Well Tempered Tuning Systems

The following tables give different well tempered systems. The first column gives the circle of 5ths, beginning on C, and with A=440 c.p.s.(A+0) as the basis. The 2nd column gives the tempering- the degree of flattening or sharpening- to the 5ths. The column shows the tempering in cents, and in fractions of a Pythagorean(P) or Syntonic(S) comma. The 3rd column gives the chromatic scale, the 4th column shows intervals of the chromatic scale in cents, and the final column gives the errors in cents compared with Just Intonation.

Kirnberger II

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+5 0 C 0 0
G+7 0
0
C# 90 -22
D+9 0
0
D 204 0
A+0 -11
-1/2S
Eb 294 -24
E-9 -11
-1/2S
E 386 0
B-7 0
0
F 498 0
F#-5 0
0
F# 590 0
C#-5 -2
-1/12P
G 702 0
Ab-3 0
0
Ab 792 -22
Eb-1 0
0
A 895 +11
Bb+1 0
0
Bb 996 0
F+3 0
0
B 1088 0
c+5 0
0
c 1200 0


Kirnberger III

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+10.5 0 C 0 0
G+7 -5.5
-1/4S
C# 90 -22
D+3.5 -5.5
-1/4S
D 193 - 11
A+0 -5.5
-1/4S
Eb 294 -22
E-3.5 -5.5
-1/4S
E 386 +0
B-1.5 0
0
F 498 0
F#+0.5 0
0
F# 590 -0
C#+0.5 -2
-1/12P
G 696.5 -5.5
G#+2.5 0
0
Ab 792 -22
Eb+4.5 0
0
A 889.5 +5.5
Bb+6.5 0
0
Bb 996 0
F+8.5 0
0
B 1088 0
C+10.5 0
0
c 1200 0


Werckmeister III(1691)

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+12 0 C 0 0
G+8 -6
-1/4P
C# 90 -22
D+4 -6
-1/4P
D 192 -12
A+0 -6
-1/4P
Eb 294 -22
E+2 0
0
E 390 +4
B+4 0
0
F 498 0
F#+0 -6
-1/4P
F# 588 -2
C#+2 0
0
G 696 -6
G#+4 0
0
Ab 792 -22
Eb+6 0
0
A 888 +4
Bb+8 0
0
Bb 996 0
F+10 0
0
B 1092 +4
C+12 0
0
c 1200 0


Bach's "Wohletemperirt" (Rediscovered By Kellner)

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+8.4 0 C 0 0
G+5.6 -4.8
-1/5P
C# 90 -22
D+2.8 -4.8
-1/5P
D 194.5 -9.5
A+0 -4.8
-1/5P
Eb 294 -22
E-2.8 -4.8
-1/5P
E 389 +3
B-0.8 0
0
F 498 0
F#-3.6 -4.8
-1/5P
F# 588 -2
C#-1.6 0
0
G 697 -5
Ab+0.4 0
0
Ab 792 -22
Eb+2.4 0
0
A 892 +8
Bb+4.4 0
0
Bb 996 0
F+6.4 0
0
B 1091 +3
c+8.4 0
0
c 1200 -14



Augustus De Morgan

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C-3 0 C 0 0
G-1.5 -0.4
-1/48P
C# 100 -17
D-0.5 -1
-1/24P
D 202.5 -8
A+0 -1.5
-3/48P
Eb 298.5 -21
E-0 -2
-1/12P
E 403 +6
B-0.5 -2.5
-5/48P
F 499 +1
F#-1.5 -3
-3/24P
F# 601.5 +11.5
C#-3 -3.5
-7/48P
G 701.5 -0.5
G#-4 3
-3/24P
Ab 799 -15
Eb-4.5 -2.5
-5/48P
A 903 +10
Bb-4.5 -2
-1/12P
Bb 998.5 +2.5
F-4 -1.5
-3/48P
B 1102.5 +14.5
C-3 -1
-1/24P
c 1200 0


Bendeler III

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+6 0 C 0 0
G+2 -6
-1/4P
C# 96 -14
D-2 -6
-1/4P
D 192 -12
A+0 0
0
Eb 294 -22
E+2 0
0
E 396 +10
B-2 -6
-1/4P
F 498 0
F#+0 0
0
F# 594 +4
C#+2 0
0
G 696 -6
G#+4 0
0
Ab 798 -16
Eb+0 -6
-1/4P
A 894 +10
Bb+2 0
0
Bb 996 0
F+4 0
0
B 1092 +4
C+6 0
0
c 1200 0