe 12 isn't the root, but has 3/2 and 7/4 as
"anchoring" harmonics.

> Perhaps... I would suspect, however, that this sort of analysis
> makes ratios into a procrustean bed we're fitting something to,
> whereas the fundamental thing we're hearing is something much
> deeper and related to the properties of scales. I think that
> 14:18:21 sounds like root-major third-fifth, not the other way
> around, with the latter being the thing which is truly
> "fundamental" - at least to a 12-EDO listener. That being said, I
> can't claim that I really know what's going on.

I agree that 14:18:21 is simply one possible tuning of "a fifth
split into major third below and minor third above" -- and the way
that type of sonority gets handled depends more on the style or
"musical grammar" than on the exact intonation!

> In the context of the scale 2 2 1 2 2 2, if I go up to the octave
> and then go two whole steps down, it sounds like a small minor
> sixth, whereas if I go up from the tonic via 2 2 1 2, it sounds
> like a large perfect fifth - to my ears, anyway.

We agree there, and I find it the same in MET-24! Going up, it's a
large fifth, completing a pentachord. And the concluding "tetrachord,"
actually a trichord, of course, 764-982-1200 in 11-EDO or 772-993-1200
in MET-24, sounds kind of strange; but that "strangeness" is localized
with the upper genus, not the pentachord that has gone before. But
descending, it sounds like those two whole-tones are part of a
tetrachord of Kurdi or Phrygian, octave-minor7-minor6, where the
expected fifth to complete the genus would be the actual 3/2. So we
seem on the same wavelength here!

> So you're saying that at first, it seems like some sort of
> stretched major chord, right? But then, once you resolve it in the
> manner you describe, your perception of the chord changes
> -retroactively-, right? I hadn't thought about this before, but I
> totally agree that this happens, though not for me this time - it
> required repeated listens for me to "decipher" what was happening.

Yes, although I know that there was another view -- like seeing first
a human face in one of those optical illusions, then two vases, but
knowing while looking at those vases it could be a face also.

> Ironically, after this discussion and a year or so's worth of
> parallel offlist discussions with Keenan Pepper, I'm now starting
> to feel like we have -too many- different leads! But the question
> that I have which lurks underneath all of this is: how are all of
> these different concepts - classical conditioning of resolutions,
> scale-based and non-ratio-based "function" assignment of intervals,
> retroactive reidentification of functions, ease of distinguishing
> intervals in a scale if they're all as far apart from one another
> as possible (haven't mentioned this yet but it's true), formation
> of a "map" of said functions (haven't talked about this yet either
> but it's also true) in a novel tuning system - what is the
> underlying theme that unifies all of these things? What are we even
> talking about? I've got a "eureka" ready to go if anyone can figure
> out what part of reality we're even discussing here...

It does get really complicated, and I would be more interested in
finding pleasing progressions, melodic or vertical, than in coming up
with some Grand Unification Theory of music.

>> This is how I heard it after four or five listens. What happened
>> with me when I listened to the segment that you describe is - at
>> first, I clicked right to the chord, and heard it as a completely
>> foreign and atonal sort of thing; couldn't figure it out at
>> all. Then I heard it resolve to the A-D-E and A-C-E, and knew I was
>> at the right place. Then I tried listening again, a few bars
>> earlier, and still it sounded very strange when that chord came
>> up. Then after a few more listens, I started hearing the chord as a
>> sort of augmented chord, which then resolved to A-D-E and then to
>> A-C-E. Now that I know what it is, I can immediately snap it into
>> that sort of augmented framework. (You may be labeling the note
>> names differently; I'm just describing what I hear.)

Your note names are fine! And your detail is really, really
interesting!

I might add that while A-C-E is unstable, at 6:7:9 it's one of the
mildest and most concordant of unstable sonorities, maybe illustrating
David Doty's observation (if I correctly remember him as the source)
that a consonance can very effectively resolve to an even greater
concord, here the 3:2 fifth Bb-F.

[On the force of accustomed diatonic patterns, e.g. in 12-EDO]

> I am personally very familiar with this phenomenon. I think that my
> problem is that when I play in radically nondiatonic scales, there
> are a lot of 12-EDO things which constantly "take priority" in my
> head. I suspect that there's some way to play things in a tuning
> system like 10-EDO, for instance, which would be so "strong" due to
> unknown factors that almost any listener hearing would have to
> interpret those in this novel "strong" way instead of some old
> 12-EDO "best I could do" sort of way. If only I knew how...

Actually, what just occurred to me is a possible example of a 12-EDO
pattern that could contribute to the success of this mode, with the
lower "pentachord" having the Battaglia effect. Again, I'm calling
this genus in a maqam context Najdi al-Tammany, because this relates
to machine, and Tammany Hall was one of the greatest political
machines of all time! The upper "tetrachord" is Hijaz.

Najdi al-Tammany "Hijaz"
|-----------------------|----------------|
F* G* Bb B C# D E* F*
0 207 438 565 772 853 1118 1200
207 231 127 207 81 265 82

With machine proper, the lower "pentachord" could really sound like
something covering a "fifth," but the upper 218-218 or 220-207 or
whatever sounded "different," at least when heard in itself, say
descending from the octave. You noted that, and I agreed.

Here, though, that upper "Hijaz tetrachord" (actually here spanning
about 428 cents) can sound rather convincing, too. Why, when that
"tetrachord" is a thirdtone narrow -- and assuming this is pure
melody, without a drone to show that the "fourth" is actually a major
third?

A possible factor: the steps are 81-265-82 cents. From my maqamic
point of view -- as an alien student, not a Near Eastern musician! --
we can make this same genus routine simply by shifting C# to C*,
getting C*-D-E*-F* or 0-150-415-498, close to 12:11-7:6-22:21.

But there's another side to this: the actual 81-265-82 cents might not
be too different from the pattern of 100-300-100 cents in 12-EDO, just
with the semitones and the minor third a bit compressed from a 12-EDO
perspective. At any rate, it sounds rather conventional by comparison
with that 218-218 or 220-207 cents thing in machine proper as the
upper "tetrachord."

> Well, I note that I hear the stretched fifth even with normal
> timbres - in the scale 2 2 1 2 2 2, it's fairly hard to avoid for
> me. What happens if you use a very relaxed timbre, like an ocarina
> sound? I know that for me, even with normal timbres, if I load up
> 2 2 1 2 2 2, play a root-third-fifth triad and transpose it up and
> down the scale, I hear something like C-E-G -> D-F-A -> E-G-C ->
> F-A-D -> etc, sort of a gospel-ish hexatonic thing. I'm aware that
> it's not really the 12-EDO version of that, but that's the
> resemblance that I sort of hear.

I've tried experimenting with a bit of two-voice counterpoint or
three-voice parallelism a bit like fauxbourdon -- except that
fauxbourdon implies some fully stable fifths to resolve to! What I
would guess so far -- too early really to tell -- is that the
"major-third-and-fifth" thing may be like a "supersaturated solution,"
not hard to experience at certain moments, but not a very settled
state.

> Did you hear the tonic chord in "tonal study in orgone" as being
> some sort of root-major third-fifth? That was also 7:9:11.

While the counterpoint and rhythmic energy captured most of my
attention -- and well! -- my overall impression was some etude or the
like in minor, although at the end I heard the touch of a major
third. Maybe I would have been a better judge in my younger years:
this dynamic range, quite typical for piano music of course, does tend
to have me either set the volume "safely" so that I miss a lot in the
quieter parts, or else give too much of my attention to adjusting the
volume setting on the fly so that I can hear the quieter things and
then turn things down for the later parts.

> Yes, I absolutely agree with this. I strongly suspect that
> microtonalists are secretly manipulating 12-EDO/western/common
> practice/etc functions and categories in their heads when they play
> in extended JI, but then ascribe those powers to something like a
> set of ratios. This is just a hunch, though; there may be some
> among us who truly do hear a completely novel set of interval
> functions which are radically unlike anything we're used to in
> 12-EDO, and where 7/6 is its own novel interval class, not just a
> sort of minor third. However, if these sorts of "enlightened"
> listeners are among us, they seem to be the silent lurker type...

As to the JI question, I'd say I mostly agree. In MET-24 we have
"near-JI" where 22:26:33:39 and 12:14:18:21 are two alternative
shadings of "minor-thirdness and minor-seventhness." They're musically
interchangeable, and I like having two subtly different choices.

There are two areas, sometimes but not necessarily connected with JI,
where "radically different" things can happen. The first is small
intervals. Having heard of LaMonte Young doing something like this, I
discovered that moving _very slowly_ back and forth above the same
drone between 16:21:24:28 and 18:24:27:32 brings out 64/63 as a
distinct melodic step. There's nothing like this in 12-EDO. And it's
very effective in MET-24, with 0-472-703-968 and 0-495-703-996 cents,
where the 64/63 is actually tempered down to around a Pythagorean
comma.

The other area is neutral intervals, and also intervals around 250,
450, 750, and 950 cents, for example; rather early on, I found that
24-EDO was "out of this world" with the latter intervals. And in
2001-2002, George Secor and I found together that certain cadences
with neutral steps were definitely distinct from anything familiar to
us -- neither of us then having a grounding in Near Eastern music.

> In terms of the actual effect caused by isoharmonic chords, which
> is what we were calling "periodicity buzz," it'd have some sort of
> effect, but the 17 cents being off would definitely make the whole
> thing not seem to buzz in sync with itself. Though, after
> everything above, it now seems to me to be rather unlikely that the
> fundamental qualia of our musical experience are ratios, or that we
> really fundamentally hear things "as" ratios at all...

The question of the actual qualia is a really intriguing one, and I'm
open to various solutions. With the isoharmonic or "isodifference"
thing as Dave Keenan calls it, he points out that the beat rates for
the differences are a good test, and I should analyze this to see what
the result is.

> I feel the same way. For me, my head has been a chaotic swirl of
> these ideas for the past year; I'm desperately trying to find a way
> to organize it all, but I'm not sure where to start. Maybe it would
> be a good idea to make a list of these various phenomena or
> something.

The machine thing is _very_ impressive, and as a melodic pentachord
perception, the Battaglia effect as I call it seems quite "portable"
to different tunings. In a 17-step notation, it would be 0-3-6-8-11
steps, or 3-3-2-3, for that pentachord. And it's thanks to you that I
played around with this "pentachord" and found 3-3-2-3|1-4-1 with that
upper 81-265-82 cents likewise sounding like a "tetrachord."

> One thing I can definitely say is that I'm very appreciative of
> your ability to introspectively identify exactly what you're
> hearing in such an unconfused manner, and to relate it to all of
> these other things you've heard before, such as pelog and maqam
> scales, and then to state all of those connections clearly enough
> that I can understand them from halfway across the country and over
> a mailing list. I think I understand your thought process very well
> when you constantly identify these intervals which you hear as
> common across tuning systems, such as a certain type of pelog fifth
> which is also sort of close to 16/11 which also makes an appearance
> in some mode of rast, or something like that.

It's interesting to draw connections -- but I realize that there are
lots that other people, including yourself, know more about! For
example, from your discussions and those of others, I know that jazz
involves some very complex harmonies where there could be all sorts of
microtonal nuances. And it's fascinating how some kinds of special
effects like the Battaglia effect in machine might port to various
tunings and styles.

> I've been writing the same sorts of descriptions of the things I
> hear on here and on other fora, except instead of relating the
> stuff I'm hearing to something like maqams or pelog, it's to
> MODMOS's of porcupine, or to 19-EDO, or to 16-EDO, or to some sort
> of extended 12-EDO jazz harmony, which are the musical systems I've
> immersed myself in daily for the past few years. However, I've at
> times been frustrated, because I rarely seem to find anyone who can
> understand a thing I'm saying.

That can be a kind of mutual predicament when people are coming from
different places. But you do communicate well, for example with that
Gospel type of progression in 11-EDO machine. Some of it is a bit like
fauxbourdon or maybe a 16th-century hymn setting, with the stepwise
motion in the bass very natural to me. And the extended jazz harmony
is one really creative field in all this!

> Some people on these lists claim to not hear a network of very
> clear interval functions, but rather they "just hear sound";
> whether that's true, or whether they're just not sure how to
> express the functions they do here, I don't know, but I do know
> that I've found the process of communication on this sort of level
> to be difficult at times.

A lot of it may be putting it (always inexactly, but still helpfully)
into words! And on the 7:9:11 thing in my piece _O Europa_, and the
radically different thing in machine, it's fascinating how we were
able to explain our experiences.

> It occurred to me in this discussion that I should perhaps not take
> for granted that I am now talking to someone who seems to think in
> exactly the same way that I do and who understands what I'm saying
> and can respond in kind, and I am grateful for that. I also think
> it's very interesting to see you going through the same sort of
> inter-tuning comparison process that I've also come to, but with a
> totally different set of tunings as a base than those which I have.

Yes, I need to remember that patterns very familiar to me are not
necessarily so to others! We may often be in much the same
situation.

> I think I can sort of see what you mean; I'm thinking of Machaut's
> Kyrie here, from the Messre de Nostre Dame, and I guess it's true
> that he never really ends a phrase on a major chord, does he? Maybe
> that's a key aspect of the signaling.

What I might say is that while some 13th-14th century composers
(putting aside some English styles) sometimes end, or better _pause_ a
phrase on a fifth split into lower major and upper minor third (to me,
"major chord" somehow implies that it's a unit of complete or stable
harmony), it is a pause or possibly an interrupted cadence rather than
a real completion. Adam de la Halle does this in the later 13th
century. We open on F-C-F, and he pauses at the end of a phrase on
G-B-D -- with the form either immediately returning to F-C-F, a
logical resolution, or else continuing in a last brief phrase taking
us to the same destination. So the signal is: "We're not through yet,
even if you were expecting we might be!"

> Major chords are always sort of the comma in the middle of a
> sentence for him, but never a period and full-stop.

Exactly!

> But I have to say that the notion that major thirds weren't just
> resolved consonances didn't really click for me until I heard
> Marcel Peres' version of Machaut's Credo; he sings it in a much
> lower register than usual, so that the major thirds start to fall
> within a critical band and sound somewhat rough, and adds lots of
> vibrato to the thirds. In other words, he does every single thing
> that you could possibly do to make a major chord not just sound
> like a static, calm, blending 4:5:6 chord. And indeed this works
> very well: just like sensory concordance can enhance grammatical
> musical consonance, Peres uses sensory discordance to enhance
> grammatical musical dissonance. That really blew my mind the first
> time I heard it; to hear the thirds suddenly becoming dissonant in
> a very real, visceral sense.

This makes sense: the lower register could do it, just a high register
can make 14:18:21 seem quite blending. Similarly, as Paul Erlich and I
once discussed, tempering a major third at 415 cents rather than 81/64
at 408 cents could maybe underscore the message that this is _not_ a
mistuned 5/4 but something else again. But these things are very much
enhancing the musical grammar. The fact that people describe thirds
either as "imperfect concords" or as "tolerable dissonances" suggests
that "pleasant but unstable" is the general idea. But for people
accustomed to a stable 4:5:6, it's the dissonance and instability that
can use some underscoring like you're describing in Peres.

[Concerning Zarlino on the diminished fifth as permissible to sound
with both notes "in the same percussion"]

> What does this mean exactly - that it doesn't have to go
> B-F -> C-F -> C-E, or something like that?

Actually it's a fairly subtle point having to do with the preparation
rather than the resolution. For example, he permits:

F F E
A B C

His point is that B-F can be sounded together, as well as in the kind
of a suspension that could also be applied to other dissonances such
as a sixth or seventh:

F E
A B C

In both uses, the freer as well as the conventionally suspended
diminished fifth (with F already sounding when B is sung), it is
expected that a consonance precedes the dissonance. And it's optional
to use another suspension at the fourth in the course of the
resolution, for example

F F E F
A B C F

And in four voices we sometimes get things like this:

F E F
B C C
F G A
D C F

So we start with a tritone above the bass on the sixth degree
of the mode, then a resolution of the highest voices B-F to C-E
elaborated by a suspension of the kind you were describing while the
bass moves to the fifth degree, and then a usual cadence with the bass
failing a fifth while an upper voice rises by a semitone. But the
tritone sonority, basically a minor third plus major sixth, can and
does also just resolve directly to a cadence on the step below:

F E
B C
F G
D C

This would be a frequent half-cadence internally, but generally not so
conclusive.

> But why, I wonder, are tritones tense?

A fair question, and the answer would seem to be, "They need not
always be so, as in Paul Erlich's decatonic with a stable 4:5:6:7
tetrad."

In historical European music, the first reason might be a perception
of the tritone as a "wolf" fourth or fifth -- something close to, but
not quite, what you wanted if you were looking for 3/2 or 4/3! I
experienced this myself in 1966 when I found out in high school how
much I enjoyed "parallel fourths and fifths" -- the names of the
intervals were new to me! -- and hit an unexpected dissonance: "Is
this piano out of tune?" Answer: "No, you just played a tritone."
So it's sort of "dissonant" by force of an odious comparison.

That didn't mean it wasn't used, only that it was considered
definitely unstable. And as thirds because increasingly stable, that
instability resolving specifically to a third for the diminished
fifth, or sixth for the augmented fourth, becomes one important
ingredient of certain 16th-century cadences. Around 1600, it gets
used in a way that results in a seventh above the bass also, something
that by around 1670 or 1680 becomes a standard part of the new scheme
of organization which would be called "tonality," where it represented
"V7-I" in the later Roman numeral system. And in this key system, the
location of the tritone was central to defining a key, if I understand
correctly.

But something like 7/5 or 11/8 might well be treated as relatively
concordant or even stable; some people have suggested that 8:11:13 may
have special properties making it quite consonant.

So, while I was focusing on historical European understandings, they
needn't apply to other styles or musics: you asked the right question!

> I've experimented with diatonic scales from 7-EDO to 5-EDO. One
> really interesting one is in 52-EDO, 10 10 1 10 10 10 1. In this
> scale, it takes me a second to hear the tiny 26 cent half steps as
> being half steps, but I inevitably catch on and the whole thing
> snaps into place. Once it does, the diminished fifth is dissonant
> and enjoys resolution to the major third, as it usually does; if I
> play I ii iii IV V vi vii? I, that vii? is definitely recognizable
> as a diminished chord to me. However, the diminished fifth is 508
> cents; it's now close to 4/3 and is in fact a better approximation
> to 4/3 than the actual perfect fourth of the scale.

First of all, just a humorous aside. I looked at 52-EDO in Scala and
said to myself, "Isn't this curious: lots of these intervals match
MET-24 within a cent or two!" Then I realized that 52-EDO was a subset
of 104-EDO, even closer to the O3 or POTE tuning!

But carrying the 12-17-22-27-32-37-42-47 series to 52 -- something I
hadn't even thought of -- is fantastic! And your result made me try
this in MET-24 with success, although a less dramatic example:

Eb D
A* Bb
520c 415c

Here a 520-cent wide fourth contracts to a 415-cent major third, with
the lower voice moving up by a comma at 23.4 cents, and the upper
voice descending by a regular semitone at 82 cents. But it sounded
quite recognizable, if not quite what Zarlino had in mind <grin>.

> Thanks,
> Mike

With thanks and best wishes for a Happy New Year,

Margo