Surprising Scarlatti

I thought that, the older the music, the better COFT would be against
meantone, but I was wrong! I've just added k082.zip to my web:

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

containing tunings of Domenico Scarlatti Sonata in F, K.082, sequenced
by John Sankey. As presented by Mr. Sankey, the tunings are fixed at:

E3 1F 3D (-8.72) Eb
EB 30 3E (-5.14) Bb
E5 30 3F (-1.98) F
E0 00 40 (0.00) C
E7 7E 3E (-3.21) G
E2 70 3D (-6.72) D
EA 30 3C (-11.47) A
E4 50 3B (-13.84) E
EC 57 3B (-13.67) B
E6 0F 3B (-15.45) F#
E1 32 3B (-14.58) C#
E8 00 3C (-12.65) G#

Which is not particularly well suited to this particular piece. I threw
5-limit COFT at it, and got:

E3 6B 46 (For pitch 3, we have bend 21.6392)
EB 7C 44 (For pitch 10, we have bend 15.7220)
E5 58 43 (For pitch 5, we have bend 11.6660)
E0 79 42 (For pitch 0, we have bend 9.3285)
E7 1C 42 (For pitch 7, we have bend 7.0378)
E2 4B 40 (For pitch 2, we have bend 1.8604)
EA 19 3F (For pitch 9, we have bend -2.5590)
E4 2B 3E (For pitch 4, we have bend -5.2652)
EC 19 3D (For pitch 11, we have bend -8.8818)
E6 2E 3B (For pitch 6, we have bend -14.6857)
E1 2F 3A (For pitch 1, we have bend -17.8358)
E8 27 3A (For pitch 8, we have bend -18.0264)

But was very surprised to find that the total "pain" tallied by COFT
was only 10% less than for the 31-tET subset, Eb thru G#. The COFT
values closely track meantone, with sharply narrowed fifths all along
the line. I'm not quite sure why this is, except for the observation
that minor thirds tend to pull fifths narrow even more than major thirds
do.

JdL

[I wrote:]
>>I thought that, the older the music, the better COFT would be against
>>meantone, but I was wrong!

[Paul Erlich:]
>Why would you have thought that?

[I:]
>>The COFT
>>values closely track meantone, with sharply narrowed fifths all along
>>the line. I'm not quite sure why this is

[Paul:]
>Well, meantone was basically designed to be a COFT for music using 7 or
>more consecutive notes on the chain of fifths. Scarlatti's music
>certainly falls into that category, as does most of the music that
>preceded him since the invention of meantone.

>Or was there something in particular that made the meantone result
>surprising?

Well... the truth is, I never realized just to what extent meantone IS
COFT (or a fairly close approximation thereof) for certain music.
All of a sudden I appreciate meantone more than I did before.

I made up a test sequence with a mix of two-note intervals in the range
Eb ... G#:

Eb Bb F C G D A E B F# C# G#
3 10 5 0 7 2 9 4 11 6 1 8

having 11 sequential fifths (Eb and Bb, Bb and F, ... C# and G#),
then 9 sequential minor thirds (Eb and C ... B and G#), then 8
sequential major thirds (Eb and G ... E and G#). All intervals are
made to be the same weight.

The sequence doesn't cross the wolf at all. The COFT that results is
very close to meantone:

E3 32 46 (For pitch 3, we have bend 20.2362)
EB 58 45 (For pitch 10, we have bend 17.9954)
E5 3F 44 (For pitch 5, we have bend 14.2321)
E0 76 42 (For pitch 0, we have bend 9.2499)
E7 68 41 (For pitch 7, we have bend 5.7575)
E2 53 40 (For pitch 2, we have bend 2.0719)
EA 2D 3F (For pitch 9, we have bend -2.0717)
E4 18 3E (For pitch 4, we have bend -5.7577)
EC 0A 3D (For pitch 11, we have bend -9.2499)
E6 41 3B (For pitch 6, we have bend -14.2320)
E1 28 3A (For pitch 1, we have bend -17.9953)
E8 4E 39 (For pitch 8, we have bend -20.2364)

Here's the fifths formed:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force
---- ------ ---- ------ -------- ------- ------- --------
3 20.24 10 18.00 51.200 701.977 697.759 -215.969
10 18.00 5 14.23 51.200 701.977 696.237 -293.913
5 14.23 0 9.25 51.200 701.977 695.018 -356.328
0 9.25 7 5.76 51.200 701.977 696.508 -280.045
7 5.76 2 2.07 51.200 701.977 696.314 -289.937
2 2.07 9 -2.07 51.200 701.977 695.856 -313.390
9 -2.07 4 -5.76 51.200 701.977 696.314 -289.957
4 -5.76 11 -9.25 51.200 701.977 696.508 -280.038
11 -9.25 6 -14.23 51.200 701.977 695.018 -356.321
6 -14.23 1 -18.00 51.200 701.977 696.237 -293.916
1 -18.00 8 -20.24 51.200 701.977 697.759 -215.980

Minor thirds:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force
---- ------ ---- ------ -------- ------- ------- --------
0 9.25 3 20.24 51.200 315.830 310.986 -248.017
7 5.76 10 18.00 51.200 315.830 312.238 -183.941
2 2.07 5 14.23 51.200 315.830 312.160 -187.917
9 -2.07 0 9.25 51.200 315.830 311.322 -230.855
4 -5.76 7 5.76 51.200 315.830 311.515 -220.943
11 -9.25 2 2.07 51.200 315.830 311.322 -230.841
6 -14.23 9 -2.07 51.200 315.830 312.160 -187.910
1 -18.00 4 -5.76 51.200 315.830 312.238 -183.951
8 -20.24 11 -9.25 51.200 315.830 310.987 -248.010

Major thirds:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force
---- ------ ---- ------ -------- ------- ------- --------
3 20.24 7 5.76 51.200 386.147 385.521 -32.028
10 18.00 2 2.07 51.200 386.147 384.077 -105.996
5 14.23 9 -2.07 51.200 386.147 383.696 -125.473
0 9.25 4 -5.76 51.200 386.147 384.992 -59.102
7 5.76 11 -9.25 51.200 386.147 384.993 -59.096
2 2.07 6 -14.23 51.200 386.147 383.696 -125.480
9 -2.07 1 -18.00 51.200 386.147 384.076 -106.006
4 -5.76 8 -20.24 51.200 386.147 385.521 -32.029

This test sequence does not achieve alignment between meantone and COFT
as close as Scarlatti's Sonata K.082, which was within 10%: the numbers
here are:

12-tET Total spring pain: 98143.234

31-tET subset: Eb thru G#: 16531.163

COFT spring pain: 13895.559

But, this sequence does reinforce the idea that meantone is well suited
to music that doesn't doesn't stretch the identity of notes or make
circles of fifths (to anticipate a response from you, Paul, extended
meantone (>12 notes) is even more capable; more numbers to follow!).

JdL