NIGEL'S WEBSITE: TEMPERAMENTS

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

Well Tempered Tuning Systems

Idealised Well Temperament

Young I (Idealised well Temperament) 1799

Young II 1800

Vallotti

Neidhardt I (1724)

Nigel Taylor's Idealised Circulating Temperament

1/8th Comma "Well Tempered"

Broadwood "Best" Tuning (1885)

Bendeler III

Charles Earl Stanhope Improved Kirnberger IIA