Optimal
Well-Temperament Examples
Polansky, Rockmore, Johnson, Repetto
4/19
J.S Bach, Well-Tempered Clavier
Selected pieces in various well-temperaments.
Historical temperaments: Young 2, Werkmeister 3
New temperaments: OWT1, OWT2 (variations on standard tuning ideas);
Septimal OWT1, Septimal OWT2 (more experimental, using septimal
ratios)
All of these (except the two historical ones) were generated
using the optimal well-temperament algorithms described in our
writing.
(Thanks to Ron Nagorcka for generating the examples).
Book 1
Book 2
Five Different
Well-temperaments
(Cents values, rounded to the nearest cent)
Note
|
W3
|
Y2
|
OWT1
|
OWT2
|
Septimal
OWT1
|
Septimal
OWT2
|
C#
|
90 |
90 |
102 |
93
|
42 |
93
|
D
|
192
|
196
|
204
|
203
|
206
|
209
|
Eb
|
294
|
294
|
297
|
296
|
272
|
262
|
E
|
390
|
392
|
396
|
397
|
386
|
386
|
F
|
498
|
498
|
498
|
498
|
491
|
490
|
F#
|
588
|
588
|
600
|
592
|
543
|
588
|
G
|
696
|
698
|
702
|
702
|
704
|
707
|
G#
|
792
|
792
|
804
|
795
|
764
|
769
|
A
|
888
|
894
|
897
|
903
|
877
|
876
|
Bb
|
996
|
996
|
996
|
997
|
977
|
978
|
B
|
1092
|
1090
|
1098
|
1091
|
1090
|
1093
|
C
|
1200
|
1200
|
1200
|
1200
|
1200
|
1200
|
OWT1 and OWT2 are similar to W3 and Y2, except that they have a
different structure of "best" keys.
Septimal OWT1 and OWT2 are created with the following weights and
ideal intervals (only the weights are different, and only slightly):
Septimal OWT1
Intervals
|
1
|
m2
|
M2
|
m3
|
M3
|
P4
|
TT
|
P5
|
m6
|
M6
|
m7
|
M7
|
Ideal intervals
|
|
100
|
204
|
267
|
386
|
498
|
600
|
702
|
800
|
900
|
969
|
1100
|
| Interval weights |
|
.0001
|
0.001
|
0.3
|
1
|
1
|
0.001
|
1
|
.001
|
0.001
|
1
|
0.001
|
| Key Weights |
1
|
0.001
|
0.001
|
0.3
|
0.001
|
1
|
0.001
|
1
|
0.001
|
0.001
|
0.001
|
0.001
|
Septimal OWT2
Intervals
|
1
|
m2
|
M2
|
m3
|
M3
|
P4
|
TT
|
P5
|
m6
|
M6
|
m7
|
M7
|
Ideal intervals
|
|
100
|
204
|
267
|
386
|
498
|
600
|
702
|
800
|
900
|
969
|
1100
|
| Interval weights |
|
0.1
|
0.001
|
0.3
|
1
|
1
|
0.001
|
1
|
.001
|
0.001
|
1
|
0.001
|
| Key Weights |
1
|
0.001
|
0.001
|
0.001
|
0.001
|
1
|
0.001
|
1
|
0.001
|
0.001
|
0.001
|
0.001
|