A. Absolute pitch of the 66 tones of the 33
bells in the 2nd tier of the 65 bells of Marquis Yi (433 B.C.) |
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in
Cent deviations relative to the western standard scale based on A4 = 440
Hz |
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D |
D# |
E |
F |
F# |
G |
G# |
A |
A# |
B |
C |
C# |
D |
D# |
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Set I |
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Octave D6-C7, 3 bells |
-40 |
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(+14)* |
-18 |
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-42 |
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(+25)* |
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(+35)* |
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Octave D5-C6, 3 bells |
-52 |
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-62 |
-10 |
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-38 |
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-29 |
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-16 |
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Octave D4-D#5, 5 bells |
-41 |
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-40 |
-18 |
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-50 |
-35 |
-59 |
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-64 |
-60 |
-42 |
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-60 |
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Set II |
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Octave D6-C7, 3 bells |
-11 |
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-15 |
-18 |
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-16 |
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-60 |
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-23 |
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Octave D5-C6, 3 bells |
-41 |
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-58 |
-21 |
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-36 |
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-51 |
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-40 |
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Octave D4-D#5, 6 bells |
-53 |
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-63 |
-34 |
-13 |
-47 |
-46 |
-44 |
(+12)* |
-74 |
-35 |
-27 |
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-46 |
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Set III |
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Octave D5-C6, 4 bells |
-24 |
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-39 |
-15 |
-38 |
-16 |
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-25 |
-58 |
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-38 |
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Octave D4-C5, 3 bells |
-50 |
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-50 |
-22 |
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-42 |
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-41 |
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-15 |
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Octave G3-D#4, 3 bells |
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-32 |
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-21 |
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-18 |
-26 |
-24 |
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-28 |
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Total mean |
-39.0 |
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-46.7 |
-19.5 |
-25.5 |
-35.4 |
-40.5 |
-41.3 |
-58.0 |
-52.0 |
-31.6 |
-31.0 |
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-44.7 |
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Standard deviation (SD) |
14.7 |
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17.1 |
6.9 |
17.7 |
12.3 |
7.8 |
15.1 |
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29.9 |
14.9 |
9.6 |
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16.0 |
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Variance |
214.9 |
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291.2 |
48 |
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150.8 |
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227.6 |
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222 |
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Weighed variance of CGDAE:
217.3 |
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Variance of F vs. weighed
variance of CGDAE: p < 0.05 (F-test, 2-tailed, F=4.527, df1=39, df2=7) |
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Series of fifths, intervals
in Cent, absolute pitch in Cent deviations from A440 |
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The six main tones in fifths |
F |
687.9 |
C |
696.2 |
G |
696.4 |
D |
697.8 |
A |
694.5 |
E |
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mean C-E: |
696.2 |
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Mean absolute pitch |
-19.5 |
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-31.6 |
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-35.4 |
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-39.0 |
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-41.3 |
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-46.7 |
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Conclusions: |
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a) Due to the significantly
small variance, F appears to
be the norm tone, from
which D and then |
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via temperated fifths
GC and AE are derived. See also outstanding precision of D-F thirds below. |
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b) All fifths in the
series CGDAE are equally tempered for the sake of improved
thirds. |
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B.
Relative pitch of the 66 tones of the 33 bells in the 2nd tier of the 65
bells of Marquis Yi (433 B.C.) |
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All
intervals, in Cent, of the thirds and fifths in main chord D-F-A-C of the
scale DEFGAC |
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Set |
I |
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II |
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III |
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Better than piano |
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Octave |
6 |
5 |
4 |
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6 |
5 |
4 |
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5 |
4 |
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thirds in % |
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Interval |
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D - F |
322 |
342 |
323 |
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293 |
320 |
319 |
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309 |
327 |
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75 |
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F - A |
(443) |
381 |
359 |
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358 |
370 |
390 |
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390 |
381 |
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50 |
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D - A |
(765) |
723 |
682 |
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651 |
690 |
709 |
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699 |
708 |
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A - C |
(310) |
313 |
299 |
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337 |
311 |
309 |
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287 |
326 |
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62.5 |
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F - C |
(753) |
694 |
658 |
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695 |
681 |
699 |
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677 |
707 |
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All 24 thirds: 62.5 |
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Conclusion: The tuning
reflects an orientation towards pure thirds, |
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not towards pure fifths.
15 of 24 thirds are more exact than the modern piano thirds, which have |
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300 instead of 316, and
400 instead of 386 Cent. |
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( )* 4 of 66 tones are possibly mistuned. Chroma
affiliation was done according to bell |
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inscriptions. However,
musical practice is uncertain. Values excluded from calculations in A. |
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