<>
| The Well Tempered Meantone | ||||
 |
 Français |
 Nederlands |
 Deutsch |
 Español |
The
hand written text and the drawn spirals
in the figure above,
come from the score of "Das wohltemperirte Clavier" by J. S. Bach.
come from the score of "Das wohltemperirte Clavier" by J. S. Bach.
The above figure with spirals, could contain indications from J. S. Bach, on how to tune a keyboard, according a hypothesis formulated by Jobin,
A slightly differing interpretation of the above "Bach spirals" can be worked out, based on equal beating characteristics of fifths and major thirds of C-major, rather than the equal ratio of fifths leading to pure thirds, such as assumed by Jobin.
Jobin: see https://www.clavecin-en-france.org/spip.php?article52
A text discussing the calculated alternatives based on interval beat calculations, for sure clear to any musicologist, possibly assisted by some mathematician, is published in the december 2020 edition of the PIANISTIK journal (https://europianofrance.org/).
It became clear that an assumption leading to perfect equal beating of five fifths and three major thirds, meets the assumed characteristics of spirals very well. An almost identical hypothesis, includes almost equal beating of six fifths and three major thirds..
A new (English) version of the above referred Pianistik text can be downloaded.
This text should be accessible for comprehensive reading by anyone interested. This text includes supplements: brief history of musical temperaments, mathematical, acoustic and physical sciences, some basic music theory, and sufficiently extensive calculations of all the models studied, to allow for their verification,.
Calvet believes a beat rate approach can make sense.
http://andrecalvet.com/v3/index.php
Key
results (diapason: A4 = 440)
5 fifths and 3 major thirds, with
perfectly equal beat
| C | C# | D | Eb | E | F | F# | G | G# | A | Bb | B |
| 262.83 |
277.07 |
293.98 |
311.90 |
329.03 |
351.22 |
369.42 |
393.27 |
415.60 |
440.00 |
468.04 |
492.57 |
| Tuning Table (A3 = 220) | |||
| A3 = 220 | Equal beating of 5 fifths and 3 major thirds | ||
| Partition F3 - F 4 |
Beats | Major Thirds | Beats |
| A4 - A3 | 0.00 | F3 - A3 | 1.95 |
| A3 - D4 | 1.95 | Gb3 - Bb3 | 12.51 |
| D4 - G3 | -1.95 | G3 - B3 | 1.95 |
| G3 - C4 | 1.95 | Ab3 - C4 | 12.32 |
| C4 - F3 | -1.17 | A3 - C#4 | 8.27 |
| F3 - F4 | 0.00 | Bb3 - D4 | 5.84 |
| F4 - Bb3 | 0.39 | B3 - D#4 | 16.17 |
| Bb3 - Eb4 | -0.39 | C4 - E4 | 1.95 |
| Eb4 - Ab3 | 0.39 | Db4 - F4 | 19.54 |
| G#3 - C#4 | 0.00 | F4 - A4 | 3.89 |
| C#4 - F#3 | 0.00 | ||
| F#3 - B3 | 0.00 | ||
| B3 - E4 | 1.95 | ||
| E4 - A3 | -1.95 | ||
| E4 - A4 | -3.89 | ||
Broekaert Johan 2021-01-05
broekaert.devriendt@gmail.com
Most recent version of the papers
:
Most recent version of the papers
:Prof. E. Amiot, A. Calvet, E. Jobin, T. Paintoux: they were kindly open for communications and interchanges of ideas, leading to this disclosure.
My daughter Hilde: she drew my attention to investigate on what musicians want (diatonic interval purity) and not on what might be someone's preferred musical temperament.