Rotation of axes

2014-01-18T02:56:44Z

98.25.253.140: /* Derivation of the rotation formula */

'''Werckmeister temperaments''' are the [[Musical tuning|tuning systems]] described by [[Andreas Werckmeister]] in his writings.<ref>Andreas Werckmeister: ''Orgel-Probe'' (Frankfurt & Leipzig 1681), excerpts in Mark Lindley, "Stimmung und Temperatur", in ''Hören, messen und rechnen in der frühen Neuzeit'' pp. 109-331, Frieder Zaminer (ed.), vol. 6 of ''Geschichte der Musiktheorie'', Wissenschaftliche Buchgesellschaft (Darmstadt 1987).</ref><ref>A. Werckmeister: Musicae mathematicae hodegus curiosus oder Richtiger Musicalischer Weg-Weiser (Quedlinburg 1686, Frankfurt & Leipzig 1687) ISBN 3-487-04080-8</ref><ref>A. Werckmeister: Musicalische Temperatur (Quedlinburg 1691), reprint edited by Rudolf Rasch ISBN 90-70907-02-X</ref> 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 (music)|comma]]. 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 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 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.

{| class="wikitable" style="text-align:center"
|Fifth ||Tempering ||Third ||Tempering
|-
|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}}

Modern authors have calculated exact mathematical values for the frequency relationships and intervals using the Pythagorean comma:

{| border = "1" cellspacing="0"
!Note
!Exact frequency relation
!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]{8}</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 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 1/3 comma, and the fifths G{{music|#}}-D{{music|#}} and E{{music|b}}-B{{music|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.

{| class="wikitable" style="text-align:center"
|Fifth ||Tempering ||Third ||Tempering
|-
|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
|}

{| border="1" cellspacing="0" cellpadding="1"
!Note
!Exact frequency relation
!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 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 1/4, and the fifth G{{music|#}}-D{{music|#}} is widened by 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"
|Fifth ||Tempering ||Third ||Tempering
|-
|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
|}

{| border="1" cellspacing="0" cellpadding="1"
!Note
!Exact frequency relation
!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 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. 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{{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 relation
!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
|}

== External sources ==
*[http://240edo.googlepages.com/equaldivisionsoflength(edl) 196-EDL & 1568-EDL and Septenarius tunings]

== References ==
<references/>

{{musical tuning}}

[[Category:Musical temperaments]]

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