Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
mNo edit summary
mNo edit summary
Line 1: Line 1:
{{context|date=February 2012}}
'''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.
'''Pointwise mutual information''' ('''PMI''')<ref name="Church1990">{{cite journal|url=http://dl.acm.org/citation.cfm?id=89095|author=Kenneth Ward Church and Patrick Hanks|year=1990|title=Word association norms, mutual information, and lexicography|journal=Comput. Linguist.|volume=16|issue=1|date=March 1990|pages=22-29}}</ref> , or '''point mutual information''', is a measure of association used in [[information theory]] and [[statistics]].


==Definition==
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 PMI of a pair of [[probability space|outcomes]] ''x'' and ''y'' belonging to [[discrete random variable]]s ''X'' and ''Y'' quantifies the discrepancy between the probability of their coincidence given their joint distribution and their individual distributions, assuming independence. Mathematically:


: <math>
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.
\operatorname{pmi}(x;y) \equiv \log\frac{p(x,y)}{p(x)p(y)} = \log\frac{p(x|y)}{p(x)} = \log\frac{p(y|x)}{p(y)}.
</math>


The [[mutual information]] (MI) of the random variables ''X'' and ''Y'' is the expected value of the PMI over all possible outcomes (with respect to the joint distribution <math>p(x,y)</math>).
==Werckmeister I (III): "correct temperament" based on 1/4 comma divisions ==


The measure is symmetric (<math>\operatorname{pmi}(x;y)=\operatorname{pmi}(y;x)</math>). It can take positive or negative values, but is zero if ''X'' and ''Y'' are [[statistical independence|independent]]. PMI maximizes when ''X'' and ''Y'' are [[perfectly associated]], yielding the following bounds:
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.


:<math>
{| class="wikitable" style="text-align:center"
-\infty \leq \operatorname{pmi}(x;y) \leq \min\left[ -\log p(x), -\log p(y) \right] .
|Fifth ||Tempering ||Third ||Tempering
</math>
|-
|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
|}


Finally, <math>\operatorname{pmi}(x;y)</math> will increase if <math>p(x|y)</math> is fixed but <math>p(x)</math>decreases.
{{Audio|Werckmeister temperament major chord on C.mid|Play major tonic chord}}


Here is an example to illustrate:
Modern authors have calculated exact mathematical values for the frequency relationships and intervals using the Pythagorean comma:
{| border="1" cellpadding="2" class="wikitable"
 
!''x''!!''y''!!''p''(''x'',&nbsp;''y'')
{| 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
|-
|-
|0||0||0.1
|G{{music|#}} ||<math>\frac{128}{81}</math> ||792
|-
|-
|0||1||0.7
|A ||<math>\frac{1024}{729} \sqrt[4]{2}</math> ||888
|-
|-
|1||0||0.15
|B{{music|b}} ||<math>\frac{16}{9}</math> ||996
|-
|-
|1||1||0.05
|B ||<math>\frac{128}{81} \sqrt[4]{2}</math> ||1092
|}
|}
Using this table we can marginalize to get the following additional table for the individual distributions:
 
{| border="1" cellpadding="2" class="wikitable"
==Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through 1/3 comma ==
! !!''p''(''x'')!!''p''(''y'')
 
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
|-
|-
|0||.8||0.25
|C-G || ^ ||C-E ||1 v
|-
|-
|1||.2||0.75
|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
|}
|}
With this example, we can compute four values for <math>pmi(x;y)</math>. Using base-2 logarithms:
 
{| cellpadding="2"
{| 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 ||
|-
|-
|pmi(x=0;y=0)||&minus;1
|D{{music|#}} ||<math>\frac{32}{27}</math> ||294 ||<math>101\frac{1}{5}</math>  ||295.0 ||
|-
|-
|pmi(x=0;y=1)||0.222392421
|E ||<math>\frac{64}{81} \sqrt[3]{4}</math> ||392 ||<math>95\frac{3}{5}</math> ||393.5 ||
|-
|-
|pmi(x=1;y=0)||1.584962501
|F ||<math>\frac{4}{3}</math> ||498 ||<math>90</math> ||498.0 ||
|-
|-
|pmi(x=1;y=1)||&minus;1.584962501
|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 ||
|}
|}


(For reference, the [[mutual information]] <math>\operatorname{I}(X;Y)</math> would then be 0.214170945)
==Werckmeister III (V): an additional temperament divided up through 1/4 comma ==


==Similarities to mutual information==
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.
Pointwise Mutual Information has many of the same relationships as the mutual information. In particular,


<math>
{| class="wikitable" style="text-align:center"
\begin{align}
|Fifth ||Tempering ||Third ||Tempering
\operatorname{pmi}(x;y) &=& h(x) + h(y) - h(x,y) \\
|-
&=& h(x) - h(x|y) \\
|C-G || - ||C-E ||2 v
&=& h(y) - h(y|x)
|-
\end{align}
|G-D || - ||C{{music|#}}-F ||4 v
</math>
|-
|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
|}


Where <math>h(x)</math> is the [[self-information]], or <math>-\log_2 p(X=x)</math>.
{| 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
|}


==Normalized pointwise mutual information (npmi)==
==Werckmeister IV (VI): the Septenarius tunings ==
Pointwise mutual information can be normalized between [-1,+1] resulting in -1 (in the limit) for never occurring together, 0 for independence, and +1 for complete [[co-occurrence]].


<math>
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".


\operatorname{npmi}(x;y) = \frac{\operatorname{pmi}(x;y)}{-\log \left[ p(x, y) \right] }
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.


</math>
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.


==Chain-rule for pmi==
{| class="wikitable" style="text-align:center"
Pointwise mutual information follows the [[Chain_rule_%28disambiguation%29|chain rule]], that is,
!Note
:<math>\operatorname{pmi}(x;yz) = \operatorname{pmi}(x;y) + \operatorname{pmi}(x;z|y)</math>
!Monochord length
 
!Exact frequency relation
This is easily proven by:
!Value in [[Cent (music)|cents]]
:<math>
|-
\begin{align}
|C || 196 || 1/1 || 0
\operatorname{pmi}(x;y) + \operatorname{pmi}(x;z|y) & {} = \log\frac{p(x,y)}{p(x)p(y)} + \log\frac{p(x,z|y)}{p(x|y)p(z|y)} \\
|-
& {} = \log \left[ \frac{p(x,y)}{p(x)p(y)} \frac{p(x,z|y)}{p(x|y)p(z|y)} \right] \\
|C{{music|#}}|| 186 || 98/93 || 91
& {} = \log \frac{p(x|y)p(y)p(x,z|y)}{p(x)p(y)p(x|y)p(z|y)} \\
|-
& {} = \log \frac{p(x,yz)}{p(x)p(yz)} \\
|D || 176(175) || 49/44(28/25) || 186(196)
& {} = \operatorname{pmi}(x;yz)
|-
\end{align}
|D{{music|#}}|| 165 || 196/165 || 298
</math>
|-
 
|E || 156 || 49/39 || 395
{{inline|date=February 2012}}
|-
|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
|}


==References==
== External sources ==
{{reflist}}
*[http://240edo.googlepages.com/equaldivisionsoflength(edl) 196-EDL & 1568-EDL and Septenarius tunings]
* {{cite web|title=Normalized (Pointwise) Mutual Information in Collocation Extraction|url=https://svn.spraakdata.gu.se/repos/gerlof/pub/www/Docs/npmi-pfd.pdf|last1=Bouma|first1=Gerloff|year=2009|publisher=Proceedings of the Biennial GSCL Conference}}
*[http://users.telenet.be/broekaert-devriendt/Index.html "Well Tempering based on the Werckmeister Definition"]
* {{cite book|last1=Fano|first1=R M|year=1961|title=Transmission of Information: A Statistical Theory of Communications|publisher=MIT Press, Cambridge, MA|chapter=chapter 2|isbn=978-0262561693}}
* Well Tempered based on Werckmeisters last book Musikalische Paradoxal-Discourse (1707) is Equal Temperament. See: http://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament


==External links==
== References ==
* [http://cwl-projects.cogsci.rpi.edu/msr/ Demo at Rensselaer MSR Server] (PMI values normalized to be between 0 and 1)
<references/>


{{musical tuning}}


[[Category:Information theory]]
[[Category:Musical temperaments]]
[[Category:Statistical dependence]]
[[Category:Entropy and information]]

Revision as of 11:35, 15 August 2014

Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings.[1][2][3] 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 thirds, 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.

Werckmeister I (III): "correct temperament" based on 1/4 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 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 ^ 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

My name: Lindsey Gavin
My age: 28
Country: Sweden
Home town: Vemdalen
Postal code: 840 92
Address: Buanvagen 79

Look into my weblog :: http://www.hostgator1centcoupon.info/

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

Note Exact frequency relation Value in cents
C 11 0
CTemplate:Music 256243 90
D 64812 192
DTemplate:Music 3227 294
E 25624324 390
F 43 498
FTemplate:Music 1024729 588
G 8984 696
GTemplate:Music 12881 792
A 102472924 888
BTemplate:Music 169 996
B 1288124 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, FTemplate:Music-CTemplate:Music, and BTemplate:Music-F are tempered narrow by 1/3 comma, and the fifths GTemplate:Music-DTemplate:Music and ETemplate:Music-BTemplate:Music 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 - 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 relation Value in cents Approximate monochord length Value in cents
C 11 0 120 0
CTemplate:Music 163841968323 82 11415 - (misprinted as 11412) 85.8
D 8923 196 10715 195.3
DTemplate:Music 3227 294 10115 295.0
E 648143 392 9535 393.5
F 43 498 90 498.0
FTemplate:Music 1024729 588 8513 590.2
G 322723 694 8015 693.3
GTemplate:Music 8192656123 784 76215 787.7
A 25624343 890 71710 891.6
BTemplate:Music 9423 1004 6715 1003.8
B 40962187 1086 64 1088.3

Werckmeister III (V): an additional temperament divided up through 1/4 comma

In Werckmeister III the fifths D-A, A-E, FTemplate:Music-CTemplate:Music, CTemplate:Music-GTemplate:Music, and F-C are narrowed by 1/4, and the fifth GTemplate:Music-DTemplate:Music is widened by 1/4 comma. The other fifths are pure. This temperament is closer to equal temperament than the previous two.

Fifth Tempering Third Tempering
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 relation Value in cents
C 11 0
CTemplate:Music 8924 96
D 98 204
DTemplate:Music 24 300
E 892 396
F 9824 504
FTemplate:Music 2 600
G 32 702
GTemplate:Music 12881 792
A 84 900
BTemplate:Music 384 1002
B 432 1098

Werckmeister IV (VI): the Septenarius tunings

This tuning is based on a division of the monochord length into 196=7×7×4 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. However this 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 relation Value in cents
C 196 1/1 0
CTemplate:Music 186 98/93 91
D 176(175) 49/44(28/25) 186(196)
DTemplate:Music 165 196/165 298
E 156 49/39 395
F 147 4/3 498
FTemplate:Music 139 196/139 595
G 131 196/131 698
GTemplate:Music 124 49/31 793
A 117 196/117 893
BTemplate:Music 110 98/55 1000
B 104 49/26 1097

External sources

References

  1. ↑ 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).
  2. ↑ A. Werckmeister: Musicae mathematicae hodegus curiosus oder Richtiger Musicalischer Weg-Weiser (Quedlinburg 1686, Frankfurt & Leipzig 1687) ISBN 3-487-04080-8
  3. ↑ A. Werckmeister: Musicalische Temperatur (Quedlinburg 1691), reprint edited by Rudolf Rasch ISBN 90-70907-02-X

Template:Musical tuning

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=63131"