Bach- and Well-Temperaments
for Western Classical Music
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A proposal for an Objective Musical Definition
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A herewith connected proposal for an Objective Identification of Temperaments

Mathematics seductively swims below the surface of music........................................ (prof. Eugene E. Helm)
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Il faut toujours savoir pourquoi........................ (Paul Valéry, 1871-1945), ....... Et comment........(my mother)

Proposed Objective Definition
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Bach- and Well-Temperaments objectively strive for an optimal compromise between:
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- Criterium 1: Maximalisation of the purity of the most important sharp thirds, C and G
- Criterium 2: Maximalisation of the purity of the flat third on E
- Criterium 3: Preservation of a maximal number of pure fifths
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The better Well-Temperaments can be defined as “Bach”-Temperaments

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.Evidence for the plausibility of the above definition can be found in a paper with same title as this web page: download (pdf document, 1.4 MB) or go to html version. Warning: Reading of the paper implies preliminary knowledge of music theory: elements of music theory are available at tuning_english.
Mogelijke aanneembaarheid van bovenstaande definitie wordt aangetoond in volgende web-pagina: (pdf document, 1.4 MB) of in de html version. Waarschuwing: Lezing van deze tekst veronderstelt voorafgaande kennis van notenleer. Eventueel kan volgende tekst vooraf ingekeken worden: "Muzikale Structuren" (pdf of html). 

 
Abstract:
This paper discusses the development of an objective definition of Bach- and Well-Temperaments based on the translation of very simple musical criteria to very simple arithmetics and the optimisation of these criteria using a large number of manual arithmetic iterations in spreadsheets. All results appear to be reproduceable analytically as well.
A "quality"-measuring method is proposed that allows for a very simple and objective identification of musical temperaments

Objective Identification of Temperaments
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Objective comparison of temperaments against Bach- or Well-temperaments, and identification of temperaments can be done by measuring the RMS value of differences in impurity of thirds, sharp and flat, and fifths.
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This can be done by using following Excel file: download...Identification_of_Temperament.xls  (273 kB; it contains macro's that are guaranteed to be harmless, unless the site of my provider should have been infected).
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The so called "PBP-, cents-, or Beat-Temperaments" in this file, are the "temperaments" that fit best with the above definition, if impurities of sharp and flat thirds and of fifth are measured by PBP, cents of beat impurity of said intervals.
We DO NOT recommend their use for musical practice; we recommend to stay with already existing  historical temperaments.
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Objective Tuning of a Temperament
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On this site we limit ourselves to the three most commonly occuring or known temperaments:
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Pitches
C
Cis
D
Es
E
F
Fis
G
Gis
A
Bes
B
1/4 comma Meantone
263.3
275.0
294.2
314.8
329.0
352.0
367.8
393.5
411.2
440.0
470.8 
491.9
Kirnberger III unequal
263.1
277.2
294.5
311.8
328.9
350.8
370.0
393.8
415.8
440.0
467.7
493.3
Equal Temperament
261,6
277,2
293,7
311,1
329,6
349,2
370,0
392,0
415.3
440.0
466.2
493.9
Data on more temperaments can be found in the Excel file: Identification_of_Temperament.xls, see tab "calculation"; (258 kB; it contains macro's that are guaranteed to be harmless, unless the site of my provider should have been infected).

A software tool that can be used for the tuning of musical instruments can be found at:... http://www.tunelab-world.com .
Many other files also, which serve the same purpose, can be found on the internet.



I want to avoid SPAM:
My e-mail address therefor has no link for automatic opening of an e-mail template.

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Johan Broekaert
Nieuwelei, 52
B 2640 Mortsel
Belgium
tel: + 32 3 455 09 85

Klik hier voor nadere kennismaking met de familie.
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Home page of  Johan and Rosette Broekaert-Devriendt, revision 2009-10