Optimal Well-Temperament Examples

Polansky, Rockmore, Johnson, Repetto

10/13


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
A minor Prelude
Bb minor Fugue
C major Prelude
F minor Prelude

Book 2
C minor Prelude
C-sharp major Fugue
E major Fugue
F major Fugue


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