definition of COFT

[Monz:]
>(BTW, I need a definition for COFT).

Happy to oblige! I wrote the following originally to Patrick Mullen,
off-list. The acronym COFT stands for "Calculated Optimum Fixed
Tuning".

First, about my general "spring" model. It is an analog of the physical
world; there, springs behave as follows: resisting force is proportional
to displacement times spring constant. The energy held by a spring is
the integral of force over distance of displacement, which works out to
being proportional to the SQUARE of displacement.

In my model, energy is the analog of "pain", refering to the displeasure
of our ears to out-of-tune intervals (chords being represented in this
model as a set of dyads, with some awareness of otonal vs. utonal
chords). Displacement is the analog of tuning motion, with the "rest"
point being exact JI.

By setting up a big spring matrix and relaxing it successively, I can
minimize "pain" and hope to approach a nicely tuned piece (it can be
shown that minimum energy is equivalent to each node of the matrix
relaxing fully; i.e., having zero net force). For each sequence read
into the program, I create three kinds of springs:

. to control tuning, I create springs between notes of different
pitches which sound together. I call these "vertical" springs.

. to control the pain of a note changing pitch (especially while
the note is sounding!), I create "horizontal" springs between
successive instances of the same pitch.

. to control drift of the entire piece's tuning over time, I create
"grounding" springs between each note and some ideal expectation
of the tuning of that note. For that ideal I've used 12-tET, but
more recently have let the ideal "float" to any constant value
(constant over the duration of the sequence, that is). This last
is closely aligned with the "COFT" value.

There's more, of course: a minor third in 12-tET is either 6/5 or 7/6,
and a major second has three possible JI values (10/9, 9/8, or 8/7).
And, even a major third might be 9/7 rather than 5/4! A part of the
program that is not always fully ideal at the moment makes the
decision about what JI intervals to shoot for in setting up the
vertical intervals in the spring matrix. At present, I do not modify
the target tuning of a spring once it is created.

Now COFT. If the tuning of a piece is fixed, then two of the
three spring types disappear: horizontal motion within a pitch is
gone, so the value of those springs is of no importance. Grounding
becomes a non-issue, because drift is impossible with a fixed tuning,
and any tuning can be moved in absolute tuning space so that average
deflection is zero (or, notes are tuned only down from 12-tET, or
whatever...). So the "only" thing left to worry about is to sum the
values of all the vertical springs and then relax them to the least pain
position.

I now attach an actual table output for one of the pieces on my web site

http://www.adaptune.com

(a piece which works particularly well in COFT), Mozart's Andante
für Orgelwalze, K.616, sequence by Hans Jakob Heldstab. The piece
is in the key of F major. This is a 5-limit target tuning; COFT for
7-limit tunings is problematical.

How to read the table: each record shows a pair of pitches, along with
their final tuning, in cents relative to 12-tET. The strength field
is an integral of loudness over time of that pair of pitches sounding
in the sequence (with some adjustment for less important intervals).
Ideal should tend to show a quasi-JI tuning for this interval (quasi
only because sometimes different interpretations of the interval
conflict to some extent in the composite shown). Actual reflects the
tunings chosen for the two notes. Force is the means of communicating
urgency of request, and is Strength times the difference of Actual and
Ideal; the force for all intervals of each note adds to zero because the
spring set has been relaxed to a state of minimum energy ("pain"). The
Pain column is proportional to (Ideal - Actual) squared times strength.

In this table 0 == C, 1 == C# and/or Db, etc. Note that the interval
most strongly represented is C to E (0 to 4), the fifth degree and the
leading tone, with strength 451.285. That either is or is not
surprising...

A reformulation of the relationships among the columns:

Force = Strength * (Actual - Ideal)
Pain = 0.5 * Strength * (Actual - Ideal)^2

Which implies,

Pain / Force = 0.5 * (Actual - Ideal)

Pain = Force * 0.5 * (Actual - Ideal)
Pain = Force * (Actual - Ideal) / 2.0

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 5.23 1 -4.94 5.832 96.923 89.831 -41.358 146.656
0 5.23 2 -2.23 46.949 200.941 192.544 -394.208 1654.979
0 5.23 3 9.09 37.199 311.141 303.866 -270.626 984.403
0 5.23 4 -7.74 451.285 386.241 387.032 356.875 141.108
0 5.23 5 5.80 226.761 498.038 500.571 574.442 727.603
0 5.23 6 -14.35 9.708 592.176 580.420 -114.132 670.889
0 5.23 7 4.54 359.893 701.937 699.319 -942.430 1233.942
0 5.23 8 15.45 41.169 813.253 810.225 -124.669 188.763
0 5.23 9 -6.33 253.599 885.373 888.445 778.823 1195.914
0 5.23 10 4.72 88.399 996.772 999.494 240.608 327.447
0 5.23 11 -9.23 9.339 1092.321 1085.540 -63.327 214.711
1 -4.94 0 5.23 5.832 1103.077 1110.169 41.358 146.656
1 -4.94 2 -2.23 3.059 99.584 102.713 9.572 14.976
1 -4.94 3 9.09 0.366 203.955 214.035 3.691 18.602
1 -4.94 4 -7.74 63.831 310.355 297.201 -839.596 5521.784
1 -4.94 5 5.80 16.209 387.663 410.740 374.065 4316.228
1 -4.94 6 -14.35 2.292 498.885 490.589 -19.014 78.867
1 -4.94 7 4.54 15.344 600.124 609.488 143.681 672.720
1 -4.94 8 15.45 0.732 701.977 720.394 13.485 124.180
1 -4.94 9 -6.33 24.272 813.326 798.614 -357.097 2626.886
1 -4.94 10 4.72 45.310 895.760 909.663 629.968 4379.358
1 -4.94 11 -9.23 0.346 996.045 995.709 -0.116 0.020
2 -2.23 0 5.23 46.949 999.059 1007.456 394.208 1654.979
2 -2.23 1 -4.94 3.059 1100.416 1097.287 -9.572 14.976
2 -2.23 3 9.09 2.323 111.876 111.322 -1.287 0.356
2 -2.23 4 -7.74 16.489 194.369 194.488 1.966 0.117
2 -2.23 5 5.80 138.697 313.790 308.027 -799.406 2303.756
2 -2.23 6 -14.35 52.589 386.459 387.876 74.502 52.773
2 -2.23 7 4.54 140.572 498.037 506.775 1228.231 5365.742
2 -2.23 8 15.45 5.110 603.257 617.681 73.711 531.601
2 -2.23 9 -6.33 159.585 701.941 695.900 -963.948 2911.292
2 -2.23 10 4.72 101.993 813.028 806.950 -619.897 1883.826
2 -2.23 11 -9.23 75.134 884.724 892.996 621.502 2570.492
3 9.09 0 5.23 37.199 888.859 896.134 270.626 984.403
3 9.09 1 -4.94 0.366 996.045 985.965 -3.691 18.602
3 9.09 2 -2.23 2.323 1088.124 1088.678 1.287 0.356
3 9.09 4 -7.74 0.566 92.079 83.166 -5.048 22.494
3 9.09 5 5.80 7.689 199.961 196.705 -25.035 40.756
3 9.09 6 -14.35 2.365 299.082 276.554 -53.284 600.189
3 9.09 7 4.54 25.943 386.837 395.453 223.525 962.928
3 9.09 8 15.45 7.423 498.023 506.359 61.878 257.914
3 9.09 9 -6.33 10.336 587.268 584.579 -27.800 37.383
3 9.09 10 4.72 10.261 701.977 695.628 -65.147 206.811
3 9.09 11 -9.23 12.282 812.395 781.674 -377.311 5795.754
4 -7.74 0 5.23 451.285 813.759 812.968 -356.875 141.108
4 -7.74 1 -4.94 63.831 889.645 902.799 839.596 5521.784
4 -7.74 2 -2.23 16.489 1005.631 1005.512 -1.966 0.117
4 -7.74 3 9.09 0.566 1107.921 1116.834 5.048 22.494
4 -7.74 5 5.80 18.108 111.736 113.539 32.642 29.421
4 -7.74 6 -14.35 1.586 202.709 193.388 -14.786 68.908
4 -7.74 7 4.54 376.137 315.014 312.287 -1025.750 1398.642
4 -7.74 8 15.45 2.173 427.706 423.193 -9.809 22.135
4 -7.74 9 -6.33 97.215 498.177 501.413 314.560 508.915
4 -7.74 10 4.72 46.695 606.077 612.462 298.140 951.786
4 -7.74 11 -9.23 23.293 701.977 698.508 -80.808 140.168
5 5.80 0 5.23 226.761 701.962 699.429 -574.442 727.603
5 5.80 1 -4.94 16.209 812.337 789.260 -374.065 4316.228
5 5.80 2 -2.23 138.697 886.210 891.973 799.406 2303.756
5 5.80 3 9.09 7.689 1000.039 1003.295 25.035 40.756
5 5.80 4 -7.74 18.108 1088.264 1086.461 -32.642 29.421
5 5.80 7 4.54 72.968 202.097 198.748 -244.348 409.120
5 5.80 8 15.45 37.645 313.133 309.654 -130.948 227.752
5 5.80 9 -6.33 335.699 386.204 387.874 560.530 467.970
5 5.80 10 4.72 63.070 498.076 498.923 53.415 22.619
5 5.80 11 -9.23 9.907 593.238 584.969 -81.922 338.712
6 -14.35 0 5.23 9.708 607.824 619.580 114.132 670.889
6 -14.35 1 -4.94 2.292 701.115 709.411 19.014 78.867
6 -14.35 2 -2.23 52.589 813.541 812.124 -74.502 52.773
6 -14.35 3 9.09 2.365 900.918 923.446 53.284 600.189
6 -14.35 4 -7.74 1.586 997.291 1006.612 14.786 68.908
6 -14.35 7 4.54 1.087 111.876 118.899 7.631 26.795
6 -14.35 9 -6.33 32.218 314.848 308.025 -219.841 750.059
6 -14.35 10 4.72 3.984 401.085 419.074 71.670 644.639
6 -14.35 11 -9.23 1.948 498.023 505.120 13.827 49.066
7 4.54 0 5.23 359.893 498.063 500.681 942.430 1233.942
7 4.54 1 -4.94 15.344 599.876 590.512 -143.681 672.720
7 4.54 2 -2.23 140.572 701.963 693.225 -1228.231 5365.742
7 4.54 3 9.09 25.943 813.163 804.547 -223.525 962.928
7 4.54 4 -7.74 376.137 884.986 887.713 1025.750 1398.642
7 4.54 5 5.80 72.968 997.903 1001.252 244.348 409.120
7 4.54 6 -14.35 1.087 1088.124 1081.101 -7.631 26.795
7 4.54 8 15.45 1.087 111.876 110.906 -1.054 0.511
7 4.54 9 -6.33 22.275 188.167 189.126 21.346 10.228
7 4.54 10 4.72 107.402 305.931 300.175 -618.188 1779.102
7 4.54 11 -9.23 99.459 386.338 386.221 -11.575 0.674
8 15.45 0 5.23 41.169 386.747 389.775 124.669 188.763
8 15.45 1 -4.94 0.732 498.023 479.606 -13.485 124.180
8 15.45 2 -2.23 5.110 596.743 582.319 -73.711 531.601
8 15.45 3 9.09 7.423 701.977 693.641 -61.878 257.914
8 15.45 4 -7.74 2.173 772.294 776.807 9.809 22.135
8 15.45 5 5.80 37.645 886.867 890.346 130.948 227.752
8 15.45 7 4.54 1.087 1088.124 1089.094 1.054 0.511
8 15.45 9 -6.33 3.005 92.079 78.220 -41.648 288.602
8 15.45 10 4.72 1.087 182.192 189.269 7.690 27.211
8 15.45 11 -9.23 6.564 288.028 275.315 -83.447 530.435
9 -6.33 0 5.23 253.599 314.627 311.555 -778.823 1195.914
9 -6.33 1 -4.94 24.272 386.674 401.386 357.097 2626.886
9 -6.33 2 -2.23 159.585 498.059 504.100 963.948 2911.292
9 -6.33 3 9.09 10.336 612.732 615.421 27.800 37.383
9 -6.33 4 -7.74 97.215 701.823 698.587 -314.560 508.915
9 -6.33 5 5.80 335.699 813.796 812.126 -560.530 467.970
9 -6.33 6 -14.35 32.218 885.152 891.975 219.841 750.059
9 -6.33 7 4.54 22.275 1011.833 1010.874 -21.346 10.228
9 -6.33 8 15.45 3.005 1107.921 1121.780 41.648 288.602
9 -6.33 10 4.72 12.072 110.641 111.050 4.930 1.007
9 -6.33 11 -9.23 10.725 191.502 197.096 59.991 167.785
10 4.72 0 5.23 88.399 203.228 200.506 -240.608 327.447
10 4.72 1 -4.94 45.310 304.240 290.337 -629.968 4379.358
10 4.72 2 -2.23 101.993 386.972 393.050 619.897 1883.826
10 4.72 3 9.09 10.261 498.023 504.372 65.147 206.811
10 4.72 4 -7.74 46.695 593.923 587.538 -298.140 951.786
10 4.72 5 5.80 63.070 701.924 701.077 -53.415 22.619
10 4.72 6 -14.35 3.984 798.915 780.926 -71.670 644.639
10 4.72 7 4.54 107.402 894.069 899.825 618.188 1779.102
10 4.72 8 15.45 1.087 1017.808 1010.731 -7.690 27.211
10 4.72 9 -6.33 12.072 1089.359 1088.950 -4.930 1.007
10 4.72 11 -9.23 3.839 85.216 86.046 3.187 1.323
11 -9.23 0 5.23 9.339 107.679 114.460 63.327 214.711
11 -9.23 1 -4.94 0.346 203.955 204.291 0.116 0.020
11 -9.23 2 -2.23 75.134 315.276 307.004 -621.502 2570.492
11 -9.23 3 9.09 12.282 387.605 418.326 377.311 5795.754
11 -9.23 4 -7.74 23.293 498.023 501.492 80.808 140.168
11 -9.23 5 5.80 9.907 606.762 615.031 81.922 338.712
11 -9.23 6 -14.35 1.948 701.977 694.880 -13.827 49.066
11 -9.23 7 4.54 99.459 813.662 813.779 11.575 0.674
11 -9.23 8 15.45 6.564 911.972 924.685 83.447 530.435
11 -9.23 9 -6.33 10.725 1008.498 1002.904 -59.991 167.785
11 -9.23 10 4.72 3.839 1114.784 1113.954 -3.187 1.323
---- ------ ---- ------ -------- -------- -------- ---------- ----------
painSum 57647.805

Monz, are your eyes glazed over yet? I hope I've provided something
more than confusion, but please feel free to ask any questions you may
have!

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
> http://www.egroups.com/message/tuning/12668

Hi John. I'll use the first part of this post as a separate
definition for 'spring', which I also need in the Dictionary
... unless you feel the need to write a definition specifically
for 'spring'. The second part will be the definition for 'COFT'.

> Now COFT. If the tuning of a piece is fixed, then two of the
> three spring types disappear: horizontal motion within a pitch
> is gone, so the value of those springs is of no importance.
> Grounding becomes a non-issue, because drift is impossible with
> a fixed tuning, and any tuning can be moved in absolute tuning
> space so that average deflection is zero (or, notes are tuned
> only down from 12-tET, or whatever...). So the "only" thing left
> to worry about is to sum the values of all the vertical springs
> and then relax them to the least pain position.

Hmmm... unless I'm misunderstanding something, you might want
to rethink some of the absolutes you're seeing here. It's
apparent to me that you're only considering 12-note fixed tunings.
A fixed tuning with many more notes can certainly lead to drift.
I'm thinking here of, say, 53-tET (maybe even 31-tET?).

The 53-tET step-size is a 'mean comma' that gives a good
approximation of both the Pythagorean and syntonic commas, and
I could forsee lots of drift. Feedback appreciated.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

[Monz wrote:]
>>Hmmm... unless I'm misunderstanding something, you might want
>>to rethink some of the absolutes you're seeing here. It's
>>apparent to me that you're only considering 12-note fixed tunings.
>>A fixed tuning with many more notes can certainly lead to drift.
>>I'm thinking here of, say, 53-tET (maybe even 31-tET?).

>>The 53-tET step-size is a 'mean comma' that gives a good
>>approximation of both the Pythagorean and syntonic commas, and
>>I could forsee lots of drift. Feedback appreciated.

As it happens, I'm aware of the wonderful properties of 53-tET.

[Paul E:]
>Monz, in this context, fixed means "only one pitch per pitch-class" or
>better yet, "only one pitch per notated note-name". So if you're
>playing a diatonic piece in 53-tET, 53-tET would not be considered a
>fixed tuning, if you use both degree 8 and degree 9 for the note "D".

>Although it is true that John's program so far only handles 12-note
>fixed tunings, there is nothing in his concept that prevents it from
>being adapted to tunings with more notes, AS LONG AS ALL THE NOTES ARE
>NOTATED DIFFERENTLY. The only reason it stops at 12 now is because all
>the MIDI files he's been retuning have only 12 pitch classes, and
>contain no way of distinguishing , say, G# from Ab, even if the
>composer originally made these distinctions in the score. If MIDI files
>did contain these distinctions, John would calculate COFTs with
>_different_ values for G# and Ab, and hence fixed tunings with more
>than 12 notes. I'll let John expound on enhancements to his program
>that he may forsee for the future . . .

Monz, my original purpose in breaking out COFT values from the spring
adaptive model was with the idea of tuning a 12-note fixed-pitch
instrument (piano? harpsichord?) for a performance of a particular
piece.

Since then, Paul, Margo, and others have made the point, as you do,
that it would make sense to calculate COFT values for more than 12
notes per octave. David Keenan, as you undoubtedly know, is also a
major proponent of G#/Ab (etc.) distinctions, and has even written code
to split out notes, given 12-tone input.

I'm part way through writing code that will (I hope some day) accomplish
this. I'm having problems with it going crazy, specifying movement of
fifths that is clearly excessive. I'm using a 31-tET model, my own
choice, but also recommended by others. Understand, the 31-tET model is
a computational convenience; what's really taking place is a tracking of
fifths in the extended circle that will yield a set of notes > 12 but
reasonable in number, for an enhanced COFT calculation. Where these
notes are actually tuned is independent of 31-tET, though it might
track it fairly closely for some sequences, depending upon their
distribution of intervals! (Paul E has made the point that meantone
"is" COFT in some meaningful way, for an unknown set of pieces).

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/12668

> [Monz:]
> >(BTW, I need a definition for COFT).
>
> Happy to oblige!

Thanks so much, John, for posting this again. This is _supposedly_
"review" (ahem), but it takes a few times!
___________ ____ __ __
Joseph Pehrson

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/12682

> I'm part way through writing code that will (I hope some day)
accomplish this. I'm having problems with it going crazy, specifying
movement of
> fifths that is clearly excessive. I'm using a 31-tET model, my own
> choice, but also recommended by others. Understand, the 31-tET
model is a computational convenience; what's really taking place is a
tracking of
> fifths in the extended circle that will yield a set of notes > 12
but
> reasonable in number, for an enhanced COFT calculation. Where
these
> notes are actually tuned is independent of 31-tET, though it might
> track it fairly closely for some sequences, depending upon their
> distribution of intervals! (Paul E has made the point that meantone
> "is" COFT in some meaningful way, for an unknown set of pieces).
>
> JdL

This is getting more and more sophisticated all the time. Now that
Schubert has been done, might itbe possible to "retune" a more modern
piece in 31 with this??... let's say something like Schoenberg's
Verklarte Nacht in just... or wouldn't that make any sense with all
the extended tertial harmonies (??) It certainly would make some
"challenging" just chord changes (??)

_________ ____ __ __ _
Joseph Pehrson

> Joseph Pehrson <josephpehrson@c...> wrote:
> http://www.egroups.com/message/tuning/12729
>
> This is getting more and more sophisticated all the time. Now
> that Schubert has been done, might itbe possible to "retune" a
> more modern piece in 31 with this??... let's say something like
> Schoenberg's Verklarte Nacht in just... or wouldn't that make
> any sense with all the extended tertial harmonies (??) It
> certainly would make some "challenging" just chord changes (??)

Joe, this is exactly the kind of thing I've had in mind for the
music of Mahler and early (pre-serial) Schoenberg.

I was going to try and sweat it out the hard way, retuning it
myself as I went along, but now maybe I'll just ask John to
feed my MIDI-files thru his program!

One thing about why there won't be as much 'modern' music
available: much of it is still under copyright. Mahler's
copyrights just expired within the last couple of decades and
those on Schoenberg's early pieces just within the past few
years; much Schoenberg is still protected. 'Classical MIDI
Archives' (on the web), the main source for MIDI files, has
a strict policy of not violating copyright.

I have about 1/3 of _Verkaerte Nacht_ sequenced so far, in 12-tET:
http://www.ixpres.com/interval/monzo/schoenberg/verk-nac.mid

-monz
http://www.ixpres.com/interval/monzo/homepage.html

[Joseph Pehrson:]
>>I was thinking more of very early works like Transfigured Night or
>>the String Quartet #2...

[Paul Erlich:]
>Certainly, it would make a lot of sense to throw these pieces at John
>deLaubenfels's adaptive tuning programs, both 5-limit and 7-limit.

Why, Paul! Can it be that you're "softening on seven"? My impression
has been that you pretty much consider a 7-limit treatment of pieces
originally written in 12-tET to be inappropriate, or am I mistaken?

[Paul:]
>Some chords might try to come out in JI, while many others (like the
>CEGAD example I love to remind people of) are inherently tempered in
>some way, and a few chords, like augmented triads and diminished
>seventh chords, are in certain respects ideal in 12-tET. John has
>already retuned pieces from the Romantic period so the difference here
>would only be one of degree.

Right - as you know I aim for 12-tET for the examples you've given,
including any chord with more than 3 fifths in a row (CEGAD has FIVE
fifths in a row, so is doubly impossible to tune "correctly").

And I'm happy to throw my procedure at any single-voice MIDI files (in
practice, this usually means piano works) from any period. I've got
a wild-sounding "Rite of Spring" piano sequence that sounds even wilder
retuned.

Unfortunately, when it comes to generating sequences to throw at my
program, I need someone else to be the keyboardist; my skills are
lacking in that regard!

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
> http://www.egroups.com/message/tuning/12761
>
> [Joseph Pehrson:]
> >> I was thinking more of very early works like Transfigured
> >> Night or the String Quartet #2...
>
> [Paul Erlich:]
> > Certainly, it would make a lot of sense to throw these pieces at
> > John deLaubenfels's adaptive tuning programs, both 5-limit and
> > 7-limit.
>
> Why, Paul! Can it be that you're "softening on seven"? My
> impression has been that you pretty much consider a 7-limit
> treatment of pieces originally written in 12-tET to be
> inappropriate, or am I mistaken?

In fact, John, in the case of early Schoenberg there's good
reason to use at least 11-limit, possibly even 13.

Schoenberg illustrated the major scale as representing harmonics
up to the 12th in his _Harmonielehre_ [1911], and the complete
12-tET chromatic scale as representing a concise 13-limit system
in the lecture 'Problems of Harmony' [1933, published in English
in _Style and Idea_].

Some problems with this have been pointed out before, by Partch
in _Genesis_, and in discussions here in the List Archives.
I've also written quite a lot about Schoenberg's intonational
ideas in my book, with more on the way.

Let's here 5-, 7-, 11-, and 13-limit versions of what I've
sequenced so far of _Verkaerte Nacht_!
http://www.ixpres.com/interval/monzo/schoenberg/verk-nac.mid

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
> --- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/12769

>
> Let's here 5-, 7-, 11-, and 13-limit versions of what I've
> sequenced so far of _Verkaerte Nacht_!
> http://www.ixpres.com/interval/monzo/schoenberg/verk-nac.mid
>

Hear, hear!
________ ____ __ __ _
Joseph Pehrson

[I wrote:]
>>Right - as you know I aim for 12-tET for the examples you've given,
>>including any chord with more than 3 fifths in a row (CEGAD has FIVE
>>fifths in a row, so is doubly impossible to tune "correctly").

[Paul E:]
>um, I count four

My mistake! I should've said, "more than 3 notes forming successive
fifths (CEGAD has FIVE notes forming successive fifths...".

[Paul:]
>Why not aim for a mild meantone as in the optimal tunings I found for
>these chain-of-fifth chords?

That could be done! I could do it without any program changes, just
by creating some new tuning options with additional tuning files
included in the mix. It might actually make a big improvement in the
results for sequences that have a lot of chords constructed that way.
The same thing could be done to give more options for augmented triads
and full diminished seventh chords.

That said, my experience is that getting the values all balanced
among different tuning files takes a lot of work, so I may not get
around to it for a while...

JdL

[Monz wrote:]
>In fact, John, in the case of early Schoenberg there's good
>reason to use at least 11-limit, possibly even 13.

>Schoenberg illustrated the major scale as representing harmonics
>up to the 12th in his _Harmonielehre_ [1911], and the complete
>12-tET chromatic scale as representing a concise 13-limit system
>in the lecture 'Problems of Harmony' [1933, published in English
>in _Style and Idea_].

>Some problems with this have been pointed out before, by Partch
>in _Genesis_, and in discussions here in the List Archives.
>I've also written quite a lot about Schoenberg's intonational
>ideas in my book, with more on the way.

>Let's here 5-, 7-, 11-, and 13-limit versions of what I've
>sequenced so far of _Verkaerte Nacht_!
>http://www.ixpres.com/interval/monzo/schoenberg/verk-nac.mid

Thanks for the info, Monz! Just downloaded your .midi file; will
listen in a moment. Are all limits presented in order?

[Paul E:]
>Don't forget 9-limit!

OK, I'm confused here. 5-limit music includes 9/8 ratios, yes? 7-limit
music can include 9/7, no? So what is "9-limit" music?

JdL

[Monz wrote:]
>Let's here 5-, 7-, 11-, and 13-limit versions of what I've
>sequenced so far of _Verkaerte Nacht_!
>http://www.ixpres.com/interval/monzo/schoenberg/verk-nac.mid

Well, I misunderstood you; that sequence is all 12-tET, as I assume
you already knew and were trying to say? It seems to have a problem:
36 minutes long, but a long, LONG silence between about 9 minutes
and 35 minutes. I'd actually forgotten that the sequencer was still
running when the last part kicked in with a blast! Nice music, though!!
I must give Schoenberg a closer look.

And speaking of nice stuff, let me plug Monzo's VERY interesting web
page, a BIG, LONG web page, at

http://www.ixpres.com/interval/monzo/schoenberg/Vienna1905.htm

This is fascinating reading! One anecdote (O.T.): Alma Schindler meets
and weds Gustav Mahler. Mahler may be a musical genius, but as a human
being, he's pretty pathetic. Alma, a talented and aspiring composer, is
pursued by Mahler, but then, to quote Monzo:

Alma's chief ambition is to be a composer, but Mahler tells her
bluntly that there will be only one great composer in their house
(and guess who that is?...). Alma tries to hold back her tears and
accept her fate stoically, but she is heartbroken, and altho she
loves him and submits to his demand, she will remain bitter about
this sacrifice for the rest of her years with Mahler.

Arggh!! Who knows what we have missed? (end of O.T.)

As for the sequence: it'd have to be tuned by hand, I assume, to be
true to Schoenberg's ideas? (I'm still very vague - did he advocate a
fixed tuning or a dynamic one?). I can't tune it because there are
different voices ("programs") sounding together.

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
> http://www.egroups.com/message/tuning/12797
>
> [Monz wrote:]
> > Let's here [_sic_: hear] 5-, 7-, 11-, and 13-limit versions
> > of what I've sequenced so far of _Verkaerte Nacht_!
> > http://www.ixpres.com/interval/monzo/schoenberg/verk-nac.mid
>
> Well, I misunderstood you; that sequence is all 12-tET, as I
> assume you already knew and were trying to say?

Oh yes, my sequence is so far still in 12-tET. It's nowhere near
finished yet, as you found out...

> It seems to have a problem:
> 36 minutes long, but a long, LONG silence between about 9 minutes
> and 35 minutes. I'd actually forgotten that the sequencer was
> still running when the last part kicked in with a blast!

I'm sorry I didn't mention that in my previous post. Approximately
the first 1/3 of the piece is sequenced, then a silence of about
a minute, then an interesting short section is sequenced, then
a long silence, then the very end of the piece is sequenced. The
total length when it's finished will be about 1/2-hour.

> Nice music, though!! I must give Schoenberg a closer look.

This early piece is still by far the most popular of all
Schoenberg's compositions. I should post the poem on a webpage
too; the music is very poetic and very evocative of the story
in the poem.

Thanks for plugging my 'BIG, LONG web page', at
http://www.ixpres.com/interval/monzo/schoenberg/Vienna1905.htm

I've always taken those words as a nice compliment, but never
dreamed that they'd be applied to a webpage... ;-)
(to the young kiddies on the List who don't get it, NEVER MIND!)

>
> This is fascinating reading!

Thanks for the great compliment, John. I've really tried to
a convey a sense of the excitement I feel as the myriad threads
of music-history intertwine closer and closer together.

I've always thought that the *interactions* between composers
in Europe, and specifically in Vienna, during the period covered
on my webpage, were important to the new developments in the
music. But tying together the different life stories on one
chronology really starts to bring out the connections.

> Mahler may be a musical genius, but as a human being, he's
> pretty pathetic.

This position can be defended by stating the facts of Mahler's
life, but it would have to be very much a one-sided presentation.
I'm sorry if this impression is conveyed too strongly by my
webpage, but it's a result of the terseness to which I tried
to limit myself, in order that the damned thing wouldn't get too
ridiculously big.

The reason we Mahler fans find him so interesting is that his
was an extremely complex personality, full of contradictions.
He was a truly warm-hearted human being, full of sincere concern
for his fellow-man (the opening sentence of Schoenberg's
memorial dedication in his _Harmonielehre_ is: "Gustav Mahler
was a saint."). At the same time, he was a cold and ruthlessly
calculating career opportunist, who could walk over anyone
who got in his way.

As I state on my webpage, while it's true that Mahler forbade
Alma to continue composing at the time they married, an affair
she had 8 years later finally woke Mahler up to the neglect
to which he subjected her, and he gleefully began listening to
her songs and subsequently encouraged her to publish them.

So in response to:

> Arggh!! Who knows what we have missed? (end of O.T.)

The answer is: the songs are out there. They've been published,
performed, and recorded (altho I've never heard them myself).

There's a new movie just being put in the can, _Bride of the Wind_.
Due out soon, it's about Alma, and of course features Mahler and
his music too.

Alma's diary was published in English in abridged form last year,
and is very fascinating reading indeed. It covers her last
teenage years and those up to her marriage to Mahler.

>
> As for the sequence: it'd have to be tuned by hand, I assume,
> to be true to Schoenberg's ideas? (I'm still very vague - did
> he advocate a fixed tuning or a dynamic one?). I can't tune it
> because there are different voices ("programs") sounding together.

Oh, that's right, I remember now that you can only do single-voice
stuff.

The *only* tuning system ever advocated by Schoenberg with any
degree of argument was 12-tET. He *did* mention the possibility
of other ETs and also 'untempered' (i.e., JI) music in his
_Harmonielehre_, but lamented that there were too few instruments
capable of playing these other tunings, and so until those
instruments were widely available, the only practical recourse
was to wring new harmonies out of 12-tET.

The logic and force of his arguments proved strong enough to
play a big part in ensuring the adoption of 12-tET in much of
the Euro-/Americo-centric music world for much of the 1900s.
But I think we've finally broken out of that box...

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
http://www.egroups.com/message/tuning/12849

> >
> > As for the sequence: it'd have to be tuned by hand, I assume,
> > to be true to Schoenberg's ideas? (I'm still very vague - did
> > he advocate a fixed tuning or a dynamic one?). I can't tune it
> > because there are different voices ("programs") sounding together.
>
>
> Oh, that's right, I remember now that you can only do single-voice
> stuff.
>

Well, this makes sense, since John has to tune every note on a
separate channel, I believe. Is that right?? Is it going to be
possible to convert the Monz sequence to a "one channel" affair, or
is the Transfigured Night project doomed??
________ ____ __ __
Joseph Pehrson

[I wrote:]
>>> As for the sequence: it'd have to be tuned by hand, I assume,
>>> to be true to Schoenberg's ideas? (I'm still very vague - did
>>> he advocate a fixed tuning or a dynamic one?). I can't tune it
>>> because there are different voices ("programs") sounding together.

[Monz:]
>>Oh, that's right, I remember now that you can only do single-voice
>>stuff.

[Joseph Pehrson:]
>Well, this makes sense, since John has to tune every note on a
>separate channel, I believe. Is that right?? Is it going to be
>possible to convert the Monz sequence to a "one channel" affair, or
>is the Transfigured Night project doomed??

They're all string voices, so it might not kill the piece completely
to be collapsed into a single channel with a single voice. If I had
time, I could enhance my program to handle sequences such as these;
there are actually plenty of GM channels for these four string
instruments. BUT, it's probably not going to happen real soon...

JdL

> [Joseph Pehrson]
> http://www.egroups.com/message/tuning/12854
>
> Well, this makes sense, since John has to tune every note on a
> separate channel, I believe. Is that right?? Is it going to be
> possible to convert the Monz sequence to a "one channel" affair,
> or is the Transfigured Night project doomed??

It just means that in order to preserve the individual instrumental
tracks, _Verklaerte Nacht_ will have to be retuned by hand.
An extremely tedious job, but I was eventually going to do it
anyway. So the project is doomed only in the sense that you'll
have to wait until I get around to doing it, which could take
decades given all the other stuff I'm hoping to do...

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
> http://www.egroups.com/message/tuning/12856
>
> [in _Verklaerte Nacht_] They're all string voices, so it might
> not kill the piece completely to be collapsed into a single
> channel with a single voice.

Good observation, John! Try it - I think it would work at least
well enough to give us an idea of what some retuned versions would
sound like.

I'd like to hear both odd- and prime-limit versions for 5 and 7.

> If I had time, I could enhance my program to handle sequences
> such as these; there are actually plenty of GM channels for these
> four string instruments. BUT, it's probably not going to happen
> real soon...

Well... _Verklaerte Nacht_ is just a *little* bit more complicated
than that: it's a *sextet*, not a quartet... 2 violins, 2 violas,
2 cellos.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

>
> JdL

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/12857

>
> It just means that in order to preserve the individual instrumental
> tracks, _Verklaerte Nacht_ will have to be retuned by hand.
> An extremely tedious job, but I was eventually going to do it
> anyway. So the project is doomed only in the sense that you'll
> have to wait until I get around to doing it, which could take
> decades given all the other stuff I'm hoping to do...
>

Well the piece was written, I believe, in 1899... so there's no real
rush to "Transfigure" it! I enjoyed the MIDI excerpt AS IS, although
the new tuning will be a real rush!
____________ _____ _ __ _
Joseph Pehrson

[I wrote:]
>>If I had time, I could enhance my program to handle sequences
>>such as these; there are actually plenty of GM channels for these
>>four string instruments. BUT, it's probably not going to happen
> real soon...

[Monz:]
>Well... _Verklaerte Nacht_ is just a *little* bit more complicated
>than that: it's a *sextet*, not a quartet... 2 violins, 2 violas,
>2 cellos.

Oops! Missed that. Still enough channels for a program more capable
of allocating them on the fly. Are there any dyads within a single
instrument? Probably are, so trying to fake it with your existing
channel assignment would NOT give good results. Must wait.

JdL