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discussing the characteristics of well tempering (
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"Well Temperaments based on the Werckmeister definition"

Improved and supplemented edition.
From Werckmeister, passing Bach, Kirnberger and Vallotti, to Equal Temperament.
Inspired by Kelletat

or go to (1.5 MB)

Summary of content:

Graphic display of some interval characteristics of widespread Well Temperaments:

Interval characteristics

Abstract :       Based on the definition of well temperaments, elaborated by Kelletat H., but derived from Werckmeister, a number of logic mathematical steps are developed, in order to obtain mathematically optimised well tempered model-“temperaments”. This leads to an objective mathematical classification of historical temperaments. Historical temperaments that fit best with the well tempered models are the same as those that are very often installed on organs, with highest established musical appreciations and notoriety. The elaborated mathematical appreciation and the musical appreciation of historical temperaments are thus in good accordance. All Well Temperaments lie qualitatively between the elaborated optimum and the Equal Temperament. They join these limits very closely and the list is well filled in small rising steps. Hence, it makes no sense to develop new well 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

Kelletat Definition, based on Werckmeister publications:

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”..

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).