View source for Werckmeister temperament

{{Short description|Tuning system described by Andreas Werckmeister}}
'''Werckmeister temperaments''' are the [[Musical tuning|tuning systems]] described by [[Andreas Werckmeister]] in his writings.<ref>{{cite book |first=A. |last=Werckmeister |author-link=Andreas Werckmeister |year=1681 |title=Orgel-Probe |lang=de |place=Frankfurt & Leipzig }}<br/>
Excerpts from Werkmeister (1861) given in<br/>
{{cite book |first=Mark |last=Lindley |year=1987 |section=Stimmung und Temperatur |title=Hören, messen und rechnen in der frühen Neuzeit |editor-first=Frieder |editor-last=Zaminer |series=Geschichte der Musiktheorie |lang=de |publisher=Wissenschaftliche Buchgesellschaft |place=Darmstadt, DE |volume=6 |pages=109–331 }}</ref><ref>{{cite book |first=A. |last=Werckmeister |author-link=Andreas Werckmeister |orig-year=1686, 1867 |year=1972 |title=Musicae mathematicae hodegus curiosus oder Richtiger Musicalischer Weg-Weiser |lang=de, la |place=Quedlinburg, DE (1686) / Frankfurt & Leipzig, DE (1687) / Hildesheim, DE (1972) |publisher=G. Olms (1972) |ISBN=3-487-04080-8 }}</ref><ref>{{cite book |first=A. |last=Werckmeister |author-link=Andreas Werckmeister |year=1983 |orig-year=1691 |edition=reprint |editor-first=Rudolf |editor-last=Rasch |title=Musicalische Temperatur |lang=de |place=Quedlinburg, DE (1691) / Utrecht, NL (1983) |publisher=Diapason Press (1983) |ISBN=90-70907-02-X |LCCN=84242846 |OCLC=11534855 }} </ref> 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&nbsp;I and quarter-comma [[meantone]] is labelled&nbsp;II. The temperament commonly known as "Werckmeister&nbsp;III" is referred to in this article as {{nobr|"Werckmeister I (III)".<ref name="Britannica - Equal Temp">{{cite web |first=Colin |last=Pykett |title=A handy guide to choosing temperaments for the practical musician |at=note&nbsp;2 |website=colinpykett.org.uk |url=http://www.colinpykett.org.uk/handy-guide-to-choosing-temperaments.htm#Refs |access-date=12 November 2023}}</ref>}}

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 (music)|comma]].{{efn|
name=tuning_mark_footnote|
Werckmeister used the [[pipe organ|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.

==Werckmeister I (III): "correct temperament" based on {{sfrac|1|4}} comma divisions ==

This tuning uses mostly pure ([[Perfect fifth|perfect]]) fifths, as in [[Pythagorean tuning]], but each of the fifths C–G, G–D, D–A and B–F{{music|#}} is made smaller, i.e. [[Musical temperament|tempered]] by {{sfrac|1|4}} 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 [[quarter-comma meantone temperament|meantone temperament]] fifths. All major thirds are reasonably close to 400&nbsp;cents and, because not all fifths are tempered, there is no [[wolf fifth]] and all 12&nbsp;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.

:{| class="wikitable" style="text-align:center"
|- style="vertical-align:bottom;"
! Fifth !! {{small|Tempering<br/>mark}}{{efn|name=tuning_mark_footnote}}
! Third !! {{small|Tempering<br/>mark}}{{efn|name=tuning_mark_footnote}}
|-
| C–G ||^ || C–E || 1 v 
|-
| G–D || ^ || C{{music|#}}–F || 4 v 
|-
| D–A || ^ || D–F{{music|#}} || 2 v 
|-
| A–E || – || D{{music|#}}–G || 3 v 
|-
| E–B || – || E–G{{music|#}} || 3 v 
|-
| B–F{{music|#}} || ^ ||F–A || 1 v 
|-
| F{{music|#}}–C{{music|#}} || – || F{{music|#}}–B{{music|b}} || 4 v 
|-
| C{{music|#}}–G{{music|#}} || – || G–B || 2 v 
|-
| G{{music|#}}–D{{music|#}} || – || G{{music|#}}–C || 4 v 
|-
| D{{music|#}}–B{{music|b}} || – || A–C{{music|#}} || 3 v 
|-
| B{{music|b}}–F || – || B{{music|b}}–D || 2 v 
|-
| F–C || – ||B–D{{music|#}} || 3 v 
|}

{{Audio|Werckmeister temperament major chord on C.mid|Play major tonic chord}}

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

:{| class="wikitable"
! Note
! Exact frequency ratio
! Value in [[cent (music)|cents]]
|-
| C ||<math>\frac{1}{1}</math> || 0
|-
| C{{music|#}} ||<math>\frac{256}{243}</math> || 90
|-
| D ||<math>\frac{64}{81} \sqrt{2}</math> || 192
|-
| D{{music|#}} ||<math>\frac{32}{27}</math> || 294
|-
| E ||<math>\frac{256}{243} \sqrt[4]{2}</math> || 390
|-
| F ||<math>\frac{4}{3}</math> || 498
|-
| F{{music|#}} ||<math>\frac{1024}{729}</math> || 588
|-
| G ||<math>\frac{8}{9}\sqrt[4]{2^3}</math> || 696
|-
| G{{music|#}} ||<math>\frac{128}{81}</math> || 792
|-
| A ||<math>\frac{1024}{729} \sqrt[4]{2}</math> || 888
|-
| B{{music|b}} ||<math>\frac{16}{9}</math> || 996
|-
| B ||<math>\frac{128}{81} \sqrt[4]{2}</math> || 1092
|}

==Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through {{sfrac|1|3}} comma ==

In '''Werckmeister II''' the fifths C–G, D–A, E–B, F{{music|#}}–C{{music|#}}, and B{{music|b}}–F are tempered narrow by {{sfrac|1|3}} comma, and the fifths G{{music|#}}–D{{music|#}} and E{{music|b}}–B{{music|b}} are widened by {{sfrac|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{{citation needed|date=November 2016}}. Following the monochord numbers the G and D are somewhat lower than their theoretical values but other notes are somewhat higher.

:{| class="wikitable" style="text-align:center" 
|- style="vertical-align:bottom;"
! Fifth !! {{small|Tempering<br/>mark}}{{efn|name=tuning_mark_footnote}}
! Third !! {{small|Tempering<br/>mark}}{{efn|name=tuning_mark_footnote}}
|-
| C–G || ^ || C–E || 1 v 
|-
| G–D || – || C{{music|#}}–F || 4 v 
|-
| D–A || ^ || D–F{{music|#}} || 1 v 
|-
| A–E || - ||D{{music|#}}–G || 2 v 
|-
| E–B || ^ ||E–G{{music|#}} || 1 v 
|-
| B–F{{music|#}} || – || F–A || 1 v 
|-
| F{{music|#}}–C{{music|#}} || ^ || F{{music|#}}–B{{music|b}} || 4 v 
|-
| C{{music|#}}–G{{music|#}} || – || G–B || 1 v 
|-
| G{{music|#}}–D{{music|#}} || v || G{{music|#}}–C || 4 v 
|-
| D{{music|#}}–B{{music|b}} || v || A–C{{music|#}} || 1 v 
|-
| B{{music|b}}–F || ^ ||B{{music|b}}–D || 1 v 
|-
| F–C || – ||B–D{{music|#}} || 3 v 
|}

:{| class="wikitable" style="text-align:center" 
! Note
! Exact frequency ratio
! Value in cents
! Approximate monochord length
! Value in cents
|-
| C ||<math>\frac{1}{1}</math> ||0 ||<math>120</math> ||0 
|-
| C{{music|#}} ||<math>\frac{16384}{19683} \sqrt[3]{2}</math> || 82 ||<math>114\frac{1}{5}</math> (misprinted as <math>114\frac{1}{2}</math>) || 85.8 
|- 
| D ||<math>\frac{8}{9} \sqrt[3]{2}</math> || 196 ||<math>107\frac{1}{5}</math> || 195.3 
|-
| D{{music|#}} ||<math>\frac{32}{27}</math> || 294 ||<math>101\frac{1}{5}</math>  || 295.0 
|-
| E ||<math>\frac{64}{81} \sqrt[3]{4}</math> || 392 ||<math>95\frac{3}{5}</math> || 393.5 
|-
| F ||<math>\frac{4}{3}</math> || 498 ||<math>90</math> || 498.0 
|-
| F{{music|#}} ||<math>\frac{1024}{729}</math> ||588 ||<math>85\frac{1}{3}</math> || 590.2 
|-
| G ||<math>\frac{32}{27} \sqrt[3]{2}</math> ||694 ||<math>80\frac{1}{5}</math> || 693.3 
|-
| G{{music|#}} ||<math>\frac{8192}{6561} \sqrt[3]{2}</math> || 784 ||<math>76\frac{2}{15}</math> || 787.7 
|-
| A ||<math>\frac{256}{243} \sqrt[3]{4}</math> ||890 ||<math>71\frac{7}{10}</math> || 891.6 
|-
| B{{music|b}} ||<math>\frac{9}{4 \sqrt[3]{2}}</math> || 1004 ||<math>67\frac{1}{5}</math> || 1003.8 
|-
| B ||<math>\frac{4096}{2187}</math> || 1086 ||<math>64</math> || 1088.3 
|}

==Werckmeister III (V): an additional temperament divided up through {{sfrac|1|4}} comma ==

In '''Werckmeister III''' the fifths D–A, A–E, F{{music|#}}–C{{music|#}}, C{{music|#}}–G{{music|#}}, and F–C are narrowed by {{sfrac|1|4}} comma, and the fifth G{{music|#}}–D{{music|#}} is widened by {{sfrac|1|4}} comma. The other fifths are pure. This temperament is closer to [[equal temperament]] than the previous two.

:{| class="wikitable" style="text-align:center" 
|- style="vertical-align:bottom;"
! Fifth !! {{small|Tempering<br/>mark}}{{efn|name=tuning_mark_footnote}}
! Third !! {{small|Tempering<br/>mark}}{{efn|name=tuning_mark_footnote}}
|-
| C–G || – || C–E || 2 v 
|-
| G–D || – || C{{music|#}}–F || 4 v 
|-
| D–A || ^ || D–F{{music|#}} || 2 v 
|-
| A–E || ^ || D{{music|#}}–G || 3 v 
|-
| E–B || – || E–G{{music|#}} || 2 v 
|-
| B–F{{music|#}} || – || F–A || 2 v 
|-
| F{{music|#}}–C{{music|#}} || ^ || F{{music|#}}–B{{music|b}} || 3 v 
|-
| C{{music|#}}–G{{music|#}} || ^ || G–B || 2 v 
|-
| G{{music|#}}–D{{music|#}} || v || G{{music|#}}–C || 4 v 
|-
| D{{music|#}}–B{{music|b}} || – || A–C{{music|#}} || 2 v 
|-
| B{{music|b}}–F || – || B{{music|b}}–D || 3 v 
|-
| F–C || ^ || B–D{{music|#}} || 3 v 
|}

:{| class="wikitable" style="text-align:center" 
! Note
! Exact frequency ratio
! Value in cents
|-
| C ||<math>\frac{1}{1}</math> || 0
|-
| C{{music|#}} ||<math>\frac{8}{9} \sqrt[4]{2}</math> || 96
|-
| D ||<math>\frac{9}{8}</math> || 204
|-
| D{{music|#}} ||<math>\sqrt[4]{2}</math> || 300
|-
| E ||<math>\frac{8}{9} \sqrt{2}</math> || 396
|-
| F ||<math>\frac{9}{8} \sqrt[4]{2}</math> || 504
|-
| F{{music|#}} ||<math>\sqrt{2}</math> || 600
|-
| G ||<math>\frac{3}{2}</math> || 702
|-
| G{{music|#}} ||<math>\frac{128}{81}</math> || 792
|-
| A ||<math>\sqrt[4]{8}</math> ||900
|-
| B{{music|b}} ||<math>\frac{3}{\sqrt[4]{8}}</math> || 1002
|-
| B ||<math>\frac{4}{3} \sqrt{2}</math> || 1098
|}

==Werckmeister IV (VI): the Septenarius tunings ==

This tuning is based on a division of the [[monochord]] length into <math>196 = 7\times 7\times 4</math> 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 number|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&nbsp;parts, which is simply a [[Transposition (music)|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 B{{music|b}}–D would be pure, but D–F{{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–F{{music|#}}, F{{music|#}}–C{{music|#}}, and B{{music|b}}–F are tempered narrow, while the fifth G{{music|#}}–D{{music|#}} is tempered wider than pure; the other fifths are pure.

:{| class="wikitable" style="text-align:center" 
! Note
! Monochord length
! Exact frequency ratio 
! Value in [[cent (music)|cents]]
|-
| C || 196 || 1/1 || 0
|-
| C{{music|#}}|| 186 || 98/93 || 91
|-
| D || 176(175) || 49/44(28/25) || 186(196)
|-
| D{{music|#}}|| 165 || 196/165 || 298
|-
| E || 156 || 49/39 || 395
|-
| F || 147 || 4/3 || 498
|-
| F{{music|#}}|| 139 || 196/139 || 595
|-
| G || 131 || 196/131 || 698
|-
| G{{music|#}}|| 124 || 49/31 || 793
|-
| A || 117 || 196/117 || 893
|-
| B{{music|b}}|| 110 || 98/55 || 1000
|-
| B || 104 || 49/26 || 1097
|}

== Footnotes ==
{{notelist}}

== References ==
{{reflist|25em}}

== External sources ==
* {{cite web |title=196-EDL, 1568-EDL, and Septenarius tunings |website=240edo.googlepages.com |type=blog |series=Equal divisions of length (EDL) |url=http://240edo.googlepages.com/equaldivisionsoflength(edl) }}
* {{cite web |first=Broekaert |last=de Vriendt |title=Well tempering based on the Werckmeister definition |website=users.telenet.be/broekaert-devriendt |type=blog |url=http://users.telenet.be/broekaert-devriendt/Index.html }}
* {{cite web |title=Well tempered based on Werckmeisters last book ''Musikalische Paradoxal-Discourse'' (1707) is equal temperament |department=18th&nbsp;century quotes on J.S.&nbsp;Bach's temperament |website=academia.edu |url=https://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament }}

{{musical tuning}}

[[Category:Musical temperaments]]