Well Tempered Tuning Systems

The following tables give examples of 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.

Idealised Well Temperament

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+7.5 0 C 0 0
G+5 -4.5
-9/48P
C# 93 -19
D+2.5 -4.5
-9/48P
D 195 -9
A+0 -4.5
-9/48P
Eb 297 -19
E-2.5 -4.5
-9/48P
E 390 +4
B-2.5 -2
-1/12P
F 500 +2
F#-1.5 -1
-1/24P
F# 591 +1
C#+0.5 0
0
G 697.5 -4.5
G#+2.5 0
0
Ab 795 -19
Eb+4.5 0
0
A 892.5 +8.5
Bb+6.5 0
0
Bb 999 +3
F+7.5 -1
-1/24P
B 1090 +2
C+7.5 -2
-1/12P
c 1200 0


Young I (Idealised well Temperament) 1799

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+6 0 C 0 0
G+4 -4
-1/6P
C# 94 -18
D+2 -4
-1/6P
D 196 -8
A+0 -4
-1/6P
Eb 298 -18
E-2 -4.8
-1/6P
E 392 +6
B-2 2
-1/12P
F 500 +2
F#-2 -2
-1/12P
F# 592 +2
C#+0 0
0
G 698 -4
Ab+2 0
0
Ab 796 -18
Eb+4 0
0
A 894 +10
Bb+6 0
0
Bb 1000 +4
F+6 -2
-1/12P
B 1092 +4
c+0 -2
-1/12P
c 1200 0


Young II 1800

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+6 0 C 0 0
G+4 -4
-1/6P
C# 90 -22
D+2 -4
-1/6P
D 196 -8
A+0 -4
-1/6P
Eb 294 -22
E-2 -4
-1/6P
E 392 +6
B-4 -4
-1/6P
F 498 0
F#-6 -4
-1/6P
F# 588 -2
C#-4 0
0
G 698 -4
G#-2 0
0
Ab 792 -22
Eb+0 0
0
A 894 +10
Bb+2 0
0
Bb 996 0
F+4 0
0
B 1090 -2
C+6 0
0
c 1200 0


Vallotti

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+6 0 C 0 0
G+4 -4
-1/6P
C# 94 -18
D+2 -4
-1/6P
D 196 -8
A+0 -4
-1/6P
Eb 298 -18
E-2 -4
-1/6P
E 392 +6
B-4 -4
-1/6P
F 502 +4
F#-2 0
0
F# 592 +2
C#+0 0
0
G 698 -4
Ab+2 0
0
Ab 796 -18
Eb+4 0
0
A 894 +10
Bb+6 0
0
Bb 1000 +4
F+8 0
0
B 1090 +2
c+6 -4
-1/6P
c 1200 0


Neidhardt I (1724)

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+6 0 C 0 0
G+4 -4
-1/6P
C# 94 -18
D+2 -4
-1/6P
D 196 -8
A+0 -4
-1/6P
Eb 296 -20
E-2 -4
-1/6P
E 392 +6
B-2 -2
-1/12P
F 498 0
F#-2 -2
-1/12P
F# 592 +2
C#+0 0
0
G 698 -4
Ab+2 0
0
Ab 796 -18
Eb+2 -2
-1/12P
A 894 +10
Bb+2 -2
-1/12P
Bb 996 0
F+4 0
0
B 1092 +4
c+6 0
0
c 1200 0


Nigel Taylor's Idealised Circulating Temperament

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+9 0 C 0 0
G+6 -5
-5/24P
C# 92 -20
D+3 -5
-5/24P
D 194 -10
A+0 -5
-5/24P
Eb 296 -16
E-3 -5
-5/24P
E 388 +2
B-1 0
-0
F 498 +0
F#-1 -2
-1/12P
F# 590 +0
C#+1 0
0
G 697 -5
Ab+3 0
0
Ab 794 -20
Eb+5 0
0
A 891 +7
Bb+7 0
0
Bb 998 +2
F+7 -2
-1/12P
B 1090 +2
C+9 0
0
c 1200 0


1/8th Comma "Well Tempered"

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+3 0 C 0 0
G+2 -3
-1/8P
C# 90 -22
D+1 -3
-1/8P
D 196 -8
A+0 -3
-1/8P
Eb 294 -22
E-1 -3
-1/8P
E 396 +10
B-2 -3
-1/8P
F 498 0
F#-3 -3
-1/8P
F# 588 -2
C#-4 3
-1/8P
G 698 -4
G#-5 -3
-1/8P
Ab 792 -22
Eb-3 0
0
A 894 +10
Bb-1 0
0
Bb 996 0
F+1 0
0
B 1090 -2
C+3 0
0
c 1200 0


Broadwood "Best" Tuning (1885)

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+6 0 C 0 0
G+4 -4
-1/6P
C# 95 -17
D+2 -4
-1/6P
D 196 -8
A+0 -4
-1/6P
Eb 295 -21
E-2 -4
-1/6P
E 392 +6
B-3 -3
-3/24P
F 499 +1
F#-1 0
0
F# 593 +3
C#+1 0
0
G 698 -4
G#+2 1
-1/24P
Ab 796 -18
Eb+1 -3
-3/24P
A 894 +10
Bb+3 0
0
Bb 997 +1
F+5 0
0
B 1091 +3
C+6 0
0
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


Charles Earl Stanhope Improved Kirnberger IIA

CIRCLE
OF FIFTHS
TEMPERING CHROMATIC
SCALE
INTERVALS
IN CENTS
ERRORS
COMP WITH
JUST SCALE
C+8.5 0 C 0 0
G+10.5 0
0
C# 91 -21
D+5.25 -7.2
-1/3S
D 197 -7
A+0 -7.2
-1/3S
Eb 295 -21
E-5.2 -7.2
-1/3S
E 386 0
B-3.2 0
0
F 498 0
F#-2.2 -1
-1/24P
F# 589 -1
C#-0.2 0
0
G 702 0
G#+1.6 0
0
Ab 793 -21
Eb-3.6 0
0
A 891.5 +7.5
Bb+4.5 1
-1/24P
Bb 996 0
F+6.5 0
0
B 1088 0
C+8.5 0
0
c 1200 0