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discussing the characteristics of well tempering (
pdf file 650 kB):

Téléchargez le document ci-dessous discutant les charactéristiques de l'accordement bien tempéré (en Anglais
; fichier pdf 650 kB):

"Well Tempering based on the Werckmeister definition"

and it's addenda:

Application of Weighted Intervals; the Path from Werckmeister to Vallotti-Tartini


Additional Considerations

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Summary of content:

Graphic display of some interval characteristics of widespread Well Temperaments:

Interval characteristics

Abstract :        Based on the definition of well temperaments published by Werckmeister (1681), a number of logic mathematical steps are developed, in order to obtain mathematically optimised well tempered model-“temperaments”. This allows to work out some kind of objective mathematical classification of historical Well Temperaments. Historical temperaments that fit best with the models, are the same as those that are very often installed on organs, with highest established musical appreciations and notoriety. The mathematical appreciation that is worked out and the musical appreciation of historical temperaments seem to be in good accordance. All historical Well Temperaments have RMS-impurities between an absolute minimum of model-“temperaments” and the absolute maximum of the equal temperament. There is close affiliation of historical temperaments with said limits, and the list is well filled, hence, it makes no sense to develop new temperaments further on. The developed RMS computing modules can be used also for an accurate recognition of historical temperaments.

Temperament Recognition: Download file Characterisation_of_Temperaments.xls

Werckmeister Definition:

“Well temperament means a mathematical-acoustic and musical-practical organisation of the tone system within the twelve steps of an octave, so that impeccable performance in all tonalities is enabled, based on the extended just intonation (natural-harmonic tone system), while striving to keep the diatonic intervals as pure as possible.

This temperament acts, while tied to given pitch ratios, as a thriftily tempered smoothing and extension of the meantone, as unequally beating half tones and as equal (equally beating) temperament”. (Orgelprobe, 1681).

Original German text:

“Wohltemperierung heißt mathematisch-akustische und praktisch-musikalischen Einrichtung von Tonmaterial innerhalb der zwölfstufigen Oktavskala zum einwandfreien Gebrauch in allen Tonarten auf der Grundlage des natürlich-harmonischen Systems mit Bestreben möglichster Reinerhaltung der diatonische Intervalle.

Sie tritt auf als proportionsgebundene, sparsam temperierende Lockerung und Dehnung des mitteltönigen Systems, als ungleichschwebende Semitonik und als gleichschwebende Temperatur.” (Orgelprobe, 1681).