# Werckmeister well-temperaments

[Joe Monzo, with Johnny Reinhard]

Andreas Werckmeister proposed a number of temperaments which fall into the category now known as "well-temperaments", also called "circulating" or "irregular" temperaments.

In contrast to the various meantone tunings, which are based on tempering by fractions of the syntonic comma, Werckmeister's well-temperaments are all based on tempering by fractions of the pythagorean comma.

These tunings are closed 12-tone systems, intended primarily for use on keyboards; thus, all pitches mapped to the black-keys can be taken as either sharps or flats.

The first two tunings which were described by Werckmeister were not considered "good": just-intonation ("Werckmeister I"), which he considered "too perfect", and an extended form of 1/4-comma meantone with more than 12 notes ("Werckmeister II"), where the keyboard had split keys, and which he considered "incorrect". He is known today primarily for his "III" temperament, which is analyzed here in detail. (More information will follow in the future about his other temperaments.)

### Werckmeister III: "Correct Temperament No. 1" [-Barbour]

In octave-equivalent terms, here's how Werckmeister III works:

The 4 "5ths" between C:G:D:A and B:F# are tuned 1/4 of a pythagorean comma narrow, and all the rest of the "5ths" are tuned to the Pythagorean 3:2 ratio.

```Werckmeister III narrow "5th" = (3/2) / ( ((2^-19)*(3^12))^(1/4) )

in monzo notation:

2^x *  3^y                                    ~cents

[-1      1]    3:2 ratio = "perfect 5th"    701.9550009
- [-19/4   3]    1/4 Pythagorean-comma      -   5.865002596
---------------                               ---------------
[ 15/4  -2]    Werckmeister #1 narrow 5th   696.0899983
```

Below is a table showing Werckmeister III tuning as a chain of "5ths".

```                                     --"5th" size--
note   comma   --vector--    cents    comma   cents
mistuning  2       3           mistuning

G#    -1      [11      -4]   792.180
>  0    701.955
C#    -1      [12      -5]    90.225
>  0    701.955
F#    -1      [13      -6]   588.270
> -1/4  696.090
B     -3/4    [ 9&1/4  -4]  1092.180
>  0    701.955
E     -3/4    [10&1/4  -5]   390.225
>  0    701.955
A     -3/4    [11&1/4  -6]   888.270
> -1/4  696.090
D     -1/2    [ 7&1/2  -4]   192.180
> -1/4  696.090
G     -1/4    [ 3&3/4  -2]   696.090
> -1/4  696.090
C       0     [ 0       0]     0.000
>  0    701.955
F       0     [ 1      -1]   498.045
>  0    701.955
Bb      0     [ 2      -2]   996.090
>  0    701.955
Eb      0     [ 3      -3]   294.135
>  0    701.955
G#     -1     [ 4      -4]   792.180
```

(Note: in this table, fractional exponents of 2 must be given for the Werckmeister 5ths, but since the scale was normally tuned first in one reference octave and then other 8ves tuned from that, any of the exponents of 2 can be adjusted +/- any integer value without affecting the properties inherent in the tuning; thus, the G# [11 -4] tuned a narrow 5th above C# is essentially the same as the G# [4 -4] tuned a 3:2 below Eb -- the exponent of 3 is -4 in both cases, and the 7-octave difference of 2(11-4) = 27 is irrelevant.)

The Werckmeister 5ths are those between C:G:D:A and B:F#/Gb. The 5ths between Gb/F#:Db/C#:Ab/G#:Eb/D#:Bb/A#:F:C and A:E:B are 3:2 ratios.

612-edo gives a superb approximation of Werckmeister III, the maximum deviation being only ~1/29-cent. Below is a graph of Werckmeister III tuning as a scale within one octave, given as 612-edo degrees. The major divisions on the y-axis quantize it into the 612-edo representations of 12-edo for comparison. Note that 612-edo divides exactly into 12, so it provides an excellent means of comparison between 12-edo and Werckmeister III without the need for decimal or fractional parts.

```      ~12edo    ~612edo skhismas
note  degrees   correction	~cents error of 612edo from Werckmeister

B      11       -4		-1/43
Bb     10       -2		-1/86
A       9       -6		-1/29
Ab/G#   8       -4		-1/43
G       7       -2		-1/86
F#      6       -6		-1/29
F       5       -1	 	-1/173
E       4       -5		-1/35
Eb      3       -3		-1/58
D       2       -4		-1/43
C#      1       -5		-1/35
C       0        0		 0
```

200-edo also gives an excellent approximation of Werckmeister III, the maximum deviation being only ~1/4-cent. Below is a graph of Werckmeister III tuning as a scale within one octave, given as 200-edo degrees. The major divisions on the y-axis quantize it into the 200-edo representations of 12-edo for comparison; note that 200-edo does not divide evenly into 12.

```   ~12edo  ~200edo kleismas
degrees    correction     ~cents error

B    11       -1+(1/3)         -1/6
Bb   10          -2/3          -1/11
A     9       -2               -1/4
G#    8       -1+(1/3)         -1/6
G     7          -2/3          -1/11
F#    6       -2               -1/4
F     5          -2/7          -1/22
E     4       -1+(3/5)         -1/4
Eb    3       -1               -1/7
D     2       -1+(1/3)         -1/6
C#    1       -1+(3/5)         -1/4
C     0        0                0
```

Below is a list of all intervals available in Werckmeister III:

``` ~cents	~612edo 	instances
skhismas

diminished 8ve / major 7th
1109.775	566	F#:F, C#:C
1103.910	563	B:Bb, A:G#, E:Eb, G#:G
1098.045	560	D:C#, Eb:D
1092.180	557	G:F#, F:E, Bb:A, C:B

minor 7th / augmented 6th
1007.820	514	D:C, A:G
1001.955	511	G:F, F#:E, E:D, C#:B
996.090	508	B:A, Eb:C#, Bb:G#, F:Eb, G#:F#, C:Bb

diminished 7th / major 6th
905.865	462	Eb:C, F#:Eb, G#:F, C#:Bb
900.000	459	B:G#, E:C#, A:F#, Bb:G, D:B
894.135	456	G:E, F:D
888.270	453	C:A

minor 6th / augmented 5th
809.775	413	A:F, E:C
803.910	410	B:G, D:Bb, F#:D
798.045	407	G:Eb, G#:E, C#:A, Eb:B
792.180	404	F:C#, Bb:F#, C:G#

"perfect" 5th
701.955	358	F#:C#, A:E, F:C, Bb:F, G#:Eb, C#:G#, Eb:Bb, E:B
696.090	355	B:F#, D:A, C:G, G:D

diminished 5th / augmented 4th ("tritone")
611.730	312	F#:C
605.865	309	B:F, A:Eb, E:Bb, C#:G
600.000	306	D:G#, G#:D
594.135	303	G:C#, Bb:E, Eb:A, F:B
588.270	300	C:F#

"perfect" 4th
503.910	257	D:G, G:C, A:D, F#:B
498.045	254	B:E, G#:C#, Bb:Eb, Eb:G#, C:F, F:Bb, C#:F#, E:A

diminished 4th / major 3rd
407.820	208	G#:C, F#:Bb, C#:F
401.955	205	B:Eb, A:C#, E:G#, Eb:G
396.090	202	D:F#, Bb:D, G:B
390.225	199	C:E, F:A

minor 3rd / augmented 2nd
311.730	159	A:C
305.865	156	D:F, E:G
300.000	153	B:D, C#:E, F#:A, G:Bb, G#:B
294.135	150	Bb:C#, F:G#, C:Eb, Eb:F#

major 2nd / diminished 3rd
203.910	104	Bb:C, F#:G#, Eb:F, C#:Eb, G#:Bb, A:B
198.045	101	B:C#, D:E, E:F#, F:G
192.180	 98	G:A, C:D

minor 2nd / augmented prime
107.820	55	B:C, A:Bb, E:F, F#:G
101.955	52	D:Eb, C#:D
96.090	49	G:G#, Eb:E, G#:A, Bb:B
90.225	46	C:C#, F:F#

unison
0.000	 0	C:C, C#:C#, D:D, Eb:Eb, E:E, F:F, F#:F#, G:G, G#:G#, A:A, Bb:Bb, B:B
```
```ANALYSIS OF WERCKMEISTER III TRIADS
cents values given for the major and minor 3rds

MAJOR
-----

with Pythagorean 3:2 ratio "perfect 5th", ~701.955 cents:

Ab, F#/Gb, and C#/Db-major are entirely Pythagorean:

~cents	~612edo skhismas

Eb	C#	G#
294.135		150
C	Bb	F
407.820		208
G#	F#	C#

A, E, and Eb-major are quite similar to <12-eq>12edo,
with the major-3rd and perfect-5th ~2 cents wider:

E	B	Bb
300.000		153
C#	G#	G
401.955		205
A	E	Eb

Bb-major has a major-3rd ~4 cents smaller than 12edo
and a minor-3rd intermediate between 12edo and meantone:

F
305.865		156
D
396.090		202
Bb

F-major has a major-3rd ~4 cents wider than JI/meantone, and
a minor-3rd ~1.5 cents wider than meantone:

C
311.730		159
A
390.225		199
F

with tempered "5th" 1/4-comma narrow, ~696.090 cents:

B-major has the pseudo-12edo major-3rd on the bottom and the
Pythagorean minor 3rd on top:

F#
294.135		150
Eb
401.955		205
B

D and G-major have a major-3rd ~4 cents smaller than 12edo
and a minor 3rd which is exactly the same as 12edo:

A	D
300.000		153
F#	B
396.090		202
D	G

C-major has a major-3rd a few cents wider than JI/meantone and
a minor-3rd intermediate between and 12edo and meantone:

G
305.865		156
E
390.225		199
C

MINOR
-----

with Pythagorean 3:2 ratio "perfect 5th", ~701.955 cents:

F, Eb, and Bb-minor are entirely Pythagorean:

~cents	~612edo skhismas

C	Bb	F
407.820		208
G#	F#	C#
294.135		150
F	Eb	Bb

G#/Ab, F#, and C#-minor are very similar to 12edo,
with the major-3rd and perfect-5th ~2 cents wider:

Eb	C#	G#
401.955		205
B	A	E
300.000		153
G#	F#	C#

E-minor has a minor-3rd intermediate between 12edo and meantone
and a major-3rd ~4 cents smaller than 12edo:

B
396.090		202
G
305.865		156
E

A-minor has a minor-3rd ~1.5 cents wider than meantone,
and a major-3rd ~4 cents wider than JI/meantone:

E
390.225		199
C
311.730		159
A

with tempered "5th" 1/4-comma narrow, ~696.090 cents:

C-minor has the Pythagorean minor 3rd on the bottom
and the pseudo-12edo major-3rd on top:

G
401.955		205
Eb
294.135		150
C

B-minor and G-minor have a minor 3rd which is exactly the same
as 12edo, and a major-3rd ~4 cents smaller than 12edo:

F#	D
396.090		202
D	Bb
300.000		153
B	G

D-minor has a minor-3rd intermediate between and 12edo and meantone,
and a major-3rd a few cents wider than JI/meantone:

A
390.225		199
F
305.865		156
D
```
. . . . . . . . .

Here is a graph comparing Werckmeister III with 1/4-comma meantone tuning:

. . . . . . . . .

### Werckmeister IV: "Correct Temperament No. 2" [-Barbour]

The basic idea of this temperament is that every other "5th" is tuned 1/3 of a pythagorean comma narrow and the ones between them are tuned to the Pythagorean 3:2 ratio.

```Werckmeister "5th" = (3/2) / ( ((2^-19)*(3^12))^(1/4) )

in monzo notation:

2^x *  3^y                                    ~cents

[-1      1]    3:2 ratio = "perfect 5th"    701.9550009
- [-19/4   4]    1/3 Pythagorean-comma      -   5.865002596
---------------                             ---------------
[ 16/3  -3]    Werckmeister #2 narrow 5th   694.1349974

2^x *  3^y                                    ~cents

[-1      1]    3:2 ratio = "perfect 5th"    701.9550009
+ [-19/4   4]    1/3 Pythagorean-comma      -   5.865002596
---------------                             ---------------
[-22/3   5]    Werckmeister #2 wide 5th     709.7750043

--"5th" size--
note   comma   --vector--  cents    comma   cents
mistuning  2       3         mistuning

G#   -1&1/3   [17&1/3  -8]  784
>   0      702
C#   -1&1/3   [18&1/3  -9]   82
>  -1/3    694
F#   -1       [13      -6]  588
>   0      702
B    -1       [14      -7] 1086
>  -1/3    694
E    -2/3     [ 8&2/3  -4]  392
>   0      702
A    -2/3     [ 9&2/3  -5]  890
>  -1/3    694
D    -1/3     [ 4&1/3  -2]  196
>   0      702
G    -1/3     [ 5&1/3  -3]  694
>  -1/3    694
C     0       [ 0       0]    0
>   0      702
F     0       [ 2      -1]  498
>  -1/3    694
Bb   +1/3     [-3&1/3   2] 1004
>  +1/3    710
Eb    0       [ 4      -3]  294
>  +1/3    710
G#   -1&1/3   [11&1/3  -8]  784
```

Again, as in the Correct Temperament No. 1 above, the exponents of 2 for G# only signify 8ve-register, and have no effect on 8ve-invariant aspects of the tuning, signified by 3-8. Note also that Barbour's Table 136 (on p 160) has an error: the tuning of G# is given as 786 cents, but should be 784; the table above agrees with Barbour's description of this tuning as "containing 5 5ths flat by 1/3 comma, 2 5ths sharp by 1/3 comma, and 5 perfect 5ths".

### Werckmeister VI: "Correct Temperament No. 4" [-Barbour], the "Septenarius"

The last tuning described by Werckmeister is known as the "Septenarius", because it is based on a string-length of 196 = 22 * 72. As Werckmeister himself wrote, this tuning is different from the three other "good" tunings because it is not based on a tempering by a division of the comma -- instead it is actually an example of an RI (rational-intonation) which is not a JI (just-intonation). In effect it is a well-temperament similar to the others, but created in a different way.

```note  string            ------------ monzo ------------   ~cents
name  length  ratio     2  3,  5  7 11, 13, 31,131,139

C      98     2/1    [ 1  0,  0  0  0,  0,  0,  0,  0>  1200.000
H     104    49/26   [-1  0,  0  2  0, -1,  0,  0,  0>  1097.124
B     110    98/55   [ 1  0, -1  2 -1,  0,  0,  0,  0>  1000.020
A     117   196/117  [ 2 -2,  0  2  0, -1,  0,  0,  0>   893.214
G#    124    49/31   [ 0  0,  0  2  0,  0, -1,  0,  0>   792.616
G     131   196/131  [ 2  0,  0  2  0,  0,  0, -1,  0>   697.544
F#    139   196/139  [ 2  0,  0  2  0,  0,  0,  0, -1>   594.923
F     147     4/3    [ 2  1,  0  0  0,  0,  0,  0,  0>   498.045
E     156    49/39   [ 0 -1,  0  2  0, -1,  0,  0,  0>   395.169
D#    165   196/165  [ 2 -1, -1  2 -1,  0,  0,  0,  0>   298.065
D     175    28/25   [ 2  0, -2  1  0,  0,  0,  0,  0>   196.198
C#    186    98/93   [ 1 -1,  0  2  0,  0, -1,  0,  0>    90.661
C     196     1/1    [ 0  0,  0  0  0,  0,  0,  0,  0>     0.000

NOTE: "D" was listed by Werckmeister as 176, but
this is apparently an error and it should be 175.
```

see Tom Dent's webpage The Septenarius: Werckmeister's mythical tuning, made reality?, and Yahoo tuning message 60610

###### REFERENCES

Werckmeister, Andreas. 1691.

Musicalische Temperatur, Oder deutlicher und warer Mathematischer Unterricht
Wie man durch Anweisung des Monochordi ein Clavier
sonderlich die Orgelwerke
Positive, Regale, Spinetten
und dergleichen wol temperirt stimen könne
damit nach heutiger manier alle Modificti in einer angenehm- und erträglichen Harmonia mögen genommen werden
Mit vorhergehender Abhandlung ... der Musicalischen Zahlen
... Welche bey Einrichtung der Temperaturen wohl in acht zu nehmen sind.

Reprint R.A. Rasch (ed.), Diapason Press, Utrecht, 1983.
Other reprint Rüdiger Pfeiffer (ed.), Die Blaue Eule, Essen, 1996, 138 pages.

Barbour, James Murray. 1951.

Tuning and Temperament: A Historical Survey.
Michigan State College Press, East Lansing.

Reprint Da Capo Press, New York, 1973, 228 pages.

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

 support level donor: \$5 USD friend: \$25 USD patron: \$50 USD savior: \$100 USD angel of tuning: \$500 USD microtonal god: \$1000 USD