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| "Il faut toujours savoir pourquoi,
..."
Paul Valéry (1871-1945) "... et comment." (my mother).
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| "Diatonic Minimum Spread
Tuning" Johan Broekaert Observation: This manuscript was submitted to the
``Journal of Mathematics and Music'' in april 2024. A
completed, strongly improved and peer reviewed © Copyright
Version of Record of this manuscript has been published, and
is available at:
``Journal of Mathematics and Music'', date and number, at: |
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The specified text concerns new fundamental insights for tuning an octave. |
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Le texte spécifié concerne de nouvelles idées fondamentales pour accorder une octave. |
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De opgegeven tekst betreft nieuwe fundamentele inzichten voor het stemmen van een octaaf. |
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Den angegebenen Text befasst grundlegende neuen Erkenntnissen zum Stimmen einer Oktave. |
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El texto specificado trata sobre nuevas ideas fundamentales para afinar una octava. |
| Abstract A temperament holding a lowest diatonic
C‑major interval beating rate spread on fifths, and major and
minor thirds was calculated. It appears that beating rate
equality is a primordial quality factor for auditory tuning. But
fortunately, tuning pitches based on comma division, ratios or
cents, can be maintained, for the thus obtained pitches are
almost identical with those obtained based on interval beating
rate calculations, and moreover, both approaches suffer from the
same degree of inaccuracies, due to sound inharmonicities. It
appears Vallotti and a new developed well tempered
meantone are by far the temperaments with lowest impurity
spread. In conclusion, it is probably right to assume that the
auditory tuning of keyboards was the basis for the development
of historical temperaments, although the results were mainly and
duly described mathematically on the basis of proportional
relationships. Text Submitted draft version; see Diatonic Minimum Spread Related topic: see "The Well Tempered Meantone" Very sincere thanks to: Prof. E. Amiot, A. Calvet, E. Jobin, T. Paintoux: they were kindly open for communications and interchanges of ideas, leading to this disclosure. G. Baroin: for the compilation of a magnificent video dealing with the subject 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. |
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| The following complements may improve intelligibility. Introduction: Auditory tuning involves initial setting of musical intervals within one octave. Those intervals inherently deviate from perfect consonance, because there are more musically significant intervals than defined notes. This led to numerous historical musical temperaments. Octaves should be perfect with a 2/1 ratio, but deviate slightly, due to sound inharmonicity. Each note in itself must be a perfect prime, ratio 1/1. . Closing Thougths: It should be clear that calculations in this paper have not taken string inharmonicities into account. All reported results therefore, are approximations only. It has become clear and evident, that all qua`N'titative temperament definitions or instructions, are approximate indications only. In general; the qua`L'itative temperament definitions or instructions should therefore be considered to be the more fundamental ones. Most mid range strings have inharmonicities about one pro mille or more, on the second partial. The auditory obtained tuned pitches in the mid range, can therefor normally be estimated to deviate some one to three cents from the theoretical and quantitative defined ones, following ratios or beating rate calculations. Keep Fourrier (1772-1837), Railsback (1938), Fletcher (1962) and Plomp (1965) into mind indeed. |
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