Yahoo Tuning Groups Ultimate Backup tuning-math Interval analysis of Pachelbel's Canon

This analysis is based upon Herman Miller's sequence, as reported on
the tuning list, available at:

http://www.io.com/~hmiller/music/warped-canon.html

The piece is in the key of D major. Four scale degrees (D#/Eb, F,
G#/Ab, and Bb) are not present.

Interval strengths shown in the table are reduced from "nominal" by
the following factors:

1: 1/32
2: 1/16
3: nominal (1/1)
4: nominal (1/1)
5: nominal (1/1)
6: 1/8
7: same as 5
8: same as 4
9: same as 3
10: same as 2
11: same as 1

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

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 9.04 1 -9.14 1.642 100.000 81.821 -29.840 271.230
0 9.04 2 4.91 6.989 200.000 195.868 -28.880 59.667
0 9.04 4 4.27 25.940 386.314 395.228 231.239 1030.680
0 9.04 6 -9.45 8.241 600.000 581.505 -152.419 1409.504
0 9.04 7 4.90 117.704 701.955 695.854 -718.114 2190.602
0 9.04 9 5.06 65.251 884.359 896.017 760.744 4434.639
0 9.04 11 -9.59 3.367 1100.000 1081.369 -62.731 584.358
1 -9.14 0 9.04 1.642 1100.000 1118.179 29.840 271.230
1 -9.14 2 4.91 32.275 100.000 114.047 453.356 3184.047
1 -9.14 4 4.27 3251.990 315.641 313.407 -7266.541 8118.509
1 -9.14 6 -9.45 1944.208 498.045 499.684 3185.654 2609.905
1 -9.14 7 4.90 85.058 600.000 614.033 1193.590 8374.620
1 -9.14 9 5.06 4741.141 813.686 814.196 2416.844 616.005
1 -9.14 11 -9.59 28.174 1000.000 999.548 -12.732 2.877
2 4.91 0 9.04 6.989 1000.000 1004.132 28.880 59.667
2 4.91 1 -9.14 32.275 1100.000 1085.953 -453.356 3184.047
2 4.91 4 4.27 47.831 200.000 199.360 -30.601 9.789
2 4.91 6 -9.45 6542.164 386.314 385.637 -4427.375 1498.101
2 4.91 7 4.90 3831.217 498.045 499.986 7436.766 7217.745
2 4.91 9 5.06 4791.038 701.955 700.149 -8650.319 7809.165
2 4.91 11 -9.59 5334.202 884.359 885.502 6096.004 3483.301
4 4.27 0 9.04 25.940 813.686 804.772 -231.239 1030.680
4 4.27 1 -9.14 3251.990 884.359 886.593 7266.541 8118.509
4 4.27 2 4.91 47.831 1000.000 1000.640 30.601 9.789
4 4.27 6 -9.45 44.845 200.000 186.277 -615.426 4222.825
4 4.27 7 4.90 746.854 315.641 300.626 -11214.333 84193.962
4 4.27 9 5.06 4049.592 498.045 500.789 11113.095 15248.559
4 4.27 11 -9.59 401.495 701.955 686.141 -6349.129 50201.623
6 -9.45 0 9.04 8.241 600.000 618.495 152.419 1409.504
6 -9.45 1 -9.14 1944.208 701.955 700.316 -3185.654 2609.905
6 -9.45 2 4.91 6542.164 813.686 814.363 4427.375 1498.101
6 -9.45 4 4.27 44.845 1000.000 1013.723 615.426 4222.825
6 -9.45 7 4.90 23.500 100.000 114.349 337.206 2419.308
6 -9.45 9 5.06 5472.479 315.641 314.513 -6177.203 3486.340
6 -9.45 11 -9.59 2105.141 498.045 499.865 3830.429 3484.846
7 4.90 0 9.04 117.704 498.045 504.146 718.114 2190.602
7 4.90 1 -9.14 85.058 600.000 585.967 -1193.590 8374.620
7 4.90 2 4.91 3831.217 701.955 700.014 -7436.766 7217.745
7 4.90 4 4.27 746.854 884.359 899.374 11214.333 84193.962
7 4.90 6 -9.45 23.500 1100.000 1085.651 -337.206 2419.308
7 4.90 9 5.06 66.333 200.000 200.163 10.838 0.885
7 4.90 11 -9.59 3727.668 386.314 385.515 -2975.734 1187.739
9 5.06 0 9.04 65.251 315.641 303.983 -760.744 4434.639
9 5.06 1 -9.14 4741.141 386.314 385.804 -2416.844 616.005
9 5.06 2 4.91 4791.038 498.045 499.851 8650.319 7809.165
9 5.06 4 4.27 4049.592 701.955 699.211 -11113.095 15248.559
9 5.06 6 -9.45 5472.479 884.359 885.487 6177.203 3486.340
9 5.06 7 4.90 66.333 1000.000 999.837 -10.838 0.885
9 5.06 11 -9.59 35.909 200.000 185.352 -525.990 3852.335
11 -9.59 0 9.04 3.367 100.000 118.631 62.731 584.358
11 -9.59 1 -9.14 28.174 200.000 200.452 12.732 2.877
11 -9.59 2 4.91 5334.202 315.641 314.498 -6096.004 3483.301
11 -9.59 4 4.27 401.495 498.045 513.859 6349.129 50201.623
11 -9.59 6 -9.45 2105.141 701.955 700.135 -3830.429 3484.846
11 -9.59 7 4.90 3727.668 813.686 814.485 2975.734 1187.739
11 -9.59 9 5.06 35.909 1000.000 1014.648 525.990 3852.335
---- ------ ---- ------ -------- -------- -------- ---------- ----------
painSum 221203.167

JdL

[I wrote:]
>>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

[Orphon Soul wrote:]
>Some pretty interesting variables you have there.
>Where are they from? Are there more?

I tune using a spring model, an analog to physical springs, in which
deviation from rest is equivalent to tuning deviation from ideal, and
energy is equivalent to "pain" of badly-tuned intervals and/or unwanted
tuning motion. I describe this model in greater detail at:

/tuning/topicId_7890.html#7890
/tuning/topicId_12668.html#12668

Though the work I'm doing today will no doubt be considered _very_
primitive in the light of future knowledge, the spring model is
extremely flexible in its application potential, or so it seems to me
today. Time will tell...

Please let me know if I can answer any further questions!

JdL

Interval analysis of Pachelbel's Canon

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

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

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