Yahoo Tuning Groups Ultimate Backup tuning (New midi)Dactyls of Pisces

"Dactyls of Pisces"by Mats �ljare

http://www.angelfire.com/mo/oljare/images/dactyl.mid

This original version was written by me using Cakewalk and Midi Relay,in 17-tone pythagorean scale.This includes schmismic thirds,but not too many of them so there is a rather unsatisfying mix of intervals in it.It is quite apparent that this is not the ideal version of the piece.So i have also put up a 12tet version,that might be the subject of an adaptive tuning:

http://www.angelfire.com/mo/oljare/images/dactyl12.mid

Glad for comments.

��
Mats �ljare
Eskilstuna,Sweden
http://www.angelfire.com/mo/oljare
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🔗Rosati <dante@pop.interport.net>

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

[Mats Öljare wrote:]
>"Dactyls of Pisces"by Mats ljare

>http://www.angelfire.com/mo/oljare/images/dactyl.mid

>This original version was written by me using Cakewalk and Midi Relay,
>in 17-tone pythagorean scale.This includes schmismic thirds,but not too
>many of them so there is a rather unsatisfying mix of intervals in it.
>It is quite apparent that this is not the ideal version of the piece.
>So i have also put up a 12tet version,that might be the subject of an
>adaptive tuning:

>http://www.angelfire.com/mo/oljare/images/dactyl12.mid

>Glad for comments.

Mats, I just e-mailed you a .zip of three tunings of your piece, one in
strict COFT and two in adaptive (5- and 7-limit). Nice li'l piece! If
you think the tunings are interesting, please feel free to post them.

The COFT I calculate for this is (pitch 0==C, 11==B):

E0 4B 3D (For pitch 0, we have target bend -7.6434)
E1 1A 42 (For pitch 1, we have target bend 6.9875)
E2 4A 3B (For pitch 2, we have target bend -13.9894)
E3 57 40 (For pitch 3, we have target bend 2.1743)
E4 11 42 (For pitch 4, we have target bend 6.7602)
E5 2A 3F (For pitch 5, we have target bend -2.1488)
E6 15 44 (For pitch 6, we have target bend 13.1738)
E7 64 3C (For pitch 7, we have target bend -10.1997)
E8 14 41 (For pitch 8, we have target bend 3.6702)
EA 47 3B (For pitch 9, we have target bend -14.0784)
EB 00 40 (For pitch 10, we have target bend 0.0093)
EC 6A 44 (For pitch 11, we have target bend 15.2843)

(The above figures represent cents deviations from 12-tET).

JdL

🔗Mats �ljare <oljare@hotmail.com>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

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

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

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

[I wrote:]
>>It's 5-limit. I don't try to attempt 7-limit COFT; the motion required
>>is too extreme for fixed tuning.

[Paul E:]
>That seems odd -- if the original 12-tET piece implies 7-limit
>harmonies, why wouldn't the COFT only be better? And aren't you
>calculating it anyway in order to find an optimal grounding point for
>the 7-limit adaptive tuning?

This point is the source of great confusion. To answer your second
question first: I do calculate a 7-limit COFT tuning before beginning
the adaptive phase of matrix relaxation, but the grounding values are
allowed to move freely as best suits the adaptive solution (as opposed
to the fixed tuning solution). So, use of COFT in adaptive tuning only
helps with convergence: the resultant values are close to the original
COFT, so have less far to move. I am perhaps guilty of spreading
confusion by referring to "adaptive tuning grounded to COFT"; this is
not quite the case. Am I being clear?

As to your first question: even if the piece does not modulate, COFT is
difficult in 7-limit. A piece in C major will have strong conflict with
the F note, for example, between its subdominant role and its role as
7:4 of G.

JdL

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

>This point is the source of great confusion. To answer your second
>question first: I do calculate a 7-limit COFT tuning before beginning
>the adaptive phase of matrix relaxation, but the grounding values are
>allowed to move freely as best suits the adaptive solution (as opposed
>to the fixed tuning solution). So, use of COFT in adaptive tuning only
>helps with convergence: the resultant values are close to the original
>COFT, so have less far to move. I am perhaps guilty of spreading
>confusion by referring to "adaptive tuning grounded to COFT"; this is
>not quite the case. Am I being clear?

So there is another fixed 12-tone tuning that you're grounding to . . .
right?

>As to your first question: even if the piece does not modulate, COFT is
>difficult in 7-limit. A piece in C major will have strong conflict with
>the F note, for example, between its subdominant role and its role as
>7:4 of G.

But it will still be a theoretical improvement over 12-tET, right? I guess
what you're saying is that 7-limit COFTs are not, in general, significantly
different from 12-tET, yes?

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

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

[Paul E:]
>So there is another fixed 12-tone tuning that you're grounding to . . .
>right?

Right. One which is found during the relaxation of the "big" matrix
(containing all the individual springs of all 3 types), after the
initial 7-limit COFT values have been found by relaxing the "little"
matrix (condensed vertical springs only).

[JdL:]
>>As to your first question: even if the piece does not modulate, COFT
>>is difficult in 7-limit. A piece in C major will have strong conflict
>>with the F note, for example, between its subdominant role and its
>role as 7:4 of G.

[Paul:]
>But it will still be a theoretical improvement over 12-tET, right?

Almost certainly!

>I
>guess what you're saying is that 7-limit COFTs are not, in general,
>significantly different from 12-tET, yes?

It depends upon the piece, of course, but in general, yes. I can give
you numbers for a specific sequence if you're interested.

You answered Joseph Pehrson's query (Re: COFTing it up); I don't need to
add anything to that.

JdL

🔗Joseph Pehrson <pehrson@pubmedia.com>

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

[Paul E wrote:]
>>I guess what you're saying is that 7-limit COFTs are not, in general,
>>significantly different from 12-tET, yes?

[I wrote:]
>It depends upon the piece, of course, but in general, yes. I can give
>you numbers for a specific sequence if you're interested.

Without waiting for your response, I got curious, so here's an example.
Naturally, larger pieces that modulate a lot tend to have ineffective
COFT in either 5- or 7-limit. So I chose a fairly small piece,
Albeniz-Godowsky "Tango", in D, sequenced by Robert Finley.

In 5-limit, the following values are calculated:
12-tET Total spring pain: 255664.214
COFT Total spring pain: 135776.656
Adaptive spring pain: 73252.229

In 7-limit, the following values are calculated:
12-tET Total spring pain: 373529.255
COFT Total spring pain: 262260.140
Adaptive spring pain: 131343.961

You can see that COFT in 7-limit is not nearly as effective as COFT in
5-limit, though, in this case at least, the improvement is non-trivial.
Of the 262 thousand, a full 169 thousand is unrecoverable internal
stress due to conflict in what is desired in an interval at different
times during the piece. (And, recently, there can be unrecoverable
vertical stress as well, when non-self-consistent vertical tuning is
used, as recently discussed).

The 7-limit true COFT bends calculated are:

E3 65 3D (For pitch 3, we have bend -7.0007)
EB 3C 3E (For pitch 10, we have bend -4.8483)
E5 3F 3F (For pitch 5, we have bend -1.6210)
E0 16 3E (For pitch 0, we have bend -5.7945)
E7 41 40 (For pitch 7, we have bend 1.6266)
E2 07 42 (For pitch 2, we have bend 6.5153) (tonic note)
EA 7F 42 (For pitch 9, we have bend 9.4827)
E4 2B 42 (For pitch 4, we have bend 7.4048)
EC 33 3F (For pitch 11, we have bend -1.9172)
E6 7D 3E (For pitch 6, we have bend -3.2512)
E1 00 40 (For pitch 1, we have bend -0.0010)
E8 68 3F (For pitch 8, we have bend -0.5956)

Wow, look at those wide fifths, and right around tonic! The dom7 (A7)
tends to pull note 9 up and note 7 down, because of the narrowed 4:7
interval.

The grounding tuning, which begins at COFT, but is allowed to relax with
the rest of the big spring matrix, is slightly different:

E3 6F 3D (For pitch 3, we have target bend -6.7596)
EB 66 3E (For pitch 10, we have target bend -3.8223)
E5 3A 3F (For pitch 5, we have target bend -1.7336)
E0 34 3E (For pitch 0, we have target bend -5.0573)
E7 57 40 (For pitch 7, we have target bend 2.1715)
E2 09 42 (For pitch 2, we have target bend 6.5556) (tonic note)
EA 71 42 (For pitch 9, we have target bend 9.1314)
E4 11 42 (For pitch 4, we have target bend 6.7708)
EC 40 3F (For pitch 11, we have target bend -1.6054)
E6 7F 3E (For pitch 6, we have target bend -3.1966)
E1 6F 3F (For pitch 1, we have target bend -0.4417)
E8 5B 3F (For pitch 8, we have target bend -0.9286)

Just for fun, here are the corresponding 5-limit values. For true COFT:

E3 50 3D (For pitch 3, we have bend -7.5276)
EB 6F 3D (For pitch 10, we have bend -6.7474)
E5 7B 3E (For pitch 5, we have bend -3.3056)
E0 76 40 (For pitch 0, we have bend 2.9354)
E7 72 41 (For pitch 7, we have bend 6.0009)
E2 17 42 (For pitch 2, we have bend 6.9088) (tonic note)
EA 56 42 (For pitch 9, we have bend 8.4545)
E4 7F 41 (For pitch 4, we have bend 6.3198)
EC 19 3F (For pitch 11, we have bend -2.5468)
E6 51 3E (For pitch 6, we have bend -4.3290)
E1 30 3F (For pitch 1, we have bend -2.0008)
E8 58 3E (For pitch 8, we have bend -4.1622)

Grounding for 5-limit adaptive tunings is:

E3 61 3D (For pitch 3, we have target bend -7.1024)
EB 13 3E (For pitch 10, we have target bend -5.8738)
E5 79 3E (For pitch 5, we have target bend -3.3414)
E0 51 40 (For pitch 0, we have target bend 2.0264)
E7 66 41 (For pitch 7, we have target bend 5.7092)
E2 18 42 (For pitch 2, we have target bend 6.9299) (tonic note)
EA 4C 42 (For pitch 9, we have target bend 8.2094)
E4 72 41 (For pitch 4, we have target bend 5.9965)
EC 2A 3F (For pitch 11, we have target bend -2.1372)
E6 55 3E (For pitch 6, we have target bend -4.2310)
E1 29 3F (For pitch 1, we have target bend -2.1588)
E8 59 3E (For pitch 8, we have target bend -4.1369)

Still doesn't look very meantonish, does it? Must just be the interval
set particular to this piece. I'll refrain from including the long
full COFT table, in which all the balances are shown, unless someone
really wants to see it.

JdL

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

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

[Paul E:]
>>>So there is another fixed 12-tone tuning that you're grounding to . . .
>>>right?

[I wrote:]
>>Right. One which is found during the relaxation of the "big" matrix
>>(containing all the individual springs of all 3 types), after the
>>initial 7-limit COFT values have been found by relaxing the "little"
>>matrix (condensed vertical springs only).

[Paul:]
>I'm confused as to why a fixed 12-tone tuning would be found during the
>relaxation of the "big" matrix (containing all the individual springs
>of all 3 types). I would have thought that that process would start
>from the COFT and proceed to the final adaptive solution.

Well... Carl L and I exchanged off-list over this point for WEEKS, or
it seemed like it. I would have grounded to COFT values during the
adaptive relaxation, and did, but through a quirk of chance, noticed
that the ideal adaptive grounding was not quite the same as the COFT
tuning. The shift is usually around 1/2 cent, so it's not even worth
getting excited about, but I went ahead and added a call during the
adaptive relaxation phase, to relax the overall grounding points as
well. I can't explain why it's different from COFT, but I'm pretty sure
it's not due to a bug (yeah, that's what they ALL say).

If the model is right (and, of course, it is "right" only to a partial
degree of refinement in any case!), I let the model dictate the results.
As I'm sure do you, I hasten to add! There just happens to be a slight
tuning improvement (==reduction in pain) from allowing modification of
the grounding points in the adaptive phase.

I just added code to optionally suppress the adjustment of grounding
tuning during adaptive relaxation. Here's the difference in the tango
5-limit:

Adaptive spring pain with grounding modification: 73252.229
Adaptive spring pain without grounding modification: 73393.843

So we're talking 0.2% improvement in total pain calculated.

Clearer now?

JdL

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

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

[I wrote:]
>>So we're talking 0.2% improvement in total pain calculated.

[Paul E:]
>0.2%, huh? So that's why you felt comfortable saying "grounded to COFT"
>rather than "grounded first to COFT and then to a slightly different
>fixed 12-tone tuning".

Right.

[JdL:]
>>but I went ahead and added a call during the
>>adaptive relaxation phase, to relax the overall grounding points as
>>well.

[Paul:]
>This would seem to be a circular problem, isn't it? Is what you're
>doing some form of iterative solution?

Yes. I relax the big spring matrix by the simplest means possible:
one node at a time. Repeat many MANY times by Monte Carlo walk;
each node gets lots of chances to re-relax after its neighbors have,
etc. I was surprised that this crude method works in a reasonable
length of time, but it does, so I haven't bothered coming up with more
sophisticated relaxation techniques (Carl is still waiting for a
relaxation refinement that'll handle infinitely rigid vertical springs
with better than zero convergence!).

In this context, the grounding targets are nodes just as the tuning at
one moment in time is a node. It takes but a few lines of code to
relax the grounding points, to get that extra 0.2% of pain reduction.
But, considering all the confusion it causes, maybe I should've skipped
it! ;->

JdL