More sweet Mozart

I listened to this piece in 5-limit COFT and could hardly believe it
wasn't adaptively tuned, so sweet did all the chords sound. See if you
agree! Mozart, Andante für Orgelwalze, K.616, sequence by Hans Jakob
Heldstab.

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

File: kv616.zip.

BTW, does anybody know what an "Orgelwalze" is?

[I wrote:]
>>I listened to this piece in 5-limit COFT and could hardly believe it
>>wasn't adaptively tuned, so sweet did all the chords sound. See if
>>you agree! Mozart, Andante für Orgelwalze, K.616, sequence by Hans
>>Jakob Heldstab.
>>
>> http://www.idcomm.com/personal/jadl/
>>
>>File: kv616.zip.

[Paul Erlich:]
>Will you please post the fifths and thirds, or at least cents
>deviations from 12-tET, of this COFT?

Be glad to; following is the complete COFT picture. BTW, the raw
tuning info is in the file kv616.txt, included in kv616.zip. The piece
is in the key of F major.

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 in fact the strength times the difference of
the actual and the 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").

Ptch Tuning Ptch Tuning Strength Ideal Actual Force
---- ------ ---- ------ -------- ------- ------- --------
0 5.23 1 -4.94 5.832 96.923 89.831 -41.358
0 5.23 2 -2.23 46.949 200.941 192.544 -394.200
0 5.23 3 9.09 37.199 311.141 303.866 -270.623
0 5.23 4 -7.74 451.285 386.241 387.032 356.726
0 5.23 5 5.80 226.761 498.038 500.571 574.484
0 5.23 6 -14.35 9.708 592.176 580.420 -114.130
0 5.23 7 4.54 359.893 701.937 699.319 -942.453
0 5.23 8 15.45 41.169 813.253 810.225 -124.665
0 5.23 9 -6.33 253.599 885.373 888.445 778.922
0 5.23 10 4.72 88.399 996.772 999.494 240.613
0 5.23 11 -9.23 9.339 1092.321 1085.540 -63.327
1 -4.94 0 5.23 5.832 1103.077 1110.169 41.358
1 -4.94 2 -2.23 3.059 99.584 102.713 9.573
1 -4.94 3 9.09 0.366 203.955 214.035 3.691
1 -4.94 4 -7.74 63.831 310.355 297.201 -839.614
1 -4.94 5 5.80 16.209 387.663 410.740 374.069
1 -4.94 6 -14.35 2.292 498.885 490.589 -19.013
1 -4.94 7 4.54 15.344 600.124 609.488 143.681
1 -4.94 8 15.45 0.732 701.977 720.394 13.486
1 -4.94 9 -6.33 24.272 813.326 798.614 -357.087
1 -4.94 10 4.72 45.310 895.760 909.663 629.973
1 -4.94 11 -9.23 0.346 996.045 995.709 -0.116
2 -2.23 0 5.23 46.949 999.059 1007.456 394.200
2 -2.23 1 -4.94 3.059 1100.416 1097.287 -9.573
2 -2.23 3 9.09 2.323 111.876 111.322 -1.287
2 -2.23 4 -7.74 16.489 194.369 194.487 1.957
2 -2.23 5 5.80 138.697 313.790 308.027 -799.402
2 -2.23 6 -14.35 52.589 386.459 387.876 74.504
2 -2.23 7 4.54 140.572 498.037 506.774 1228.200
2 -2.23 8 15.45 5.110 603.257 617.681 73.711
2 -2.23 9 -6.33 159.585 701.941 695.901 -963.911
2 -2.23 10 4.72 101.993 813.028 806.950 -619.908
2 -2.23 11 -9.23 75.134 884.724 892.996 621.492
3 9.09 0 5.23 37.199 888.859 896.134 270.623
3 9.09 1 -4.94 0.366 996.045 985.965 -3.691
3 9.09 2 -2.23 2.323 1088.124 1088.678 1.287
3 9.09 4 -7.74 0.566 92.079 83.166 -5.048
3 9.09 5 5.80 7.689 199.961 196.705 -25.035
3 9.09 6 -14.35 2.365 299.082 276.554 -53.284
3 9.09 7 4.54 25.943 386.837 395.453 223.521
3 9.09 8 15.45 7.423 498.023 506.359 61.878
3 9.09 9 -6.33 10.336 587.268 584.579 -27.796
3 9.09 10 4.72 10.261 701.977 695.628 -65.147
3 9.09 11 -9.23 12.282 812.395 781.674 -377.311
4 -7.74 0 5.23 451.285 813.759 812.968 -356.726
4 -7.74 1 -4.94 63.831 889.645 902.799 839.614
4 -7.74 2 -2.23 16.489 1005.631 1005.513 -1.957
4 -7.74 3 9.09 0.566 1107.921 1116.834 5.048
4 -7.74 5 5.80 18.108 111.736 113.539 32.652
4 -7.74 6 -14.35 1.586 202.709 193.389 -14.785
4 -7.74 7 4.54 376.137 315.014 312.287 -1025.650
4 -7.74 8 15.45 2.173 427.706 423.193 -9.808
4 -7.74 9 -6.33 97.215 498.177 501.413 314.631
4 -7.74 10 4.72 46.695 606.077 612.463 298.158
4 -7.74 11 -9.23 23.293 701.977 698.508 -80.800
5 5.80 0 5.23 226.761 701.962 699.429 -574.484
5 5.80 1 -4.94 16.209 812.337 789.260 -374.069
5 5.80 2 -2.23 138.697 886.210 891.973 799.402
5 5.80 3 9.09 7.689 1000.039 1003.295 25.035
5 5.80 4 -7.74 18.108 1088.264 1086.461 -32.652
5 5.80 7 4.54 72.968 202.097 198.748 -244.366
5 5.80 8 15.45 37.645 313.133 309.654 -130.951
5 5.80 9 -6.33 335.699 386.204 387.874 560.601
5 5.80 10 4.72 63.070 498.076 498.923 53.407
5 5.80 11 -9.23 9.907 593.238 584.969 -81.924
6 -14.35 0 5.23 9.708 607.824 619.580 114.130
6 -14.35 1 -4.94 2.292 701.115 709.411 19.013
6 -14.35 2 -2.23 52.589 813.541 812.124 -74.504
6 -14.35 3 9.09 2.365 900.918 923.446 53.284
6 -14.35 4 -7.74 1.586 997.291 1006.611 14.785
6 -14.35 7 4.54 1.087 111.876 118.898 7.631
6 -14.35 9 -6.33 32.218 314.848 308.025 -219.835
6 -14.35 10 4.72 3.984 401.085 419.074 71.670
6 -14.35 11 -9.23 1.948 498.023 505.120 13.826
7 4.54 0 5.23 359.893 498.063 500.681 942.453
7 4.54 1 -4.94 15.344 599.876 590.512 -143.681
7 4.54 2 -2.23 140.572 701.963 693.226 -1228.200
7 4.54 3 9.09 25.943 813.163 804.547 -223.521
7 4.54 4 -7.74 376.137 884.986 887.713 1025.650
7 4.54 5 5.80 72.968 997.903 1001.252 244.366
7 4.54 6 -14.35 1.087 1088.124 1081.102 -7.631
7 4.54 8 15.45 1.087 111.876 110.906 -1.053
7 4.54 9 -6.33 22.275 188.167 189.126 21.356
7 4.54 10 4.72 107.402 305.931 300.176 -618.175
7 4.54 11 -9.23 99.459 386.338 386.221 -11.566
8 15.45 0 5.23 41.169 386.747 389.775 124.665
8 15.45 1 -4.94 0.732 498.023 479.606 -13.486
8 15.45 2 -2.23 5.110 596.743 582.319 -73.711
8 15.45 3 9.09 7.423 701.977 693.641 -61.878
8 15.45 4 -7.74 2.173 772.294 776.807 9.808
8 15.45 5 5.80 37.645 886.867 890.346 130.951
8 15.45 7 4.54 1.087 1088.124 1089.094 1.053
8 15.45 9 -6.33 3.005 92.079 78.220 -41.647
8 15.45 10 4.72 1.087 182.192 189.269 7.690
8 15.45 11 -9.23 6.564 288.028 275.315 -83.448
9 -6.33 0 5.23 253.599 314.627 311.555 -778.922
9 -6.33 1 -4.94 24.272 386.674 401.386 357.087
9 -6.33 2 -2.23 159.585 498.059 504.099 963.911
9 -6.33 3 9.09 10.336 612.732 615.421 27.796
9 -6.33 4 -7.74 97.215 701.823 698.587 -314.631
9 -6.33 5 5.80 335.699 813.796 812.126 -560.601
9 -6.33 6 -14.35 32.218 885.152 891.975 219.835
9 -6.33 7 4.54 22.275 1011.833 1010.874 -21.356
9 -6.33 8 15.45 3.005 1107.921 1121.780 41.647
9 -6.33 10 4.72 12.072 110.641 111.049 4.926
9 -6.33 11 -9.23 10.725 191.502 197.095 59.987
10 4.72 0 5.23 88.399 203.228 200.506 -240.613
10 4.72 1 -4.94 45.310 304.240 290.337 -629.973
10 4.72 2 -2.23 101.993 386.972 393.050 619.908
10 4.72 3 9.09 10.261 498.023 504.372 65.147
10 4.72 4 -7.74 46.695 593.923 587.537 -298.158
10 4.72 5 5.80 63.070 701.924 701.077 -53.407
10 4.72 6 -14.35 3.984 798.915 780.926 -71.670
10 4.72 7 4.54 107.402 894.069 899.824 618.175
10 4.72 8 15.45 1.087 1017.808 1010.731 -7.690
10 4.72 9 -6.33 12.072 1089.359 1088.951 -4.926
10 4.72 11 -9.23 3.839 85.216 86.046 3.187
11 -9.23 0 5.23 9.339 107.679 114.460 63.327
11 -9.23 1 -4.94 0.346 203.955 204.291 0.116
11 -9.23 2 -2.23 75.134 315.276 307.004 -621.492
11 -9.23 3 9.09 12.282 387.605 418.326 377.311
11 -9.23 4 -7.74 23.293 498.023 501.492 80.800
11 -9.23 5 5.80 9.907 606.762 615.031 81.924
11 -9.23 6 -14.35 1.948 701.977 694.880 -13.826
11 -9.23 7 4.54 99.459 813.662 813.779 11.566
11 -9.23 8 15.45 6.564 911.972 924.685 83.448
11 -9.23 9 -6.33 10.725 1008.498 1002.905 -59.987
11 -9.23 10 4.72 3.839 1114.784 1113.954 -3.187

JdL

[Paul Erlich:]
>So, if I'm understanding correctly, John deLaubenfels' COFT which makes
>Mozart's K616 sound almost just comes out with the following fifths:

>Ab
>693.6
>Eb
>695.6
>Bb
>701.1
>F
>699.4
>C
>699.3
>G
>693.2
>D
>695.9
>A
>698.6
>E
>698.5
>B
>694.9
>F#
>709.4
>C#(Db)
>720.4
>Ab

Correct. You're reading these values directly from the table, I hope,
and not going through the bother of calculating them? They're given
in the "Actual" column.

http://www.egroups.com/message/tuning/10466

[Paul:]
>Fascinating! Anyone see any similarities to any recorded historical
>tuning? I would guess, from this result, that the piece (which is in
>F) has no G#s, but has both C#s and Dbs, with C#s predominating. Anyone
>have access to Mozart's original notation of this piece?

>I would also surmise that Ab-Eb and G-D occur quite rarely as
>simultaneities, with B-F# somewhat rare too and Eb-Bb and D-A uncommon
>as well. Meanwhile, all major thirds from Ab-C through D-F# are all
>quite pure, except Bb-D @393› and Eb-G @395.5› (these major thirds must
>appear less often in the harmonies of the piece).

The representation of each interval is given in the "Strength" column
of the table. Some of the less-than-ideal thirds ARE fairly present,
but, given the model, this tuning set is the best possible tradeoff.

>The benefit over a straight meantone is quite tangible in the increased
>purity of the more-used fifths.

Since posting these results I've added code to the program to quantify
the "pain" of a number of 12-note fixed tunings. Some numbers:

5-limit COFT 94059.211

12-tET 262051.252

31-tET subset: Db thru F# 167045.100

31-tET subset: Ab thru C# 133869.159

31-tET subset: Eb thru G# 207664.471

31-tET subset: Bb thru D# 259414.234

This makes the COFT look not THAT much better than Ab-C# meantone. Part
of the equation is an unrecoverable "internal stress" of 36411.406 due
to conflicting desires at different times for a particular interval.
Subtracting this number from all the values above gives the COFT a
relatively better look than the raw numbers do.

By comparison, the full 5-limit adaptive quasi-JI overtop of COFT
results in:

After relaxing, Total spring pain: 44463.314

Which can be broken down as:

Final vertical spring pain: 23091.045
Final horizontal spring pain: 4579.759
Final grounding spring pain: 16792.510

But, to my ears, the COFT version sounds almost as sweet as the adaptive
version.

JdL

Interesting. If you decide to include like calculations for other 12-note
fixed tunings, would you please consider Werckmeister III?

>Since posting these results I've added code to the program to quantify
>the "pain" of a number of 12-note fixed tunings. Some numbers: ...

John Thaden
Little Rock, Arkansas, USA
http://www.flash.net/~jjthaden

[I wrote:]
>>Since posting these results I've added code to the program to quantify
>>the "pain" of a number of 12-note fixed tunings. Some numbers:
>>
>> 5-limit COFT 94059.211
>>
>> 12-tET 262051.252
>>
>> 31-tET subset: Db thru F# 167045.100
>>
>> 31-tET subset: Ab thru C# 133869.159
>>
>> 31-tET subset: Eb thru G# 207664.471
>>
>> 31-tET subset: Bb thru D# 259414.234

[Paul Erlich:]
>Clearly, the COFT wins here because it compromises the tuning of note 2
>between C# and Db. What would be interesting is if you could
>distinguish the D-flats from the C-sharps and use a 13-tone, rather
>than 12-tone, subset of 31-tET. That might come out better than a
>12-tone COFT, but it might not, since a subset 31-tET still fails to
>distinguish between the much-used and little-used fifths (not counting
>the wolf). Of course, a 13-tone COFT would do better than either, and
>might be so smooth-sounding that a subsequent adaptive relaxation might
>not even result in an audible improvement.

You may be right. I'm still working on code that distinguishes C# from
Db (etc.), without spinning like a top around the circle of fifths.
Once that piece of functionality is available, we'll have some
interesting tunings, I'm sure! I'm actually somewhat skeptical that
it'll be less jarring, however, because once the ear has heard the
tuning of, say, a Db, the much flatter C# may sound mis-tuned, howbeit
better aligned vertically at that moment. We shall see!

JdL

[John Thaden:]
>Interesting. If you decide to include like calculations for other
>12-note fixed tunings, would you please consider Werckmeister III?

I'll add it to my list of things to do :-) Right now, I generate the
fixed tunings I use from either COFT or n-tET, but I could add others
or even allow them to be read in from external files. Lessee: going
back to a post by Wim Hoogewerf from May 4, Werckmeister III apparently
has the following deviation from 12-tET, in cents:

C 0
C# -10
D -8
D# -6
E -10
F -2
F# -12
G -4
G# -8
A -12
Bb -4
B -8

If this is wrong, would somebody please correct me? Boy, that seems
like a STRANGE set of numbers! I'm now curious to see how much "pain"
it causes... Any recommendations for particular pieces to throw at it?

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jadl@i...> wrote:
> [John Thaden:]
> >Interesting. If you decide to include like calculations for other
> >12-note fixed tunings, would you please consider Werckmeister III?
>
>
> I'll add it to my list of things to do :-) Right now, I generate
the
> fixed tunings I use from either COFT or n-tET, but I could add
others
> or even allow them to be read in from external files. Lessee: going
> back to a post by Wim Hoogewerf from May 4, Werckmeister III
apparently
> has the following deviation from 12-tET, in cents:
>
> C 0
> C# -10
> D -8
> D# -6
> E -10
> F -2
> F# -12
> G -4
> G# -8
> A -12
> Bb -4
> B -8
>
> If this is wrong, would somebody please correct me? Boy, that
seems
> like a STRANGE set of numbers!

It's not all that strange -- the fifths C-G, G-D, D-A, and B-F# are
each tempered by 1/4 Pythagorean comma, while the other fifths are
just. It works wonderfully for music in keys like F major and A
minor, with the tempered B-F# mitigating some potentially dissonant
thirds when modulating in the usual, dominant direction.

> I'm now curious to see how much "pain"
> it causes... Any recommendations for particular pieces to throw at
it?

The Bach and Mozart examples would be appropriate, since they both
demand a "circulating" temperament (if one restricts oneself to 12
pitches), which is what Werckmeister III was designed to be.