Author Archives: Siemen

The Extended Meantone Style of Gesualdo

Gesualdo titlepageThe author re-examines the syntax of Gesualdo’s musical style in the light of the intonational practices of his time. A theoretical structure with which to describe his harmonic practices, and those of other 16th Century composers, is developed. The historical context is presented as well as a series of analyses of Gesualdo’s music.

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Toward a Theory of Meantone (and 31-E.T.) Harmony

MeantoneTheory titlepageThe author develops a syntax for meantone harmony based on the structural characteristics of the temperament. These features include a description of the Regions with their boundary functions, and also the three genera. An abstract topology of triad progression is introduced, as well as an introduction to the organization of functional tables for 31-ET.

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Notes on an Approach to Musical Composition in 31-E.T.

31Composition titlepageThe purpose of this article is to examine certain aspects of theory which concert practical musical composition in 31-E.T. (or extended meantone). The principles which are put forth may also be used to advantage with other equal divisions, including 12-E.T., 19-E.T., and so on. This article was meant to compliment the other articles which I have written on meantone theory. Those articles are: TOWARD A THEORY OF MEANTONE HARMONY, THE MEANTONE SERIES OF CYCLICAL TEMPERAMENTS, THE MATRIX MODEL OF MUSICAL HARMONY, and THE EXTENDED MEANTONE STYLE OF GESUALDO. These articles form a unified approach with no (I hope!) internal contradictions.

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31-E.T. SYMMETRY, a brief study. With Commentary on the Fokker-Genera

31Symmetry titlepageSome representative examples of symmetrical structures are presented, along with a comparison between the types of symmetry in 31-E.T. and other equal-tempered systems. The Euler-Fokker genera are interpreted as a means of generating symmetrical structures. Lastly, an alternative method of generating the most important symmetrical functions is outlined.

An Abbreviated Dictionary of 31-et Harmonies

31etdictionary titlepageThis collection of harmonies is an expansion of the set presented in my paper Notes on Composition. The need for this information arose out of very practical concerns. I have been playing the enharmonic guitar, and I wanted to work out the finger-positions for chords. Consequently I needed a reference of useful 31-et harmonies. Of course, this information could also be applied to keyboards or other instruments capable of chord functions. The collection of harmonies shown in Notes are presented in a most concise but abstract way. My aim here is to present the information in such a manner that it can have practical applications for real music-making.

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Studies in the Transition from Medieval to Renaissance Tuning Paradigms

transition titlepageThe author explains how tendencies within the structure of the Pythagorean chromatic scale supported the transfer to Just Intonation; however, certain features of Just Intonation necessitated  the application of some form of temperament. The meantone temperament formed the best compromise for the musical styles that were being explored. The author aims at a co-relation between tuning paradigm and associated musical style.

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Technical Drawings of the Geometric Mean

These diagrams show how the Geometric Mean can be used to find the fret positions for various temperaments. There is mention of it in various papers, namely

Terpstra-GeometricMeanGenerating4ETTerpstra-GeometricMeanGenerating5ET Terpstra-GeometricMeanGenerating7ET Terpstra-GeometricMeanGenerating12ET Terpstra-GeometricMeanGenerating19ET Terpstra-GeometricMeanGenerating31ET Terpstra-GeometricMeanGenerating53ET

Aristoxenus and Equal Temperament – a reinterpretation

I Aristoxenus titlepagewas repeatedly told that one cannot set a tempered harmony on a monochord by classical means. However, I realised that one exception exists, the complex of multiple divisions 12-et, 24-et, 36-et, 72-et, and 144-et. These very systems are the ones clearly described by Aristoxenus. This realisation prompted the following study of Aristoxenus.

I wrote this paper very long ago. I still stand by most of it, but a couple of my conjectures have shifted over the years. In the paper I presumed that Pythagoreans did much of the research into irrational ratios. In spite of the example of Hippasus, I now feel that the Pythagoreans were generally conservatives and the bulwark of anti-temperament sentiment. Work on irrationals was likely done by anti-pythagorean elements, such as Democritus.

Also, I conjecture that Philolaus the Pythagorean supports 53-et. Now I no longer believe that to be the case. My argument there was simple enough. Since he knew that the 8:9 whole tone is nine commas in size, he must have known that the ditone equals eighteen commas, the fourth twenty-two and the octave fifty-three. He may well have thought that his commas were all the same size, when they were likely not to be so. I now think that Philolaus probably abhorred the very idea of temperament.

For my current thinking on this topic, see my recent essay HOMAGE TO PTOLEMY (a companion paper to ARISTOXENUS) under the heading MONOCHORD. See also RESTORING THE MUSE under the philosophy heading.

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Irregular Temperaments and the Division of the Ditonic Comma

WellTemperaments titlepageThe author presents an analysis of the structure of irregular temperaments, with special emphasis on well-temperaments or circulating temperaments. Roughly 200 scales are examined, of which about a third are historical and the rest are original to the author. A systematic method is presented for the conversion of various regular temperaments (meantone temperaments) into circulating temperaments. Thus the historical material is embedded within the wider context of scalar architecture. The aim is an overview, leading to the understanding and control of design characteristics.

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