Another Bach thought...

As if I didn't stir up enough muck when I brought up WTC... I'm sure this
thread has been brought up as well...

What about Bach's "Chromatic Fantasia and Fugue"? (I'm sure I've
incorrectly remembered the title). It seems to me, as I remember when I
played this piece, that enharmonics within the course of the piece abound...
thus seeming to demand some sort of system closer to 12tET than any kind of
mean-tone. I haven't studied the work, so Werkmeister could be what was
intended...but it seems to me that it certainly doesn't lend itself to any
of the "performance retunings" that have been described...

Also -- just another thought that I've had, in my novitiate thinking about
tuning... how in the heck is our ear willing to fudge a chord like G-B-d-f
in 12tET to be an approximation of 4:5:6:7??? I mean, notwithstanding the
"B", that "f" is a long shot off of where it should be. What's the
explanation? Is it cultural? Is our ear that forgiving?

I'm trying to formulate a project on how to derive "tonal" functions in
systems in what I guess would be called "n-tone equal temperament", where n
is an arbitrary number of subdivisions of the octave. Been done, once
again, I'm sure. But, trying to figure out a system, whereby one can figure
out when the ear finally says, "Eh, that's close enough", is maddening.

Eric

--- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
wrote:

> What about Bach's "Chromatic Fantasia and Fugue"? (I'm sure I've
> incorrectly remembered the title). It seems to me, as I remember
when I
> played this piece, that enharmonics within the course of the piece
abound...
> thus seeming to demand some sort of system closer to 12tET than any
kind of
> mean-tone.

Har. You'd think that Shostakovich wouldn't work in anything not
close to 12et, but you'd be wrong. I'm uploading the 10th symphony
right now, and unlike the Bach it really does makes one wonder if the
piece really wants to have all the sweet harmony asbru tuning is
giving it, but asbru certainly does give it.

> Also -- just another thought that I've had, in my novitiate
thinking about
> tuning... how in the heck is our ear willing to fudge a chord like
G-B-d-f
> in 12tET to be an approximation of 4:5:6:7??? I mean,
notwithstanding the
> "B", that "f" is a long shot off of where it should be. What's the
> explanation? Is it cultural? Is our ear that forgiving?

Paul has been working on extending what he calls "harmonic entropy"
to chords. The G-B-D-F in 12-et could have contributions from
1-5/4-3/2-7/4, 1-5/4-3/2-9/5, 1-9/7-3/2-9/5, 1-5/4-3/2-16/9, etc.

hi, eric!

--- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
wrote:
> As if I didn't stir up enough muck when I brought up WTC... I'm sure
this
> thread has been brought up as well...
>
> What about Bach's "Chromatic Fantasia and Fugue"? (I'm sure I've
> incorrectly remembered the title). It seems to me, as I remember
when I
> played this piece, that enharmonics within the course of the piece
abound...
> thus seeming to demand some sort of system closer to 12tET than any
kind of
> mean-tone. I haven't studied the work, so Werkmeister could be what
was
> intended...but it seems to me that it certainly doesn't lend itself
to any
> of the "performance retunings" that have been described...

this work was for organ, right?

> Also -- just another thought that I've had, in my novitiate thinking
about
> tuning... how in the heck is our ear willing to fudge a chord like
G-B-d-f
> in 12tET to be an approximation of 4:5:6:7???

hmm . . . well i don't know if it really is, and i don't think the
fudging is ever totally black or white.

> I'm trying to formulate a project on how to derive "tonal" functions
in
> systems in what I guess would be called "n-tone equal temperament",
where n
> is an arbitrary number of subdivisions of the octave. Been done,
once
> again, I'm sure. But, trying to figure out a system, whereby one
can figure
> out when the ear finally says, "Eh, that's close enough", is
maddening.

i don't think you can draw a sharp line, but certainly you can explore
*local valleys* of dissonance, the bottoms of which are usually but
not always JI chords, and determine, say, which of your
equal-temperament chords falls into which valleys; you can do many
other things too . . . there was quite a bit of this kind of activity
on this list, which then split off into the harmonic entropy list
(since the subject wasn't popular here) -- if you have time, check out
that list and search the archives of this list too.

the premise of harmonic entropy theory is that chords, just like
individual musical notes, will tend to be understood in terms of a
fundamental "root" pitch (not necessarily present as a note in the
chord) to the extent to which the notes in the chord approximate a
harmonic series over that fundamental pitch. this is terhardt's view
of harmony, and is echoed by pierce and parncutt as well. i look
forward to discussing this further on the harmonic entropy list, if
you like.

>
> Eric

-paul

>> What about Bach's "Chromatic Fantasia and Fugue"?
//
>this work was for organ, right?

It is most frequently performed on the harpsichord.

-Carl

I like to think of 12tet as a "phonemic" system, where the ear does indeed assume
that each interval/note stands in for a variety of just intervals/notes. In choral music
written in 12tet-notation, singers produce "allophones" for the 12 pitches based on
function--but, from the insider's persepctive, the singers aren't conscious that they
are producing more than 12 distinct pitches, just as a Spanish speaker does not
contrast [s] and [z], although both souns occur in the language, but analyzes them
both as a phoneme /s/. JI notation is more "phonetic"-- it takes an outsider's
perspective and writes the tones as they actually sound, not simply according to their
function. It's sort of like Paanini's Devanaagari script, which writes Sanskrit nearly
phonetically, even blurring the margin of words.

A tagential question is: which system is more clear? Well, phonemic notation is
probably clearer for some JI music, especially for 5-limit and below: clearer from the
perspective that you can give a score in conventional notation to a intonation-savvy
choir and say "sing this in just intonation" and they'll do a pretty good job (although
there will likely be flatting). Phonetic notation is *mathematically* more clear, that's
certain and if being theoretically explicit is a goal, it's the best choice. For 7-limit JI it
would be hard to be purely phonemic.

If the musicians can be assumed to have an excellent background in theory (not
always true) and the music is tonal, you can probably write in 7- or even 11-limit JI
with only a few added accidentals, relying on the musicians to analyze the score and
make "just decisions". -Justin (who got a bit off topic, but oh well)

> the premise of harmonic entropy theory is that chords, just like
> individual musical notes, will tend to be understood in terms of a
> fundamental "root" pitch (not necessarily present as a note in the
> chord) to the extent to which the notes in the chord approximate a
> harmonic series over that fundamental pitch. this is terhardt's view
> of harmony, and is echoed by pierce and parncutt as well. i look
> forward to discussing this further on the harmonic entropy list, if
> you like.
>
> >
> > Eric
>
> -paul

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:
> I like to think of 12tet as a "phonemic" system, where the ear does
indeed assume
> that each interval/note stands in for a variety of just
intervals/notes. In choral music
> written in 12tet-notation, singers produce "allophones" for the 12
pitches based on
> function--but, from the insider's persepctive, the singers aren't
conscious that they
> are producing more than 12 distinct pitches, just as a Spanish
speaker does not
> contrast [s] and [z], although both souns occur in the language,
but analyzes them
> both as a phoneme /s/. JI notation is more "phonetic"-- it takes an
outsider's
> perspective and writes the tones as they actually sound, not simply
according to their
> function.

yes, but in what segment of musical performing practice? even when
you find that a group is *adaptively* tuning each chord to just
intonation within itself, for chords where this is possible such as
major and minor triads, it is typically still not possible to fairly
render a passage in *strict* JI, because of certain commas that are
being elided -- or perhaps even more importantly, melodic tendencies
that are shaping the intonation.

> A tagential question is: which system is more clear? Well, phonemic
notation is
> probably clearer for some JI music, especially for 5-limit and
below: clearer from the
> perspective that you can give a score in conventional notation to a
intonation-savvy
> choir and say "sing this in just intonation" and they'll do a
pretty good job (although
> there will likely be flatting).

or sharping, or arbitrary pitch-shifting (which tend to sound like
poor singing). the problem is even worse when you want to do 5-limit
things that 12-equal can't express, for example eliding
the "porcupine comma" of 250:243, as happens in herman miller's
music . . . then you have to *notate* it as drifting by a semitone
even when the composer intended no such thing . . .

> Phonetic notation is *mathematically* more clear, that's
> certain and if being theoretically explicit is a goal, it's the
best choice. For 7-limit JI it
> would be hard to be purely phonemic.

if you use 31-equal or even 22-equal notation for 7-limit, you're not
much worse off than using 12-equal for 5-limit notation, in terms of
being able to clearly express just chords and the like.

> If the musicians can be assumed to have an excellent background in
theory (not
> always true) and the music is tonal, you can probably write in 7-
or even 11-limit JI
> with only a few added accidentals, relying on the musicians to
analyze the score and
> make "just decisions". -Justin (who got a bit off topic, but oh
well)

72-equal (with only 3 sizes of alterations from 12-equal) as a sort
of "shortcut" notation for 11-limit ji has been advocated by ted mook
and has been argued much around here, easily hundreds of posts on
this one subject can be found in the archives. joseph pehrson is
having practical success with this system right now, with some of the
finest musicians around . . .

>
> yes, but in what segment of musical performing practice? even when
> you find that a group is *adaptively* tuning each chord to just
> intonation within itself, for chords where this is possible such as
> major and minor triads, it is typically still not possible to fairly
> render a passage in *strict* JI, because of certain commas that are
> being elided -- or perhaps even more importantly, melodic tendencies
> that are shaping the intonation.

Well, you could always allow (for, say C Major) D and D- to coexist in the same tonal
piece and, when moving from II to V move the D from one part to another to cover
the shift or actually ask the singers to master 'comma tweaking'. I think it IS possible
to sing in strict JI with practice-- it's certainly a lot easier than singing in 'strict' 12tet!
At least the JI intervals are sensitive to detuning, while the ET intervals can go all over
the place leading to rapid uncontrolled drift (unless the choir members all have
perfect pitch, and that would be lamentable).

>
> or sharping, or arbitrary pitch-shifting (which tend to sound like
> poor singing). the problem is even worse when you want to do 5-limit
> things that 12-equal can't express, for example eliding
> the "porcupine comma" of 250:243, as happens in herman miller's
> music . . . then you have to *notate* it as drifting by a semitone
> even when the composer intended no such thing . . .

You'll have to explain the significance of the pocupine comma to me and how it
relates to Herman Miller.
>
> if you use 31-equal or even 22-equal notation for 7-limit, you're not
> much worse off than using 12-equal for 5-limit notation, in terms of
> being able to clearly express just chords and the like.

I'm sure that's the case, but ideally it would be better not just to be 'not much worse
off' but pretty darn close.

>
> 72-equal (with only 3 sizes of alterations from 12-equal) as a sort
> of "shortcut" notation for 11-limit ji has been advocated by ted mook
> and has been argued much around here, easily hundreds of posts on
> this one subject can be found in the archives. joseph pehrson is
> having practical success with this system right now, with some of the
> finest musicians around . . .

The math is there, but I think you can hear when things aren't exactly 'right on'. I
personally would rather not approximate anything, but go ahead and write it all out
"phonetically"... although I acknowledge that "phonemic" writing with some
instructions to the players can arrive at the same result. -Justin

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:
> >
> > yes, but in what segment of musical performing practice? even
when
> > you find that a group is *adaptively* tuning each chord to just
> > intonation within itself, for chords where this is possible such
as
> > major and minor triads, it is typically still not possible to
fairly
> > render a passage in *strict* JI, because of certain commas that
are
> > being elided -- or perhaps even more importantly, melodic
tendencies
> > that are shaping the intonation.
>
> Well, you could always allow (for, say C Major) D and D- to coexist
in the same tonal
> piece and, when moving from II to V move the D from one part to
another to cover
> the shift or actually ask the singers to master 'comma tweaking'.

always . . . but the actual musical examples are typically far
thornier than this . . . not to mention that it's irrelevant for pre-
tonal music!

but i think the result is musically preferable when they don't have
to tweak commas at all! for example, if each note is permitted to
have only *two* interpretations 6 cents apart, an entire work (well,
a european common practice work from the years 1450-1800) could be
performed with all the triads in perfect vertical ji. the notation
would be standard, and be interpreted as 1/4-comma meantone (as it
was, anyway, by default for some time) with the 6-cent latitude
for "chord justification". see

http://sonic-arts.org/dict/adaptiveji.htm

> > or sharping, or arbitrary pitch-shifting (which tend to sound
like
> > poor singing). the problem is even worse when you want to do 5-
limit
> > things that 12-equal can't express, for example eliding
> > the "porcupine comma" of 250:243, as happens in herman miller's
> > music . . . then you have to *notate* it as drifting by a
semitone
> > even when the composer intended no such thing . . .
>
> You'll have to explain the significance of the pocupine comma to me
and how it
> relates to Herman Miller.

well, as with the more familiar commas, a chord progression in JI can
have starting and ending points that are this comma apart. the
problem is, when the consonant intervals are notated in 12-equal, the
resulting progression will drift, notationally, by a semitone. a more
dramatic example is graham breed's infamous blackjack chord
progression. consisting of a cycle of 7 chords, the JI version of the
progression drifts by 2401:2400, which is less than 1 cent (each time
you go around the cycle). but if the consonant intervals are all
notated in 12-equal, the progression appears to drift by an entire
semitone each time you go around the cycle. now, if a JI composer
wants to write a piece which drifts by a small amount or not at all,
why should he or she be restricted to progressions which show this
behavior when stuffed into 12-equal?

> > if you use 31-equal or even 22-equal notation for 7-limit, you're
not
> > much worse off than using 12-equal for 5-limit notation, in terms
of
> > being able to clearly express just chords and the like.
>
> I'm sure that's the case, but ideally it would be better not just
to be 'not much worse
> off' but pretty darn close.

well, as i attempted to convey above, you can always contrive
situations that aren't even close, for any given equal, linear, or
planar temperament when used as a notational basis for JI.

> > 72-equal (with only 3 sizes of alterations from 12-equal) as a
sort
> > of "shortcut" notation for 11-limit ji has been advocated by ted
mook
> > and has been argued much around here, easily hundreds of posts on
> > this one subject can be found in the archives. joseph pehrson is
> > having practical success with this system right now, with some of
the
> > finest musicians around . . .
>
> The math is there, but I think you can hear when things aren't
>exactly 'right on'.

the idea, at least for chords, is that the players will adaptively
adjust to make them exactly right on.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> 72-equal (with only 3 sizes of alterations from 12-equal) as a sort
> of "shortcut" notation for 11-limit ji has been advocated by ted
mook
> and has been argued much around here, easily hundreds of posts on
> this one subject can be found in the archives. joseph pehrson is
> having practical success with this system right now, with some of
the
> finest musicians around . . .

If you are picky, I've pointed out from time to time that 9 nominals,
which can be attached the obvious way to the 5-line staff, along with
a half-step symbol and a 21/20 or quarter-tone (36/35) symbol,
suffice to notate what is effectively 11-limit just intonation. The
score would look like a normal musical score, and could easily be
produced by programs like Sibelius or Finale, but could not be read
as one, since it would seem like staring madness.

No good or even bad musicians have tried it, and maybe none ever
will, but it remains true that 11-limit is notatable with fewer bells
and whistles than Secor-Keenan saggital, Johnston, etc.

>I've pointed out from time to time that 9 nominals,

Representing what basic scale?

>along with a half-step symbol and a 21/20 or
>quarter-tone (36/35) symbol,

IOW, the regulary half-step b or #, plus *either* a 36/35
*or* a 21/20 symbol?

>suffice to notate what is effectively 11-limit just intonation.

Which temperament is this? Oh wait, ennealimmal, right?

-Carl

Yes, but would the tuning of the 11-limit intervals be transparent to anyone other
than the composer? -i.e., even if the players could play the piece perfectly reading the
notation, could they easily run an analysis of the theory behind it? -Justin
>
> If you are picky, I've pointed out from time to time that 9 nominals,
> which can be attached the obvious way to the 5-line staff, along with
> a half-step symbol and a 21/20 or quarter-tone (36/35) symbol,
> suffice to notate what is effectively 11-limit just intonation. The
> score would look like a normal musical score, and could easily be
> produced by programs like Sibelius or Finale, but could not be read
> as one, since it would seem like staring madness.
>
> No good or even bad musicians have tried it, and maybe none ever
> will, but it remains true that 11-limit is notatable with fewer bells
> and whistles than Secor-Keenan saggital, Johnston, etc.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I've pointed out from time to time that 9 nominals,
>
> Representing what basic scale?

9-equal

> >along with a half-step symbol and a 21/20 or
> >quarter-tone (36/35) symbol,
>
> IOW, the regulary half-step b or #, plus *either* a 36/35
> *or* a 21/20 symbol?
>
> >suffice to notate what is effectively 11-limit just intonation.
>
> Which temperament is this? Oh wait, ennealimmal, right?

It's ennealimmal if you leave off the half-step, where it notates 7-
limit; add the half-step symbol and you have hemiennealimmal notating
the 11-limit.

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:

> Yes, but would the tuning of the 11-limit intervals be transparent
to anyone other
> than the composer?

It would make no sense until you learned the system. However, this is
a general problem, not specific to this notation.

-i.e., even if the players could play the piece perfectly reading the
> notation, could they easily run an analysis of the theory behind
it? -Justin

If they learned it.

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...>
wrote:
> Yes, but would the tuning of the 11-limit intervals be transparent
to anyone other
> than the composer? -i.e., even if the players could play the piece
perfectly reading the
> notation, could they easily run an analysis of the theory behind
it? -Justin

sure -- all it takes is knowing how each of the primes is mapped by
9-equal degrees and the one or two alteration amounts. then, every
time you see a chord, you can quickly recognize any 11-limit
consonances it may contain, determine whether it fits a harmonic
series pattern and what the root is, and of course common-tone
relationships with nearby chords in the score will be readily
visible. there's always the possibility, though, that the odd JI
composer might *want* to make use of tiny "ragisma" shifts or drifts,
in which case this system will fall short . . .

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

there's always the possibility, though, that the odd JI
> composer might *want* to make use of tiny "ragisma" shifts or
drifts,
> in which case this system will fall short . . .

I've done ragisma drifts. After you've done one, you can't hear any
difference anyway.