Yahoo Tuning Groups Ultimate Backup tuning calculated optimum fixed tuning

JdL wrote,

>As a spin-off of the new spring model for adaptive tuning, I've written
>a function that attempts to calculate the optimum fixed tuning for an
>entire sequence.

John, I just had a brilliant flash. Why not use this optimum fixed tuning as
the grounding for the adaptively-tuned version???!!! I'm amazed I didn't
think of this before.

>I thought the first results for the Bach/Busoni piece, in D minor, might
>be interesting (values are cents from 12-tET):

> For pitch 0 6.6164
> For pitch 1 -10.1093
> For pitch 2 0.1304
> For pitch 3 5.3325
> For pitch 4 0.4689
> For pitch 5 13.5001
> For pitch 6 -11.9440
> For pitch 7 1.6882
> For pitch 8 -10.4004
> For pitch 9 1.2251
> For pitch 10 10.7833
> For pitch 11 -7.2912

Please fill us in on the details of how this is calculated.

Should I assume pitch 0 is C? Then what you have is:

Eb-Bb sharp by about 1/6 comma
Bb-F a little sharp
F-C tempered by about 4/10 comma!
C-G tempered by about 1/3 comma
G-D tempered by about 1/6 comma
D-A nearly just
A-E tempered by about 1/8 comma
E-B tempered by almost 1/2 comma !!!
B-F# tempered by about 3/10 comma
F#-C# nearly just
C#-G# tempered by about 1/10 comma

Fascinating! Sort of a mutated temperament ordinaire. I'd have to say,
though that if B-F# and especially F-C occur as simultaneities anywhere in
the piece, you'd have an interval outside the historical range of
acceptability.

>Is anyone interesting in throwing sequence(s) at this? As with adaptive
>tuning, it needs to be a single voice. I'm thinking it'd be better to
>stick with 5-limit rather than try to shoehorn 7-limit onto fixed
>tuning!

What I'd be interested in is finding an optimal 22-tone fixed tuning for a
piece of mine, with 7-limit harmony being the goal.

>BTW, I was able to turn drift correction completely off, except for an
>averaging of the final bends to center the numbers (the 12 values above
>should add up to zero). There is no "inward squeezing" on the
>deviations other than that caused by the intervals themselves.

I have no idea what that means. If I knew how you calculated the optimum
fixed tuning, I might begin to understand what that's all about.

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Herman Miller <hmiller@io.com>

🔗Paul Hahn <Paul-Hahn@library.wustl.edu>

🔗John A. deLaubenfels <jadl@idcomm.com>

So I'm sitting in the, uhhh, meditation room this morning, and suddenly
think, "Hey! The calculated optimum fixed tuning values could be used
as grounding points for adaptive tuning!" Then, about .1 second later,
I thought, "I'll bet Paul E has already posted that recommendation, and
it's waiting in this morning's digest."

Sure enough!

[Paul Erlich, TD 514.13:]
>John, I just had a brilliant flash. Why not use this optimum fixed
>tuning as the grounding for the adaptively-tuned version???!!! I'm
>amazed I didn't think of this before.

Well, how long did you have from my post to yours? That seems short
enough...

[JdL:]
>>I thought the first results for the Bach/Busoni piece, in D minor,
>>might be interesting (values are cents from 12-tET):
>> For pitch 0 6.6164
>> For pitch 1 -10.1093
>> For pitch 2 0.1304
>> For pitch 3 5.3325
>> For pitch 4 0.4689
>> For pitch 5 13.5001
>> For pitch 6 -11.9440
>> For pitch 7 1.6882
>> For pitch 8 -10.4004
>> For pitch 9 1.2251
>> For pitch 10 10.7833
>> For pitch 11 -7.2912

[Paul:]
>Please fill us in on the details of how this is calculated.

Really simply. From the full sequence and its springs, established as
before, I produce a concentrated spring set that sums all the individual
vertical springs, both in strength and desired tuning. Springs to
ground and horizontal springs I don't copy across, the latter since
there will be no horizontal motion. What results is 12 x 11 springs of
widely varied strengths (well, that counts each spring twice; there are
really 66...). I relax the nodes, very quick and easy to do, then print
out the results.

[Paul:]
>Should I assume pitch 0 is C?

Yes.

[Paul:]
>Then what you have is:
>
>Eb-Bb sharp by about 1/6 comma
>Bb-F a little sharp
>F-C tempered by about 4/10 comma!
I calculate .41 comma
>C-G tempered by about 1/3 comma
>G-D tempered by about 1/6 comma
>D-A nearly just
>A-E tempered by about 1/8 comma
>E-B tempered by almost 1/2 comma !!!
I calculate .41 comma
>B-F# tempered by about 3/10 comma
I calculate .31 comma
>F#-C# nearly just
>C#-G# tempered by about 1/10 comma

>Fascinating! Sort of a mutated temperament ordinaire. I'd have to say,
>though that if B-F# and especially F-C occur as simultaneities anywhere
>in the piece, you'd have an interval outside the historical range of
>acceptability.

I hear ya. I'm going back now to look at the strength of the springs
across those intervals; i.e., the extent to which they are actually
played in the piece. Results:

G#-Eb (?) 14.618
Eb-Bb 489.265
Bb-F 3185.846
F-C 604.659
C-G 999.440
G-D 5196.238
D-A 15395.276
A-E 11018.787
E-B 1753.688
B-F# 1031.748
F#-C# 777.062
C#-G# 26.072

Of course, other springs connect other intervals!

[JdL:]
>>Is anyone interesting in throwing sequence(s) at this? As with
>>adaptive tuning, it needs to be a single voice. I'm thinking it'd be
>>better to stick with 5-limit rather than try to shoehorn 7-limit onto
>>fixed tuning!

[Paul:]
>What I'd be interested in is finding an optimal 22-tone fixed tuning
>for a piece of mine, with 7-limit harmony being the goal.

Whew! Does the sequence begin as 22-tET, 12-tET, or something else?

[Herman Miller, TD 514.23:]
>Here's a couple you might want to try:
>http://www.io.com/~hmiller/music/dragons.mid
>http://www.io.com/~hmiller/music/rriladeni-harp.mid

>Those two are fairly conservative harmonically, and should work out
>well with a 5-limit tuning. For something a bit more exotic, you can
>see what this one comes up with:

>http://www.io.com/~hmiller/music/galticeran.mid

I have downloaded them & will post the results soon!

JdL

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

>>Please fill us in on the details of how this is calculated.

>Really simply. From the full sequence and its springs, established as
>before, I produce a concentrated spring set that sums all the individual
>vertical springs, both in strength and desired tuning. Springs to
>ground and horizontal springs I don't copy across, the latter since
>there will be no horizontal motion. What results is 12 x 11 springs of
>widely varied strengths (well, that counts each spring twice; there are
>really 66...). I relax the nodes, very quick and easy to do, then print
>out the results.

Cool! Now (back to an earlier thread) couldn't you use a similar approach to
fix all vertical triads in exact JI and use a concentrated spring set to get
an adaptive JI retuning?

>>Fascinating! Sort of a mutated temperament ordinaire. I'd have to say,
>>though that if B-F# and especially F-C occur as simultaneities anywhere
>>in the piece, you'd have an interval outside the historical range of
>>acceptability.

>I hear ya. I'm going back now to look at the strength of the springs
>across those intervals; i.e., the extent to which they are actually
>played in the piece. Results:

> G#-Eb (?) 14.618
> Eb-Bb 489.265
> Bb-F 3185.846
> F-C 604.659
> C-G 999.440
> G-D 5196.238
> D-A 15395.276
> A-E 11018.787
> E-B 1753.688
> B-F# 1031.748
> F#-C# 777.062
> C#-G# 26.072

>Of course, other springs connect other intervals!

Hmmm. I forgot to mention, E-B in your tuning is even worse than F-C and
B-F#! So these three fifths/fouths do occur with significant frequency in
the piece -- that means you'll have to modify your algorithm before you can
say it comes up with a reasonable fixed tuning.

For a lot of these problems minimax may be a better approach than least
squares. Know anything about linear programming?

>>What I'd be interested in is finding an optimal 22-tone fixed tuning
>>for a piece of mine, with 7-limit harmony being the goal.

>Whew! Does the sequence begin as 22-tET, 12-tET, or something else?

22-tET (though it doesn't exist in MIDI form yet). You can hear me playing
it (very badly) on the Tuning Punks site.

>[Herman Miller, TD 514.23:]
>>Here's a couple you might want to try:
>>http://www.io.com/~hmiller/music/dragons.mid
>>http://www.io.com/~hmiller/music/rriladeni-harp.mid

>>Those two are fairly conservative harmonically, and should work out
>>well with a 5-limit tuning. For something a bit more exotic, you can
>>see what this one comes up with:

>>http://www.io.com/~hmiller/music/galticeran.mid

>I have downloaded them & will post the results soon!

A fixed 15-tone tuning for Herman's "Porcupine Overture" progression would
be fascinating as well. A sort of 250:243-pump, if you've been following.

🔗John A. deLaubenfels <jadl@idcomm.com>

Just listened to the Bach/Busoni in COFT and it's really lovely!
One of the things that pops out is a calculation of the pain reduction
compared to 12-tET:

doCOFT(): initial pain 7065118.255
After adjustment, Total spring pain: 1934120.469

It's up on my web site, at

http://www.idcomm.com/personal/jadl/

In sharp contrast, the Schubert d894-1 file showed much less gain:

doCOFT(): initial pain 9886826.930
After adjustment, Total spring pain: 6258629.009

and corrections only in the +/-5 cent range. I haven't listened to
it yet, or posted it.

[Herman Miller, TD 514.23:]
>Here's a couple you might want to try:
>http://www.io.com/~hmiller/music/dragons.mid
>http://www.io.com/~hmiller/music/rriladeni-harp.mid

>Those two are fairly conservative harmonically, and should work out
>well with a 5-limit tuning. For something a bit more exotic, you can
>see what this one comes up with:

>http://www.io.com/~hmiller/music/galticeran.mid

Really nice sequences!! May I post the retunings? Here are the raw
numbers:

dragons:
doCOFT(): initial pain 1345509.489
For pitch 0, we have bend -2.6411 (E0 16 3F)
For pitch 1, we have bend 12.9157 (E1 0A 44)
For pitch 2, we have bend -2.2710 (E2 25 3F)
For pitch 3, we have bend 9.6005 (E3 04 43)
For pitch 4, we have bend -9.6027 (E4 7C 3C)
For pitch 5, we have bend 0.3847 (E5 0F 40)
For pitch 6, we have bend -11.1101 (E6 3F 3C)
For pitch 7, we have bend -2.9668 (E7 08 3F)
For pitch 8, we have bend 11.9587 (E8 63 43)
For pitch 9, we have bend -0.5753 (EA 69 3F)
For pitch 10, we have bend 6.3515 (EB 00 42)
For pitch 11, we have bend -12.0442 (EC 19 3C)
After adjustment, Total spring pain: 339661.291

(big deflections, good pain recovery)

rriladeni-harp:
doCOFT(): initial pain 691792.809
For pitch 0, we have bend 5.1192 (E0 4F 41)
For pitch 1, we have bend -15.9663 (E1 7B 3A)
For pitch 2, we have bend 8.2285 (E2 4C 42)
For pitch 3, we have bend -0.7889 (E3 61 3F)
For pitch 4, we have bend -0.7597 (E4 62 3F)
For pitch 6, we have bend 0.3208 (E6 0C 40)
For pitch 7, we have bend 7.6607 (E7 35 42)
For pitch 8, we have bend -12.9788 (E8 73 3B)
For pitch 9, we have bend 6.1699 (EA 79 41)
For pitch 10, we have bend 1.0906 (EB 2C 40)
For pitch 11, we have bend 1.9040 (EC 4D 40)
After adjustment, Total spring pain: 310475.968

(not so much pain recovery)

galticeran:
doCOFT(): initial pain 2218789.229
For pitch 0, we have bend -0.6434 (E0 66 3F)
For pitch 1, we have bend 4.2976 (E1 2D 41)
For pitch 2, we have bend 6.7661 (E2 11 42)
For pitch 3, we have bend -4.8488 (E3 3C 3E)
For pitch 4, we have bend -7.9945 (E4 3D 3D)
For pitch 5, we have bend 3.3638 (E5 08 41)
For pitch 6, we have bend 1.9290 (E6 4E 40)
For pitch 7, we have bend 2.2861 (E7 5C 40)
For pitch 8, we have bend -13.1980 (E8 6B 3B)
For pitch 9, we have bend 4.9792 (EA 49 41)
For pitch 10, we have bend 4.2512 (EB 2C 41)
For pitch 11, we have bend -1.1883 (EC 50 3F)
After adjustment, Total spring pain: 1556130.771

(even less pain recovery)

I've listened to the COFT'd versions but not the 12-tET yet. They sound
nice!

JdL

🔗David C Keenan <d.keenan@uq.net.au>

In TD 514.9 "John A. deLaubenfels" <jadl@idcomm.com> wrote:

>As a spin-off of the new spring model for adaptive tuning, I've written
>a function that attempts to calculate the optimum fixed tuning for an
>entire sequence. This could, in theory, be used before a performance,
>to tune a harpsichord, piano, clavichord, or other instrument with
>similar capabilities and restrictions.

One would also want to know what maximum deviations from these "optimum" values occur when the piece is adaptively retuned. It may be, for example, that one pitch12 occurs equally often as Ab and G# (i.e. occurrences clustered about two widely different pitches), and so the "optimum" fixed tuning is halfway between, but useless (giving two wolf fifths). In the piece referred to below, it looks like pitch 3 may have this problem to some degree. This appears to be Eb but it probably plays the role of D# at least once. Otherwise I'd expect it to have an offset of 10 to 14 cents, not 5 cents. A 13 tone fixed tuning would probably be much better.

A D# seems unlikely since the piece is in D minor. Could your adaptive retuner be erroneously assigning it such a role. But in any case, there _will_ be pieces that need more than 12 notes (by meantone spelling).

>I thought the first results for the Bach/Busoni piece, in D minor, might
>be interesting (values are cents from 12-tET):
>
> For pitch 0 6.6164
> For pitch 1 -10.1093
> For pitch 2 0.1304
> For pitch 3 5.3325
> For pitch 4 0.4689
> For pitch 5 13.5001
> For pitch 6 -11.9440
> For pitch 7 1.6882
> For pitch 8 -10.4004
> For pitch 9 1.2251
> For pitch 10 10.7833
> For pitch 11 -7.2912

Apart from the anomaly of pitch 3 (Eb/D#), the above bears some resemblance to 1/5-comma meantone. It also bears some resemblance to the following 5-limit lattice.

B F# C# G#
C G D A E (B)
Eb Bb F (C)

As Paul E. said, the fifths to the parenthesised B and C are pretty bad.

>Is anyone interesting in throwing sequence(s) at this?

It might be more useful to really understand what it's doing to this piece first.

>I'm thinking it'd be better to
>stick with 5-limit rather than try to shoehorn 7-limit onto fixed
>tuning!

_12-tone_ fixed tuning, certainly.

>If there is no existing name for something like this, I propose
>Calculated Optimum Fixed Tuning, or COFT for short.

I don't think "calculated" adequately distinguishes it. Most optimum fixed tunings are calculated. The main thing is that it is optimised for a particular piece. How about "piece-optimised fixed tuning" or something that means the same.

>So I'm sitting in the, uhhh, meditation room this morning, and suddenly
>think, "Hey! The calculated optimum fixed tuning values could be used
>as grounding points for adaptive tuning!" Then, about .1 second later,
>I thought, "I'll bet Paul E has already posted that recommendation, and
>it's waiting in this morning's digest."
>
>Sure enough!

You're not the only two. I thought it was kind of obvious. ... And wrong!

The whole idea that there are only 12 notes is wrong (in general). So how could it make sense to spring them to _any_ fixed set of 12. JdL's algorithm is generally far superior to mine. But mine didn't have this failing. It was based on sliding the 12 available notes up and down an extended-meantone chain of fifths. It was real-time, not leisure. It would begin with a standard Eb to G# tuning of the available 12-notes and if certain things happened it would revise the G# to become an Ab, or the Eb to become a D# etc.

It would shift like crazy with diesis pumps, but at least it wouldn't make the mistake of trying to get D# and Eb to agree with each other when they didn't need to (because they were widely separated in time and modulation). A spring grounding them both to any single fixed pitch will also act as an unwanted spring between them.

John, you've gotta figure out how to minimise the grounding.

What went wrong with the idea of grounding only one note (e.g. the tonic)? Or a few notes. Seemed very sensible to me. If a piece ends up having several unconnected (by springs) parts, surely you still only need one grounding spring per unconnected part. But I can't believe many pieces would be unconnected.

In fact, strictly speaking, all you need to fix is the pitch of one _occurrence_ of one note in each unconnected part (say the first note). Drift would be controlled only by the "horizontal" springs between occurrences of the same note (same MIDI note number modulo 12). The strength of these horizontal springs would be inversely proportional to the difference in time between the two notes.

Thus we would hope to control drift of the tonic but allow Eb to "drift" to become D#.

Incidentally is there some accepted shorter terminology to distinguish between a C (for example) in any octave at any time, a C in a particular octave at any time and a C in a particular octave at a particular time (in a piece).

I guess "pitch class" and "pitch" are precise for the first two. But "note" and "tone" seem to be used interchangeably and to be used for all three categories. Hence my use of "note occurrence" for the third category.

-- Dave Keenan
http://dkeenan.com

🔗Herman Miller <hmiller@io.com>

🔗John A. deLaubenfels <jadl@idcomm.com>

[I wrote:]
>>As a spin-off of the new spring model for adaptive tuning, I've
>>written a function that attempts to calculate the optimum fixed tuning
>>for an entire sequence. This could, in theory, be used before a
>>performance, to tune a harpsichord, piano, clavichord, or other
>>instrument with similar capabilities and restrictions.

[Paul Hahn, TD 515.2:]
>You might be interested in contacting John Sankey and/or Bill Sethares,
>who have profiled entire Scarlatti sonatas for frequency, duration, and
>prominence of 5-limit intervals, then produced customized temperaments
>for specific sonatas that reduce the RMS errors. John, Bill, are you
>out there? Am I accurately characterizing your work?

Aha! Then I may not have any standing to choose a name for the process.
Do you know what they call(ed) it? I know that Bill is or was on the
list; how about John Sankey? Does anyone have an e-mail address for
him?

[Paul Erlich, TD 515.10:]
>Now (back to an earlier thread) couldn't you use a similar approach to
>fix all vertical triads in exact JI and use a concentrated spring set
>to get an adaptive JI retuning?

Sure. Slightly different specifics to the programming. Not sure when
I'll get around to it, but it's on my list!

[Paul:]
>Hmmm. I forgot to mention, E-B in your tuning is even worse than F-C
>and B-F#! So these three fifths/fouths do occur with significant
>frequency in the piece -- that means you'll have to modify your
>algorithm before you can say it comes up with a reasonable fixed
>tuning.

Well, I was all set to post back, "Please listen to the piece and tell
me if you still feel that way," but you saved me the trouble with your
later post (quoted below).

[Paul:]
>For a lot of these problems minimax may be a better approach than least
>squares. Know anything about linear programming?

Can't say as I do. What can you tell me about it?

[Paul, referring to his own piece:]
>22-tET (though it doesn't exist in MIDI form yet). You can hear me
>playing it (very badly) on the Tuning Punks site.

Oh, is that the piece? It's kyool! Not badly played, either, to my
recollection.

[Paul, TD 515.21:]
>So (tentatively) an appropriate unequal fixed 12-tone tuning can
>provide a relative pain-reduction from 12-tET that is twice as great in
>the Bach than in the Schubert. Cool! I'd bet Bach would fall in-between
>Schubert and, say, Byrd, in this regard.

Bring on the Byrd, and let's find out!

[JdL:]
>>Just listened to the Bach/Busoni in COFT and it's really lovely!

[Paul:]
>At 2:45, where I thought I heard the quartertone before, I thought I
>heard a shift again. You've trained me well! Of course, after I
>listened to it again, it became clear that there was no shift, the
>harmony is just funny there (some kind of suspended leading tone
>there).

Funny! The ear is a strange device, no doubt!

[Paul:]
>>>E-B tempered by almost 1/2 comma !!!
[JdL:]
>> I calculate .41 comma

[Paul:]
>The difference of the deviations is
>
>0.4689-(-7.2912) = 7.7601
>
>700-7.7601 = 692.2399�
>
>That's 9.7151 flat, or .45 comma.

Quite right. My math error.

[Paul:]
>I found this performance very beautiful -- listening to it in
>"background mode", none of these three over-tempered fifths/fourths
>called attention to themselves.

I didn't notice them either! And I have to confess (puhLEEZE don't
quote this against me) that I enjoyed the comfort of knowing that the
tuning was stable, at least as variety from what I usually listen to!

>They must only occur in pretty fast passages, yes?

I haven't tracked 'em down, but I'd guess yes.

[David Keenan, TD 516.8:]
>One would also want to know what maximum deviations from these
>"optimum" values occur when the piece is adaptively retuned. It may be,
>for example, that one pitch12 occurs equally often as Ab and G# (i.e.
>occurrences clustered about two widely different pitches), and so the
>"optimum" fixed tuning is halfway between, but useless (giving two wolf
>fifths). In the piece referred to below, it looks like pitch 3 may have
>this problem to some degree. This appears to be Eb but it probably
>plays the role of D# at least once. Otherwise I'd expect it to have an
>offset of 10 to 14 cents, not 5 cents. A 13 tone fixed tuning would
>probably be much better.

Your point is well taken. I address this more completely below.

[Dave:]
>A D# seems unlikely since the piece is in D minor. Could your adaptive
>retuner be erroneously assigning it such a role. But in any case, there
>_will_ be pieces that need more than 12 notes (by meantone spelling).

Ideally, yes. As for the D#, it is tuned as a result of actual
intervals in the piece; I'll try to get a table that summarizes all
intervals succinctly, for better analysis.

[Dave:]
>Apart from the anomaly of pitch 3 (Eb/D#), the above bears some
>resemblance to 1/5-comma meantone. It also bears some resemblance to
>the following 5-limit lattice.
>
> B F# C# G#
>C G D A E (B)
> Eb Bb F (C)
>
>As Paul E. said, the fifths to the parenthesised B and C are pretty
>bad.

On paper they're lousy. Can you find them in the sequence?

[JdL:]
>>Is anyone interesting in throwing sequence(s) at this?

>It might be more useful to really understand what it's doing to this
>piece first.

Poo. We'll analyze as we go along.

[JdL:]
>>I'm thinking it'd be better to stick with 5-limit rather than try to
>>shoehorn 7-limit onto fixed tuning!

[Dave:]
>_12-tone_ fixed tuning, certainly.

That's implicit for me; as you know, my biggest love in tuning is
7-limit adaptive.

[JdL:]
>>If there is no existing name for something like this, I propose
>>Calculated Optimum Fixed Tuning, or COFT for short.

[Dave:]
>I don't think "calculated" adequately distinguishes it. Most optimum
>fixed tunings are calculated.

Yeeessss...

>The main thing is that it is optimised for a particular piece. How
>about "piece-optimised fixed tuning" or something that means the same.

My objection is that I envision concatenating two or more sequences that
will be played in a single concert, and getting results optimized for
the set of pieces. How about "Concert Optimized Fixed Tuning"?

[JdL:]
>>So I'm sitting in the, uhhh, meditation room this morning, and
>>suddenly think, "Hey! The calculated optimum fixed tuning values
>>could be used as grounding points for adaptive tuning!" Then, about
>>.1 second later, I thought, "I'll bet Paul E has already posted that
>>recommendation, and it's waiting in this morning's digest."
>>
>>Sure enough!

[Dave:]
>You're not the only two. I thought it was kind of obvious.

Well, yeah. Anyway, I was being slow.

[Dave:]
>... And wrong!

Possibly. I'm undecided on the merits as of now.

[Dave:]
>The whole idea that there are only 12 notes is wrong (in general).

True... But there are a wealth of 12-note instruments that are used to
play music I'm interested in. I want to reach out to the musicians who
play them. Please listen to the Bach/Busoni and tell me if you don't
think it's a pretty good compromise, ambiguous note(s) and all.

[Dave:]
>So how could it make sense to spring them to _any_ fixed set of 12.
>JdL's algorithm is generally far superior to mine. But mine didn't have
>this failing. It was based on sliding the 12 available notes up and
>down an extended-meantone chain of fifths. It was real-time, not
>leisure. It would begin with a standard Eb to G# tuning of the
>available 12-notes and if certain things happened it would revise the
>G# to become an Ab, or the Eb to become a D# etc.

>It would shift like crazy with diesis pumps, but at least it wouldn't
>make the mistake of trying to get D# and Eb to agree with each other
>when they didn't need to (because they were widely separated in time
>and modulation). A spring grounding them both to any single fixed pitch
>will also act as an unwanted spring between them.

>John, you've gotta figure out how to minimise the grounding.

Well, you and Paul E are more obsessed with it than I am, that's for
certain. I agree that grounding needs to be kept as weak as possible,
so that it does not distort the natural tendencies of the piece, yet
be strong enough to control massive shifting of the center of tuning of
the piece.

[Dave:]
>What went wrong with the idea of grounding only one note (e.g. the
>tonic)? Or a few notes. Seemed very sensible to me.

Nothing went wrong, well, not quite true: grounding one note
artificially distorts motion for that note compared to all others. The
idea is still very much alive, but I'm distracted with many new fronts
at the moment.

[Dave:]
>If a piece ends up having several unconnected (by springs) parts,
>surely you still only need one grounding spring per unconnected part.
>But I can't believe many pieces would be unconnected.

>In fact, strictly speaking, all you need to fix is the pitch of one
>_occurrence_ of one note in each unconnected part (say the first note).
>Drift would be controlled only by the "horizontal" springs between
>occurrences of the same note (same MIDI note number modulo 12). The
>strength of these horizontal springs would be inversely proportional to
>the difference in time between the two notes.

I do construct my horizontal springs in much that way. But my
experience is that constant gentle drift-reducing pressure is superior
to widely separated hard pressure. The question is still very open,
however.

>Thus we would hope to control drift of the tonic but allow Eb to
>"drift" to become D#.

To a great extent, it already CAN, not in the COFT version, to be sure,
but in the adaptive versions.

>Incidentally is there some accepted shorter terminology to distinguish
>between a C (for example) in any octave at any time, a C in a
>particular octave at any time and a C in a particular octave at a
>particular time (in a piece).

Not that I know of.

Dave, I wouldn't write the death-knell for your tuning methods, as you
almost seem to by speaking of them in the past tense. I'm sure they
have a lot to offer that my own methods don't. Why don't you grow them
with other techniques (like springs, perhaps?). The tower of tuning
needs as wide a base as possible!!

[Herman Miller, TD 516.12:]
>Yes, go ahead and post the retunings. I'm listening to the dragons.mid
>with the optimized tuning now and it sounds nice!

Kyool! I'll post all 3 as soon as I can! Maybe I'll throw in some
adaptive options if you don't mind...

JdL

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Herman Miller <hmiller@io.com>

🔗John A. deLaubenfels <jadl@idcomm.com>

In TD 516.8, David Keenan asked for more complete analysis of the
Bach/Busoni piece for which I posted proposed tuning values. For the
note most in question, Eb/D#, the program calculates the tradeoffs as
follows:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force
---- ------ ---- ------ -------- ------- ------- --------
3 5.33 6 -11.94 219.009 303.733 282.723 -4601.308
3 5.33 11 -7.29 141.376 813.217 787.376 -3653.252
3 5.33 9 1.23 184.981 611.174 595.893 -2826.763
3 5.33 1 -10.11 89.212 1000.347 984.558 -1408.585
3 5.33 8 -10.40 14.618 498.023 484.267 -201.073
3 5.33 2 0.13 17.171 1097.006 1094.798 -37.920

3 5.33 5 13.50 173.967 203.955 208.167 732.910
3 5.33 10 10.78 489.265 701.940 705.451 1717.931
3 5.33 7 1.69 506.407 388.457 396.356 4000.115
3 5.33 0 6.62 514.849 889.090 901.284 6277.878

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 composit shown). Actual reflects the
tunings chosen for the two notes. Force is the means of communicating
urgency of request; the force for each note adds to zero because the
matrix has been relaxed.

Most tending to push tuning down, into D# character:

F# (note 6)
B (note 11)
A (note 9)

Most tending to push tuning up, into Eb character:

C (note 0)
G (note 7)
Bb (note 10)

The whole picture for all pairs of notes sounding in the piece:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force
---- ------ ---- ------ -------- ------- ------- --------
0 6.62 1 -10.11 62.550 103.970 83.274 -1294.489
0 6.62 2 0.13 857.590 200.111 193.514 -5657.387
0 6.62 3 5.33 514.849 310.910 298.716 -6277.878
0 6.62 4 0.47 1656.173 386.147 393.853 12762.124
0 6.62 5 13.50 604.659 498.023 506.884 5357.831
0 6.62 6 -11.94 426.936 609.951 581.440 -12172.632
0 6.62 7 1.69 999.440 701.977 695.072 -6901.390
0 6.62 8 -10.40 22.893 813.853 782.983 -706.715
0 6.62 9 1.23 2039.630 887.599 894.609 14296.864
0 6.62 10 10.78 166.936 1000.171 1004.167 666.992
0 6.62 11 -7.29 35.996 1088.124 1086.092 -73.130
1 -10.11 0 6.62 62.550 1096.030 1116.726 1294.489
1 -10.11 2 0.13 350.512 105.792 110.240 1558.762
1 -10.11 3 5.33 89.212 199.653 215.442 1408.585
1 -10.11 4 0.47 6419.054 314.277 310.578 -23745.149
1 -10.11 5 13.50 335.258 401.493 423.609 7414.808
1 -10.11 6 -11.94 777.062 498.037 498.165 99.921
1 -10.11 7 1.69 885.206 609.131 611.798 2360.805
1 -10.11 8 -10.40 26.072 700.869 699.709 -30.248
1 -10.11 9 1.23 7929.251 813.480 811.334 -17014.416
1 -10.11 10 10.78 933.616 892.087 920.893 26892.829
1 -10.11 11 -7.29 193.546 1004.061 1002.818 -240.513
2 0.13 0 6.62 857.590 999.889 1006.486 5657.387
2 0.13 1 -10.11 350.512 1094.208 1089.760 -1558.762
2 0.13 3 5.33 17.171 102.994 105.202 37.920
2 0.13 4 0.47 2462.839 201.348 200.339 -2485.886
2 0.13 5 13.50 9024.502 315.409 313.370 -18400.861
2 0.13 6 -11.94 8972.938 386.254 387.926 15002.644
2 0.13 7 1.69 5196.238 498.103 501.558 17952.383
2 0.13 8 -10.40 327.650 608.950 589.469 -6382.873
2 0.13 9 1.23 15395.276 701.972 701.095 -13507.493
2 0.13 10 10.78 6153.107 813.588 810.653 -18056.786
2 0.13 11 -7.29 3045.160 885.437 892.578 21746.185
3 5.33 0 6.62 514.849 889.090 901.284 6277.878
3 5.33 1 -10.11 89.212 1000.347 984.558 -1408.585
3 5.33 2 0.13 17.171 1097.006 1094.798 -37.920
3 5.33 5 13.50 173.967 203.955 208.167 732.910
3 5.33 6 -11.94 219.009 303.733 282.723 -4601.308
3 5.33 7 1.69 506.407 388.457 396.356 4000.115
3 5.33 8 -10.40 14.618 498.023 484.267 -201.073
3 5.33 9 1.23 184.981 611.174 595.893 -2826.763
3 5.33 10 10.78 489.265 701.940 705.451 1717.931
3 5.33 11 -7.29 141.376 813.217 787.376 -3653.252
4 0.47 0 6.62 1656.173 813.853 806.147 -12762.124
4 0.47 1 -10.11 6419.054 885.723 889.422 23745.149
4 0.47 2 0.13 2462.839 998.652 999.661 2485.886
4 0.47 5 13.50 281.698 111.100 113.031 543.998
4 0.47 6 -11.94 258.790 193.962 187.587 -1649.671
4 0.47 7 1.69 3389.264 309.857 301.219 -29276.670
4 0.47 8 -10.40 1005.091 386.190 389.131 2956.264
4 0.47 9 1.23 11018.787 498.063 500.756 29673.896
4 0.47 10 10.78 601.789 608.184 610.314 1281.786
4 0.47 11 -7.29 1753.688 701.933 692.240 -16999.177
5 13.50 0 6.62 604.659 701.977 693.116 -5357.831
5 13.50 1 -10.11 335.258 798.507 776.391 -7414.808
5 13.50 2 0.13 9024.502 884.591 886.630 18400.861
5 13.50 3 5.33 173.967 996.045 991.833 -732.910
5 13.50 4 0.47 281.698 1088.900 1086.969 -543.998
5 13.50 6 -11.94 6.674 92.079 74.556 -116.945
5 13.50 7 1.69 442.154 191.870 188.188 -1627.642
5 13.50 8 -10.40 143.575 298.426 276.100 -3205.497
5 13.50 9 1.23 6119.987 386.586 387.725 6970.492
5 13.50 10 10.78 3185.846 498.034 497.283 -2390.136
5 13.50 11 -7.29 134.598 608.783 579.209 -3980.615
6 -11.94 0 6.62 426.936 590.049 618.560 12172.632
6 -11.94 1 -10.11 777.062 701.963 701.835 -99.921
6 -11.94 2 0.13 8972.938 813.746 812.074 -15002.644
6 -11.94 3 5.33 219.009 896.267 917.277 4601.308
6 -11.94 4 0.47 258.790 1006.038 1012.413 1649.671
6 -11.94 5 13.50 6.674 1107.921 1125.444 116.945
6 -11.94 7 1.69 263.914 111.680 113.632 515.251
6 -11.94 8 -10.40 24.651 203.955 201.544 -59.429
6 -11.94 9 1.23 6034.368 315.401 313.169 -13464.940
6 -11.94 10 10.78 108.746 397.618 422.727 2730.576
6 -11.94 11 -7.29 1031.748 498.023 504.653 6840.550
7 1.69 0 6.62 999.440 498.023 504.928 6901.390
7 1.69 1 -10.11 885.206 590.869 588.202 -2360.805
7 1.69 2 0.13 5196.238 701.897 698.442 -17952.383
7 1.69 3 5.33 506.407 811.543 803.644 -4000.115
7 1.69 4 0.47 3389.264 890.143 898.781 29276.670
7 1.69 5 13.50 442.154 1008.130 1011.812 1627.642
7 1.69 6 -11.94 263.914 1088.320 1086.368 -515.251
7 1.69 8 -10.40 4.573 111.876 87.911 -109.600
7 1.69 9 1.23 2281.449 202.946 199.537 -7777.218
7 1.69 10 10.78 3353.983 313.970 309.095 -16349.260
7 1.69 11 -7.29 2473.306 386.470 391.020 11255.853
8 -10.40 0 6.62 22.893 386.147 417.017 706.715
8 -10.40 1 -10.11 26.072 499.131 500.291 30.248
8 -10.40 2 0.13 327.650 591.050 610.531 6382.873
8 -10.40 3 5.33 14.618 701.977 715.733 201.073
8 -10.40 4 0.47 1005.091 813.810 810.869 -2956.264
8 -10.40 5 13.50 143.575 901.574 923.900 3205.497
8 -10.40 6 -11.94 24.651 996.045 998.456 59.429
8 -10.40 7 1.69 4.573 1088.124 1112.089 109.600
8 -10.40 9 1.23 168.040 111.128 111.625 83.534
8 -10.40 10 10.78 38.683 203.955 221.184 666.479
8 -10.40 11 -7.29 828.115 313.361 303.109 -8489.629
9 1.23 0 6.62 2039.630 312.401 305.391 -14296.864
9 1.23 1 -10.11 7929.251 386.520 388.666 17014.416
9 1.23 2 0.13 15395.276 498.028 498.905 13507.493
9 1.23 3 5.33 184.981 588.826 604.107 2826.763
9 1.23 4 0.47 11018.787 701.937 699.244 -29673.896
9 1.23 5 13.50 6119.987 813.414 812.275 -6970.492
9 1.23 6 -11.94 6034.368 884.599 886.831 13464.940
9 1.23 7 1.69 2281.449 997.054 1000.463 7777.218
9 1.23 8 -10.40 168.040 1088.872 1088.375 -83.534
9 1.23 10 10.78 546.942 107.942 109.558 883.804
9 1.23 11 -7.29 798.764 197.055 191.484 -4450.471
10 10.78 0 6.62 166.936 199.829 195.833 -666.992
10 10.78 1 -10.11 933.616 307.913 279.107 -26892.829
10 10.78 2 0.13 6153.107 386.412 389.347 18056.786
10 10.78 3 5.33 489.265 498.060 494.549 -1717.931
10 10.78 4 0.47 601.789 591.816 589.686 -1281.786
10 10.78 5 13.50 3185.846 701.966 702.717 2390.136
10 10.78 6 -11.94 108.746 802.382 777.273 -2730.576
10 10.78 7 1.69 3353.983 886.030 890.905 16349.260
10 10.78 8 -10.40 38.683 996.045 978.816 -666.479
10 10.78 9 1.23 546.942 1092.058 1090.442 -883.804
10 10.78 11 -7.29 67.875 110.740 81.926 -1955.772
11 -7.29 0 6.62 35.996 111.876 113.908 73.130
11 -7.29 1 -10.11 193.546 195.939 197.182 240.513
11 -7.29 2 0.13 3045.160 314.563 307.422 -21746.185
11 -7.29 3 5.33 141.376 386.783 412.624 3653.252
11 -7.29 4 0.47 1753.688 498.067 507.760 16999.177
11 -7.29 5 13.50 134.598 591.217 620.791 3980.615
11 -7.29 6 -11.94 1031.748 701.977 695.347 -6840.550
11 -7.29 7 1.69 2473.306 813.530 808.980 -11255.853
11 -7.29 8 -10.40 828.115 886.639 896.891 8489.629
11 -7.29 9 1.23 798.764 1002.945 1008.516 4450.471
11 -7.29 10 10.78 67.875 1089.260 1118.074 1955.772

JdL