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