Yahoo Tuning Groups Ultimate Backup tuning-math Refinement (?) of "true 5-limit" adaptive tuning

The term "true 5-limit" is Paul E's; it refers to targeting intervals
solely on the basis of their 12-tET interval rather than playing a more
complex contextual game using tuning files. Thus,

3 semitones -> 6:5 target interval (wider than 12-tET)
4 semitones -> 5:4 target interval (narrower than 12-tET)
5 semitones -> 4:3 target interval (slightly narrower than 12-tET)

With inversions (7, 8, 9 semitones) treated as expected (all my tuning
is octave-invariant, so an interval is "never" greater than 6
semitones).

A dominant 7th chord thus stacks a 6:5 minor third above the fifth
degree of 3:2, so that the 7th degree winds up about 9/10 of root above,
rather than 8/9 of root when I use my 5-limit tuning files.

In previous work, source intervals of 1, 2, and 6 semitones were sprung
very weakly, essentially to say "don't care" about how thy are tuned.

Recently, Robert Walker has provided feedback on his preferences for
works of C.P.E Bach and J.C. Bach (flute & harpsichord) that I've
retuned for him. His feeling is that, when the fifth degree is strongly
present in a dom 7, the 7th degree should be sharp (approx 9/10 of root)
but when the fifth degree is not present (or only very weakly present),
the 7th degree should be a bit flatter (more like 8/9 of root).

At first I thought a new set of tuning files would be necessary to
accomplish this, but then realized that a weak, but not negligible,
tritone spring would also address the goal. A tritone _must_ be
targeted to 600 cents, because we know nothing of inversion(s) in the
tuning-file free approach.

So, if the fifth is missing, for example, C-E-Bb, then the Bb becomes
targeted to 386 + 600 cents = 986 cents, slightly flatter than 8/9
of root above. But if the tritone spring is fairly weak, then when
the fifth _is_ present, the 7th degree is pushed back up.

The following analysis is for the notes C-E-G-Bb in equal volume.
Major and minor thirds, fourths, and their inversions all have full
nominal strength in their vertical springs. Tritones have 1/8
nominal strength (2.560 rather than 20.480). The second analysis
has the fifth degree missing. Note the different location of the
7th degree (note 10) compared to root (note 0): 187.8 cents in
the full chord, 208.7 cents in the C-E-Bb chord.

I'm giving major seconds 1/16 nominal strength and 200 cent target.

halfest3 cseven:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 0.08 4 -12.81 20.480 386.314 387.104 16.190 6.400
0 0.08 7 0.49 20.480 701.955 700.405 -31.749 24.609
0 0.08 10 12.24 1.280 1000.000 1012.155 15.558 94.556
4 -12.81 0 0.08 20.480 813.686 812.896 -16.190 6.400
4 -12.81 7 0.49 20.480 315.641 313.300 -47.939 56.108
4 -12.81 10 12.24 2.560 600.000 625.051 64.130 803.251
7 0.49 0 0.08 20.480 498.045 499.595 31.749 24.609
7 0.49 4 -12.81 20.480 884.359 886.700 47.939 56.108
7 0.49 10 12.24 20.480 315.641 311.750 -79.688 155.035
10 12.24 0 0.08 1.280 200.000 187.845 -15.558 94.556
10 12.24 4 -12.81 2.560 600.000 574.949 -64.130 803.251
10 12.24 7 0.49 20.480 884.359 888.250 79.688 155.035
---- ------ ---- ------ -------- -------- -------- ---------- ----------
painSum 1139.960

halfest3 csevenn5:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 7.30 4 -5.84 27.307 386.314 386.861 14.949 4.092
0 7.30 10 -1.46 1.707 1000.000 991.241 -14.949 65.471
4 -5.84 0 7.30 27.307 813.686 813.139 -14.949 4.092
4 -5.84 10 -1.46 3.413 600.000 604.380 14.949 32.736
10 -1.46 0 7.30 1.707 200.000 208.759 14.949 65.471
10 -1.46 4 -5.84 3.413 600.000 595.620 -14.949 32.736
---- ------ ---- ------ -------- -------- -------- ---------- ----------
painSum 102.299

These are analyses with vertical considerations only. In real sequences,
horizontal and grounding considerations will alter the results somewhat
at any given location.

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 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.

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

Naming convention: I use

es2 - previously done, with negligible 1,2,6 springs
es3 - this new spring set, as described above

JdL

🔗Paul Erlich <paul@stretch-music.com>

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> >>At first I thought a new set of tuning files would be necessary to
> >>accomplish this, but then realized that a weak, but not
negligible,
> >>tritone spring would also address the goal. A tritone _must_ be
> >>targeted to 600 cents, because we know nothing of inversion(s) in
the
> >>tuning-file free approach.
>
> [Paul E:]
> >I don't understand this . . . I thought your program recognized
> >octave-equivalence, and hence inversions, independently of the
tuning
> >files.
>
> Your statement is correct. But, for example, in 7-limit, a tritone
is
> NOT 600 cents, yet the program knows which side of the inversion
it's
> on because of pattern-matching between the notes on the the
desirable
> intervals specified by the tuning file.

This I understand . . .

>Am I being clear?

Still confused as to the original statement.
>
> >And by 'weak', do you mean 'strength approaches zero'?
>
> That would be 'negligible'. Here, 'weak' means 1/8 of nominal. Do
the
> tables make sense?
>
Very much so. I guess I was just wondering if your concept here was
similar to those cases where you needed to put a negligible strength
on some spring for computational reasons (i.e., the matrix
wouldn't 'relax' correctly otherwise) but conceptually the strength
was really zero.

🔗John A. deLaubenfels <jdl@adaptune.com>

[I wrote:]
>>>>At first I thought a new set of tuning files would be necessary to
>>>>accomplish this, but then realized that a weak, but not negligible,
>>>>tritone spring would also address the goal. A tritone _must_ be
>>>>targeted to 600 cents, because we know nothing of inversion(s) in
>>>>the tuning-file free approach.

[Paul E:]
>>>I don't understand this . . . I thought your program recognized
>>>octave-equivalence, and hence inversions, independently of the tuning
>>>files.

[JdL:]
>>Your statement is correct. But, for example, in 7-limit, a tritone is
>>NOT 600 cents, yet the program knows which side of the inversion it's
>>on because of pattern-matching between the notes on the the desirable
>>intervals specified by the tuning file.

[Paul:]
>This I understand . . .

[JdL:]
>>Am I being clear?

[Paul:]
>Still confused as to the original statement.

OK, you're groking on the fact that the tritone is a very special case,
yes? That is, I can target two semitones to, say, 190 cents, which
means that the inversion, 10 semitones, will be targeted to 1010 cents.
And so on for every interval EXCEPT the tritone. By symmetry, it can
ONLY be targeted to 600 cents. If I target 590 cents, I am
simultaneously targeting 610 cents. Yes?

[Paul:]
>>>And by 'weak', do you mean 'strength approaches zero'?

[JdL:]
>>That would be 'negligible'. Here, 'weak' means 1/8 of nominal. Do
>>the tables make sense?

[Paul:]
>Very much so. I guess I was just wondering if your concept here was
>similar to those cases where you needed to put a negligible strength
>on some spring for computational reasons (i.e., the matrix
>wouldn't 'relax' correctly otherwise) but conceptually the strength
>was really zero.

In the er3/es3, all spring strengths are greater than negligible. And
in truth I could have done the er2/es2 with zero-strength springs on 1,
2, and 6; that won't hurt the matrix (it's only infinite strength
springs that are trouble-makers!). I just left the 1% springs in place
to see those other intervals represented in the table.

JdL

🔗Paul Erlich <paul@stretch-music.com>

🔗John A. deLaubenfels <jdl@adaptune.com>

[I wrote:]
>>In the er3/es3, all spring strengths are greater than negligible. And
>>in truth I could have done the er2/es2 with zero-strength springs on
>>1, 2, and 6; that won't hurt the matrix (it's only infinite strength
>>springs that are trouble-makers!). I just left the 1% springs in
>>place to see those other intervals represented in the table.

[Paul:]
>So what you're saying is, in practice, you could use zero-strength
>springs on 1, 2, and 6 for most MIDI files, since the size of the
>tritone would be determined by various horizontal & grounding
>considerations . . . but for this single-chord example, you needed to
>use a finite-strength spring because otherwise, there would be no
>solution . . . ? In that case, I would favor using zero-strength
>springs in practice . . .

Are we clear that the 0% and the 1% springs (the latter represented
by er2/es2 files) would be almost identical in result, and that both
will be different from the er3/es3 I've been describing? Are you making
a statement about your preference as to one group vs. the other, or
commenting on how you would do the former group?

[Paul:]
>P.S. Why is this discussion on this list? I thought it was voted that
>this list should not remain separate . . . but it seems to be
>growing! A monster! Perhaps I'd feel like I was respecting the vote
>if I at least transferred ownership of this list to someone
>else . . .

If you are determined not to host this list any more, then please do
hand it off. I'd rather keep my long, detailed posts off the fat list
if possible. Maybe I'll start a boring-details@yahoogroups.com.

JdL

🔗jpehrson@rcn.com

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗jpehrson@rcn.com

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

/tuning-math/message/145

> [I wrote:]
> >>If you are determined not to host this list any more, then please
do
> >>hand it off. I'd rather keep my long, detailed posts off the fat
> >>list if possible. Maybe I'll start a boring-details@y...
>
> [Joseph Pehrson:]
> >What I was thinking is that this list could include *BOTH* the
> >salient details and *ALSO* expressions of exuberance!
>
> >Let's say, after a long detailed post, somebody *else* could just
> >write, "Dig it!"
>
> Oh, I agree! When I say "boring details", I just mean that I know
> _some_ will be bored, but I'm excited, and if others are as well,
it's
> great to hear it!
>
> >Right now, one can't do that on the "big" list, since somebody is
> >always counting the words so that people with slow connections
don't
> >have to download so much!
>
> I suppose if this list ever gets real fat, we'll have to be careful
> here as well.
>
> JdL

I suppose that's true, John... but at the moment it still isn't the
case... so I'm for lots of big numbers and stuff that I can't
understand (well I'll get PART of it!) and EXUBERANCE!

_________ ________ ________
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

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

> [Paul:]
> >So what you're saying is, in practice, you could use zero-strength
> >springs on 1, 2, and 6 for most MIDI files, since the size of the
> >tritone would be determined by various horizontal & grounding
> >considerations . . . but for this single-chord example, you needed
to
> >use a finite-strength spring because otherwise, there would be no
> >solution . . . ? In that case, I would favor using zero-strength
> >springs in practice . . .
>
> Are we clear that the 0% and the 1% springs (the latter represented
> by er2/es2 files) would be almost identical in result, and that
both
> will be different from the er3/es3 I've been describing?

Sure.

> Are you making
> a statement about your preference as to one group vs. the other, or
> commenting on how you would do the former group?

I'm asking a question, and you didn't answer (see above).
>
> [Paul:]
> >P.S. Why is this discussion on this list? I thought it was voted
that
> >this list should not remain separate . . . but it seems to be
> >growing! A monster! Perhaps I'd feel like I was respecting the
vote
> >if I at least transferred ownership of this list to someone
> >else . . .
>
> If you are determined not to host this list any more, then please do
> hand it off.

Are you volunteering?

🔗John A. deLaubenfels <jdl@adaptune.com>

[Paul wrote:]
>>>So what you're saying is, in practice, you could use zero-strength
>>>springs on 1, 2, and 6 for most MIDI files, since the size of the
>>>tritone would be determined by various horizontal & grounding
>>>considerations . . . but for this single-chord example, you needed to
>>>use a finite-strength spring because otherwise, there would be no
>>>solution . . . ? In that case, I would favor using zero-strength
>>>springs in practice . . .

[I wrote:]
>>Are we clear that the 0% and the 1% springs (the latter represented
>>by er2/es2 files) would be almost identical in result, and that both
>>will be different from the er3/es3 I've been describing?

[Paul:]
>Sure.

[JdL:]
>>Are you making
>>a statement about your preference as to one group vs. the other, or
>>commenting on how you would do the former group?

[Paul:]
>I'm asking a question, and you didn't answer (see above).

I will be glad to do my best to answer your question(s). You can help
me do so by clarifying just what it is you _are_ asking! Anyway, I
thought I had answered that question in the post before: yes. Though I
haven't run it, I'm quite sure it'll work. Guess I don't see the point
of worrying about 0% vs. 1% strength, however, unless the results are
audibly different, which I DON'T think they'd be! Would you like to
perform a blind A/B test on a piece of your choosing?

Speaking of questions unanswered, did I finally make sense about the
symmetry of the tritone forcing it to target 600 cents? Not for the
first time, I was trying to communicate something that seemed SO clear,
but apparently doing a poor job of it.

[Paul:]
>>>P.S. Why is this discussion on this list? I thought it was voted that
>>>this list should not remain separate . . . but it seems to be
>>>growing! A monster! Perhaps I'd feel like I was respecting the vote
>>>if I at least transferred ownership of this list to someone
>>>else . . .

[JdL:]
>>If you are determined not to host this list any more, then please do
>>hand it off.

[Paul:]
>Are you volunteering?

Ohmigosh, it's FAQ editor all over again! ;-> What are the duties?

JdL