Werckmeister temperament
Encyclopedia
Werckmeister temperaments are the tuning systems
described by Andreas Werckmeister
in his writings . The tuning systems are confusingly 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 tunings I (III), II (IV) and III (V) were presented graphically by a cycle of fifths and a list of major third
s, giving the temperament of each in fractions of a comma
. Werckmeister used the organbuilder
's notation of ^ for a downwards tempered or narrowed interval and v for an upward tempered or widened one. (This appears counterintuitive - it is based on the use of a conical tuning tool which would reshape the ends of the pipes.) A pure fifth is simply a dash. Werckmeister was not explicit about whether the syntonic comma
or Pythagorean comma
was meant: the difference between them, the so-called schisma
, is almost inaudible and he stated that it could be divided up among the fifths.
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.
) fifths, as in Pythagorean tuning
, but each of the fifths C-G, G-D, D-A and B-F is made smaller, i.e. tempered
by 1/4 comma. 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.
Modern authors have calculated exact mathematical values for the frequency relationships and intervals:
than the previous two.
length into
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. However this produces a musically bad effect because the fifth G-D would then be very flat (more than half a comma); the third B-D would be pure, but D-F 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-F, F-C, and B-F are tempered narrow, while the fifth G-D is tempered wider than pure; the other fifths are pure.
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
described by Andreas Werckmeister
Andreas Werckmeister
Andreas Werckmeister was an organist, music theorist, and composer of the Baroque era.-Life:Born in Benneckenstein, Germany, Werckmeister attended schools in Nordhausen and Quedlinburg. He received his musical training from his uncles Heinrich Christian Werckmeister and Heinrich Victor Werckmeister...
in his writings . The tuning systems are confusingly 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
Monochord
A monochord is an ancient musical and scientific laboratory instrument. The word "monochord" comes from the Greek and means literally "one string." A misconception of the term lies within its name. Often a monochord has more than one string, most of the time two, one open string and a second string...
. The monochord labels start from III since just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
is labelled I and quarter-comma meantone is labelled II.
The tunings I (III), II (IV) and III (V) were presented graphically by a cycle of fifths and a list of major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
s, giving the temperament of each in fractions of a comma
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
. Werckmeister used the organbuilder
Pipe organ
The pipe organ is a musical instrument that produces sound by driving pressurized air through pipes selected via a keyboard. Because each organ pipe produces a single pitch, the pipes are provided in sets called ranks, each of which has a common timbre and volume throughout the keyboard compass...
's notation of ^ for a downwards tempered or narrowed interval and v for an upward tempered or widened one. (This appears counterintuitive - it is based on the use of a conical tuning tool which would reshape the ends of the pipes.) A pure fifth is simply a dash. Werckmeister was not explicit about whether the syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
or Pythagorean comma
Pythagorean comma
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
was meant: the difference between them, the so-called schisma
Schisma
In music, the schisma is the ratio between a Pythagorean comma and a syntonic comma and equals 32805:32768, which is 1.9537 cents...
, is almost inaudible and he stated that it could be divided up among the fifths.
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 1/4 comma divisions
This tuning uses mostly pure (perfectPerfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
) fifths, as in Pythagorean tuning
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, but each of the fifths C-G, G-D, D-A and B-F is made smaller, i.e. tempered
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
by 1/4 comma. 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 | Tempering | Third | Tempering |
| C-G | ^ | C-E | 1 v |
| G-D | ^ | C-F | 4 v |
| D-A | ^ | D-F | 2 v |
| A-E | - | D-G | 3 v |
| E-B | - | E-G | 3 v |
| B-F | ^ | F-A | 1 v |
| F-C | - | F-B | 4 v |
| C-G | - | G-B | 2 v |
| G-D | - | G-C | 4 v |
| D-B | - | A-C | 3 v |
| B-F | - | B-D | 2 v |
| F-C | - | B-D | 3 v |
Modern authors have calculated exact mathematical values for the frequency relationships and intervals:
| Note | Exact frequency relation | Value in cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
|---|---|---|
| C |  |
0 |
| C |  |
90 |
| D |  |
192 |
| D |  |
294 |
| E |  |
390 |
| F |  |
498 |
| F |  |
588 |
| G |  |
696 |
| G |  |
792 |
| A |  |
888 |
| B |  |
996 |
| B |  |
1092 |
Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through 1/3 comma
In Werckmeister II the fifths C-G, D-A, E-B, F-C, and B-F are tempered narrow by 1/3 comma, and the fifths G-D and E-B are widened by 1/3 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 | Tempering | Third | Tempering |
| C-G | ^ | C-E | 1 v |
| G-D | - | C-F | 4 v |
| D-A | ^ | D-F | 1 v |
| A-E | - | D-G | 2 v |
| E-B | ^ | E-G | 1 v |
| B-F | - | F-A | 1 v |
| F-C | ^ | F-B | 4 v |
| C-G | - | G-B | 1 v |
| G-D | v | G-C | 4 v |
| D-B | v | A-C | 1 v |
| B-F | ^ | B-D | 1 v |
| F-C | - | B-D | 3 v |
| Note | Exact frequency relation | Value in cents | Approximate monochord length | Value in cents | |
|---|---|---|---|---|---|
| C |  |
0 |  |
0 | |
| C |  |
82 |  - (misprinted as  ) |
85.8 | |
| D |  |
196 |  |
195.3 | |
| D |  |
294 |  |
295.0 | |
| E |  |
392 |  |
393.5 | |
| F |  |
498 |  |
498.0 | |
| F |  |
588 |  |
590.2 | |
| G |  |
694 |  |
693.3 | |
| G |  |
784 |  |
787.7 | |
| A |  |
890 |  |
891.6 | |
| B |  |
1004 |  |
1003.8 | |
| B |  |
1086 |  |
1088.3 |
Werckmeister III (V): an additional temperament divided up through 1/4 comma
In Werckmeister III the fifths D-A, A-E, F-C, C-G, and F-C are narrowed by 1/4, and the fifth G-D is widened by 1/4 comma. The other fifths are pure. This temperament is closer to equal temperamentEqual temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
than the previous two.
| Fifth | Tempering | Third | Tempering |
| C-G | - | C-E | 2 v |
| G-D | - | C-F | 4 v |
| D-A | ^ | D-F | 2 v |
| A-E | ^ | D-G | 3 v |
| E-B | - | E-G | 2 v |
| B-F | - | F-A | 2 v |
| F-C | ^ | F-B | 3 v |
| C-G | ^ | G-B | 2 v |
| G-D | v | G-C | 4 v |
| D-B | - | A-C | 2 v |
| B-F | - | B-D | 3 v |
| F-C | ^ | B-D | 3 v |
| Note | Exact frequency relation | Value in cents |
|---|---|---|
| C |  |
0 |
| C |  |
96 |
| D |  |
204 |
| D |  |
300 |
| E |  |
396 |
| F |  |
504 |
| F |  |
600 |
| G |  |
702 |
| G |  |
792 |
| A |  |
900 |
| B |  |
1002 |
| B |  |
1098 |
Werckmeister IV (VI): the Septenarius tunings
This tuning is based on a division of the monochordMonochord
A monochord is an ancient musical and scientific laboratory instrument. The word "monochord" comes from the Greek and means literally "one string." A misconception of the term lies within its name. Often a monochord has more than one string, most of the time two, one open string and a second string...
length into
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 rationalRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
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
Transposition
Transposition may refer to:Mathematics* Transposition , a permutation which exchanges two elements and keeps all others fixed* Transposition, producing the transpose of a matrix AT, which is computed by swapping columns for rows in the matrix AGames* Transposition , different moves or a different...
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. However this produces a musically bad effect because the fifth G-D would then be very flat (more than half a comma); the third B-D would be pure, but D-F 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-F, F-C, and B-F are tempered narrow, while the fifth G-D is tempered wider than pure; the other fifths are pure.
| Note | Monochord length | Exact frequency relation | Value in cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
|---|---|---|---|
| C | 196 | 1/1 | 0 |
| C | 186 | 98/93 | 91 |
| D | 176(175) | 49/44(28/25) | 186(196) |
| D | 165 | 196/165 | 298 |
| E | 156 | 49/39 | 395 |
| F | 147 | 4/3 | 498 |
| F | 139 | 196/139 | 595 |
| G | 131 | 196/131 | 698 |
| G | 124 | 49/31 | 793 |
| A | 117 | 196/117 | 893 |
| B | 110 | 98/55 | 1000 |
| B | 104 | 49/26 | 1097 |