A. Absolute pitches of the 66 tones of the 33 bells in the 2nd tier of the 65 bells of Marquis Yi (433 B.C.)
in Cent deviations relative to the standard western scale based on A4 = 440 Hz
D D# E F F# G G# A A# B C C# D D#
Set I
Octave D6-C7, 3 bells -40 (+14) -18 -42 (+25) (+35) Octave excluded from analysis, due to obvious mistuning of three tones
Octave D5-C6, 3 bells -52 -62 -10 -38 -29 -16
Octave D4-D#5, 5 bells -41 -40 -18 -50 -35 -59 -64 -60 -42 -60
Set II
Octave D6-C7, 3 bells -11 -15 -18 -16 -60 -23
Octave D5-C6, 3 bells -41 -58 -21 -36 -51 -40
Octave D4-D#5, 6 bells -53 -63 -34 -13 -47 -46 -44 (+12) -74 -35 -27 -46
Set III
Octave D5-C6, 4 bells -24 -39 -15 -38 -16 -25 -58 -38
Octave D4-C5, 3 bells -50 -50 -22 -42 -41 -15
Octave G3-D#4, 3 bells -32 -21 -18 -26 -24 -28 Octave excluded from analysis, due to incompleteness
Total mean -39.0 -46.7 -19.5 -35.4 -41.3 -31.6 Accidentals excluded from further analysis, due to incompleteness
Temperation of fifths - intervals in Cent, absolute pitches in Cent deviations from A440
The six main tones in fifths F 687.9 C 696.2 G 696.4 D 697.8 A 694.5 E mean C-E: 696.2
Mean absolute pitches -19.5 -31.6 -35.4 -39.0 -41.3 -46.7
B. The 42 tones from table A that enter the analysis
D E F G A C
-52 -62 -10 -38 -29 -16
-41 -40 -18 -50 -59 -60
-11 -15 -18 -16 -60 -23
-41 -58 -21 -36 -51 -40
-53 -63 -34 -47 -44 -35
-24 -39 -15 -16 -25 -38
-50 -50 -22 -42 -41 -15
C. Five tuning models and deviations of the bells' intervals from the model intervals
pyth D E F G A C
0 204 204 90 294 204 498 204 702 294 996 Results per octave
mean SD median
14 62 32 5 19 26 22 19
3 32 36 13 5 18 15 13
8 7 2 48 43 22 22 8
21 47 19 19 17 25 13 19
14 39 17 1 15 17 14 15
Result for
1st interval (D-E)
19 34 5 13 7 16 12 13 mean
All 35 interval deviations: 4 38 24 3 32 20 16 24 11.9
mean 20.5
SD 15.6
median 17
just D E F G A C
0 182 182 134 316 182 498 204 702 316 1018 Results per octave
mean SD median
8 18 10 5 3 9 6 8
19 12 14 13 17 15 3 14
14 37 20 48 21 28 14 21
1 3 3 19 5 6 7 3
8 5 5 1 7 5 3 5
Result for
1st interval (D-E)
3 10 17 13 29 14 10 13 mean
All 35 interval deviations: 18 6 2 3 10 8 6 6 10.1
mean 12.2
SD 10.3
median 10
12ET D E F G A C
0 200 200 100 300 200 500 200 700 300 1000 Results per octave
mean SD median
10 52 28 9 13 22 18 13
1 22 32 9 1 13 14 9
4 3 2 44 37 18 21 4
17 37 15 15 11 19 10 15
10 29 13 3 9 13 10 10
Result for
1st interval (D-E)
15 24 1 9 13 12 8 13 mean
All 35 interval deviations: 0 28 20 1 26 15 14 20 8.1
mean 16.1
SD 13.3
median 13
1/3komma for FCGDAE D E F G A C
0 190 190 126 316 190 505 190 695 316 1010 Results per octave
mean SD median
0 26 18 19 3 13 11 18
11 4 22 1 17 11 8 11
6 29 12 34 21 21 11 21
7 11 5 5 5 6 3 5
0 3 3 13 7 5 5 3
Result for
1st interval (D-E)
5 2 9 1 29 9 11 5 mean
All 35 interval deviations: 10 2 10 11 10 9 4 10 5.7
mean 10.6
SD 9.1
median 9.4
695.2 for CGDAE on pure D-F D E F G A C
0 192 192 123 316 188 504 192 696 311 1008 Results per octave
mean SD median
2 29 16 17 2 13 11 16
9 1 20 1 12 9 8 9
4 26 14 36 26 21 13 26
9 14 3 7 0 7 5 7
2 6 1 11 2 4 4 2
Result for
1st interval (D-E)
7 1 11 1 24 9 10 7 mean
All 35 interval deviations: 8 5 8 9 15 9 4 8 5.9
mean 10.3
SD 9.2
median 8.2
D. Cross matrix showing the results, in level of significance, of comparing the differences in fitness of the five models
Test: t-test
pyth just 12ET 1/3ko 696.2
mean 20.5 12.2 16.1 10.6 10.3
SD 15.6 10.3 13.3 9.1 9.2
median 17 10 13 9.4 8.2
phyt
just 0.011
12ET 0.209 0.178
1/3ko 0.002 0.493 0.048
696.2 0.001 0.418 0.038 0.888