Beat Rate Charts

The following charts give the beat rates for a number of temperaments. The 1/3rd and 1/4 comma charts are more for interest only than practical keyboard application, with the exception of some suitable harpsichords, or virginals. The beat rates are calculated on A=440Hz, are in beats (cycles) per second, and setting the bearings on the octave beginning with middle C.

Kirnberger III
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2.45 C-E 0
G-D -3.66 G-B +6.12
D-A -2.74 D-F# +9.17
A-E -4.09 A-C# +18.09
E-B 0 E-G# +18.68
B-F# 0 B-Eb +28.02
F#-C# -1.25 F#-Bb +23.11
C#-Ab 0 C#-F +17.33
Ab-Eb 0 Ab-C +25.99
Eb-Bb 0 Eb-G +14.6
Bb-F 0 Bb-D +14.58
F-C 0 F-A +5.46


Kirnberger II(1779)

This tuning contains more just intervals than any other circulating temperament. The two heavily tempered 5ths beat quite fast. Any tuner unused to fast beat rates can work an octave lower than usual. Set A above middle C to 440Hz; tune the A below to 220Hz; temper the 5ths D-A (-2.74) and A-E (-4.09). The two cent schisma can be absorbed by the other ten 5ths.

INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G 0 C-E 0
G-D 0 G-B 0
D-A -5.48 D-F# 0
A-E -8.17 A-C# +11.2
E-B 0 E-G# +18.62
B-F# 0 B-Eb +27.93
F#-C# -1.25 F#-Bb +20.95
C#-Ab 0 C#-F +17.27
Ab-Eb 0 Ab-C +25.91
Eb-Bb 0 Eb-G +19.43
Bb-F 0 Bb-D +29.15
F-C 0 F-A +10.90


YOUNG II BEAT RATES

Any competent piano tuner can set this temperament; begin by setting the A as usual, set the A(220 c.p.s.)below, then work using equal beat rates in the octave lower than normal, for the six tempered 5ths.

INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -1.78 C-E +4.46
G-D -2.66 G-B +6.67
D-A -1.99 D-F# +4.99
A-E -2.98 A-C# +12.46
E-B -2.23 E-G# +13.07
B-F# -3.33 B-Eb +25.17
F#-C# 0 F#-Bb +23.05
C#-Ab 0 C#-F +17.28
Ab-Eb 0 Ab-C +25.93
Eb-Bb 0 Eb-G +15.89
Bb-F 0 Bb-D +18.52
F-C 0 F-A +9.91


YOUNG I (Idealised) BEAT RATES
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -1.78 C-E +4.46
G-D -2.66 G-B +8.9
D-A -1.98 D-F# +8.31
A-E -3 A-C# +17.46
E-B -1.11 E-G# +16.82
B-F# -1.67 B-Eb +28.01
F#-C# 0 F#-Bb +23.10
C#-Ab 0 C#-F +15.74
Ab-Eb 0 Ab-C +21.24
Eb-Bb 0 Eb-G +12.37
Bb-F 1.58 Bb-D +13.27
F-C 1.19 F-A +7.93


Idealised Well Temperament Beat Rates
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2 C-E +2.97
G-D -2.99 G-B +7.23
D-A -2.24 D-F# +8.33
A-E -3.35 A-C# +18.08
E-B -1.11 E-G# +17.75
B-F# -0.83 B-Eb +30.03
F#-C# 0 F#-Bb +23.10
C#-Ab 0 C#-F +16.54
Ab-Eb 0 Ab-C +22.43
Eb-Bb 0 Eb-G +12.82
Bb-F -0.79 Bb-D +13.25
F-C -1.19 F-A +6.45


Rameau's Modified Mean Tone
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2.45 C-E 0
G-D -3.66 G-B 0
D-A -2.74 D-F# +4.58
A-E -4.09 A-C# +13.71
E-B -3.06 E-G# +15.4
B-F# 0 B-Eb +38.49
F#-C# 0 F#-Bb +36.64
C#-Ab 0 C#-F +24.42
Ab-Eb +3.87 Ab-C +30.09
Eb-Bb +3.89 Eb-G +12.84
Bb-F -3.05 Bb-D +2.22
F-C -3.27 F-A 0


Nigel Taylor's Circulating Temperament
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2.22 C-E +1.49
G-D -3.33 G-B +7.79
D-A -2.49 D-F# +8.33
A-E -3.72 A-C# +18.69
E-B 0 E-G# +18.67
B-F# -1.67 B-Eb +28.03
F#-C# 0 F#-Bb +23.11
C#-Ab 0 C#-F +15.75
Ab-Eb 0 Ab-C +23.62
Eb-Bb 0 Eb-G +13.27
Bb-F -1.58 Bb-D +13.25
F-C 0 F-A +6.94


1/4 COMMA MEAN TONE BEAT RATES
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2.45 C-E 0
G-D -3.66 G-B 0
D-A -2.74 D-F# 0
A-E -4.09 A-C# 0
E-B -3.06 E-G# 0
B-F# -4.58 B-Eb +59.03
F#-C# -3.42 F#-Bb +44.14
C#-Ab -2.56 C#-F +33
Ab-Eb +25.69 Ab-C +49.35
Eb-Bb -2.93 Eb-G 0
Bb-F -4.38 Bb-D 0
F-C -3.27 F-A 0


1/3 COMMA MEAN TONE BEAT RATES
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -3.27 C-E -5.45
G-D -4.89 G-B 8.15
D-A -3.65 D-F# -6.09
A-E -5.45 A-C# -9.09
E-B -4.07 E-G# -6.79
B-F# -6.09 B-Eb +79.81
F#-C# -4.55 F#-Bb +59.61
C#-Ab -3.4 C#-F +44.52
Ab-Eb +39.91 Ab-C +66.51
Eb-Bb -3.93 Eb-G -6.55
Bb-F -5.87 Bb-D -9.78
F-C -4.38 F-A -7.3


8X1/8 WELL TEMPERAMENT BEAT RATES
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -1.33 C-E +7.44
G-D -1.99 G-B +11.13
D-A -1.47 D-F# +8.33
A-E -2.23 A-C# +12.43
E-B -1.67 E-G# +9.31
B-F# -2.51 B-Eb +18.15
F#-C# -1.88 F#-Bb +16.74
C#-Ab -2.8 C#-F +14.87
Ab-Eb 0 Ab-C +25.88
Eb-Bb 0 Eb-G +16.74
Bb-F 0 Bb-D +21.13
F-C 0 F-A +12.89


WERCKMEISTER III BEAT RATES
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2.67 C-E +2.98
G-D -3.99 G-B +11.15
D-A -2.99 D-F# +8.33
A-E 0 A-C# +19.97
E-B 0 E-G# +14.97
B-F# -5 B-Eb +22.46
F#-C# 0 F#-Bb +23.1
C#-Ab 0 C#-F +17.34
Ab-Eb 0 Ab-C +26.01
Eb-Bb 0 Eb-G +14.17
Bb-F 0 Bb-D +13.26
F-C 0 F-A +3.97


JEAN LE ROND D'ALEMBERT (1752) BEAT RATES
INTERVALS 5THS(c.p.s.) INTERVALS MAJ. 3RDS
C-G -2.45 C-E 0
G-D -3.66 G-B +0.80
D-A -2.74 D-F# +1.19
A-E -4.09 A-C# +2.69
E-B -2.66 E-G# +2.68
B-F# -3.98 B-Eb +16.65
F#-C# -2.98 F#-Bb +21.96
C#-Ab -2.23 C#-F +23.57
Ab-Eb +2.99 Ab-C +46.0
Eb-Bb +2.25 Eb-G +25.87
Bb-F +3.38 Bb-D +25.86
F-C +2.54 F-A +9.69