Werckmeister temperament

Template:Short description Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings. The tuning systems are numbered in two different ways: The first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord. The monochord labels start from III since just intonation is labelled I and quarter-comma meantone is labelled II. The temperament commonly known as "Werckmeister III" is referred to in this article as Template:Nobr

The tunings I (III), II (IV) and III (V) were presented graphically by a cycle of fifths and a list of major thirds, giving the temperament of each in fractions of a comma.Template:Efn

The last "Septenarius" tuning was not conceived in terms of fractions of a comma, despite some modern authors' attempts to approximate it by some such method. Instead, Werckmeister gave the string lengths on the monochord directly, and from that calculated how each fifth ought to be tempered.

Werckmeister I (III): "correct temperament" based on Template:Sfrac comma divisions

This tuning uses mostly pure (perfect) fifths, as in Pythagorean tuning, but each of the fifths C–G, G–D, D–A and B–FTemplate:Music is made smaller, i.e. tempered by Template:Sfrac comma. No matter if the Pythagorean comma or the syntonic comma is used, the resulting tempered fifths are for all practical purposes the same as meantone temperament fifths. All major thirds are reasonably close to 400 cents and, because not all fifths are tempered, there is no wolf fifth and all 12 notes can be used as the tonic.

Werckmeister designated this tuning as particularly suited for playing chromatic music ("ficte"), which may have led to its popularity as a tuning for J. S. Bach's music in recent years.

Fifth	Template:SmallTemplate:Efn	Third	Template:SmallTemplate:Efn
C–G	^	C–E	1 v
G–D	^	CTemplate:Music–F	4 v
D–A	^	D–FTemplate:Music	2 v
A–E	–	DTemplate:Music–G	3 v
E–B	–	E–GTemplate:Music	3 v
B–FTemplate:Music	^	F–A	1 v
FTemplate:Music–CTemplate:Music	–	FTemplate:Music–BTemplate:Music	4 v
CTemplate:Music–GTemplate:Music	–	G–B	2 v
GTemplate:Music–DTemplate:Music	–	GTemplate:Music–C	4 v
DTemplate:Music–BTemplate:Music	–	A–CTemplate:Music	3 v
BTemplate:Music–F	–	BTemplate:Music–D	2 v
F–C	–	B–DTemplate:Music	3 v

Because a quarter of the Pythagorean comma is $\sqrt[4]{\frac{531441}{524288}}$, or $\frac{27}{32}\sqrt[4]{2}$, it is possible to calculate exact mathematical values for the frequency relationships and intervals:

Note	Exact frequency ratio	Value in cents
C	$\frac{1}{1}$	0
CTemplate:Music	$\frac{256}{243}$	90.225
D	$\frac{64}{81}\sqrt{2}$	192.180
DTemplate:Music	$\frac{32}{27}$	294.135
E	$\frac{256}{243}\sqrt[4]{2}$	390.225
F	$\frac{4}{3}$	498.045
FTemplate:Music	$\frac{1024}{729}$	588.270
G	$\frac{8}{9}\sqrt[4]{2^3}$	696.090
GTemplate:Music	$\frac{128}{81}$	792.180
A	$\frac{1024}{729}\sqrt[4]{2}$	888.270
BTemplate:Music	$\frac{16}{9}$	996.090
B	$\frac{128}{81}\sqrt[4]{2}$	1092.180

Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through Template:Sfrac comma

In Werckmeister II the fifths C–G, D–A, E–B, FTemplate:Music–CTemplate:Music, and BTemplate:Music–F are tempered narrow by Template:Sfrac comma, and the fifths GTemplate:Music–DTemplate:Music and ETemplate:Music–BTemplate:Music are widened by Template:Sfrac comma. The other fifths are pure. Werckmeister designed this tuning for playing mainly diatonic music (i.e. rarely using the "black notes"). Most of its intervals are close to sixth-comma meantone. Werckmeister also gave a table of monochord lengths for this tuning, setting C=120 units, a practical approximation to the exact theoretical values. Following the monochord numbers the G and D are somewhat lower than their theoretical values but other notes are somewhat higher.

Fifth	Template:SmallTemplate:Efn	Third	Template:SmallTemplate:Efn
C–G	^	C–E	1 v
G–D	–	CTemplate:Music–F	4 v
D–A	^	D–FTemplate:Music	1 v
A–E	–	DTemplate:Music–G	2 v
E–B	^	E–GTemplate:Music	1 v
B–FTemplate:Music	–	F–A	1 v
FTemplate:Music–CTemplate:Music	^	FTemplate:Music–BTemplate:Music	4 v
CTemplate:Music–GTemplate:Music	–	G–B	1 v
GTemplate:Music–DTemplate:Music	v	GTemplate:Music–C	4 v
DTemplate:Music–BTemplate:Music	v	A–CTemplate:Music	1 v
BTemplate:Music–F	^	BTemplate:Music–D	1 v
F–C	–	B–DTemplate:Music	3 v

Note	Exact frequency ratio	Value in cents	Approximate monochord length	Value in cents
C	$\frac{1}{1}$	0	$120$	0
CTemplate:Music	$\frac{16384}{19683}\sqrt[3]{2}$	82.405	$114\frac{1}{5}$ (misprinted as $114\frac{1}{2}$)	85.766
D	$\frac{8}{9}\sqrt[3]{2}$	196.090	$107\frac{1}{5}$	195.275
DTemplate:Music	$\frac{32}{27}$	294.135	$101\frac{1}{5}$	294.990
E	$\frac{64}{81}\sqrt[3]{4}$	392.180	$95\frac{3}{5}$	393.542
F	$\frac{4}{3}$	498.045	$90$	498.045
FTemplate:Music	$\frac{1024}{729}$	588.270	$85\frac{1}{3}$	590.224
G	$\frac{32}{27}\sqrt[3]{2}$	694.135	$80\frac{2}{5}$	693.320
GTemplate:Music	$\frac{8192}{6561}\sqrt[3]{2}$	784.360	$76\frac{2}{15}$	787.721
A	$\frac{256}{243}\sqrt[3]{4}$	890.225	$71\frac{7}{10}$	891.587
BTemplate:Music	$\frac{9}{4\sqrt[3]{2}}$	1003.910	$67\frac{1}{5}$	1003.802
B	$\frac{4096}{2187}$	1086.315	$64$	1088.269

Werckmeister III (V): an additional temperament divided up through Template:Sfrac comma

In Werckmeister III the fifths D–A, A–E, FTemplate:Music–CTemplate:Music, CTemplate:Music–GTemplate:Music, and F–C are narrowed by Template:Sfrac comma, and the fifth GTemplate:Music–DTemplate:Music is widened by Template:Sfrac comma. The other fifths are pure. This temperament is closer to equal temperament than the previous two.

Fifth	Template:SmallTemplate:Efn	Third	Template:SmallTemplate:Efn
C–G	–	C–E	2 v
G–D	–	CTemplate:Music–F	4 v
D–A	^	D–FTemplate:Music	2 v
A–E	^	DTemplate:Music–G	3 v
E–B	–	E–GTemplate:Music	2 v
B–FTemplate:Music	–	F–A	2 v
FTemplate:Music–CTemplate:Music	^	FTemplate:Music–BTemplate:Music	3 v
CTemplate:Music–GTemplate:Music	^	G–B	2 v
GTemplate:Music–DTemplate:Music	v	GTemplate:Music–C	4 v
DTemplate:Music–BTemplate:Music	–	A–CTemplate:Music	2 v
BTemplate:Music–F	–	BTemplate:Music–D	3 v
F–C	^	B–DTemplate:Music	3 v

Note	Exact frequency ratio	Value in cents
C	$\frac{1}{1}$	0
CTemplate:Music	$\frac{8}{9}\sqrt[4]{2}$	96.090
D	$\frac{9}{8}$	203.910
DTemplate:Music	$\sqrt[4]{2}$	300
E	$\frac{8}{9}\sqrt{2}$	396.090
F	$\frac{9}{8}\sqrt[4]{2}$	503.910
FTemplate:Music	$\sqrt{2}$	600
G	$\frac{3}{2}$	701.955
GTemplate:Music	$\frac{128}{81}$	792.180
A	$\sqrt[4]{8}$	900
BTemplate:Music	$\frac{3}{\sqrt[4]{8}}$	1001.955
B	$\frac{4}{3}\sqrt{2}$	1098.045

Werckmeister IV (VI): the Septenarius tunings

This tuning is based on a division of the monochord length into $196=7\times 7\times 4$ parts. The various notes are then defined by which 196-division one should place the bridge on in order to produce their pitches. The resulting scale has rational frequency relationships, so it is mathematically distinct from the irrational tempered values above; however in practice, both involve pure and impure sounding fifths. Werckmeister also gave a version where the total length is divided into 147 parts, which is simply a transposition of the intervals of the 196-tuning. He described the Septenarius as "an additional temperament which has nothing at all to do with the divisions of the comma, nevertheless in practice so correct that one can be really satisfied with it".

One apparent problem with these tunings is the value given to D (or A in the transposed version): Werckmeister writes it as "176", but the value is suspect: It produces a musically bad effect because the fifth G–D would then be very flat (more than half a comma); the third BTemplate:Music–D would be pure, but D–FTemplate:Music would be more than a comma too sharp – all of which contradict the rest of Werckmeister's writings on temperament. In the illustration of the monochord division, the number "176" is written one place too far to the right, where 175 should be. Therefore it is conceivable that the number 176 is a mistake for 175, which gives a musically much more consistent result. Both values are given in the table below.

In the tuning with D=175, the fifths C–G, G–D, D–A, B–FTemplate:Music, FTemplate:Music–CTemplate:Music, and BTemplate:Music–F are tempered narrow, while the fifth GTemplate:Music–DTemplate:Music is tempered wider than pure; the other fifths are pure.

Note	Monochord length	Exact frequency ratio	Value in cents
C	196	1/1	0
CTemplate:Music	186	98/93	90.661
D	176 (175)	49/44 (28/25)	186.334 (196.198)
DTemplate:Music	165	196/165	298.065
E	156	49/39	395.169
F	147	4/3	498.045
FTemplate:Music	139	196/139	594.923
G	131	196/131	697.544
GTemplate:Music	124	49/31	792.616
A	117	196/117	893.214
BTemplate:Music	110	98/55	1000.020
B	104	49/26	1097.124

External sources

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