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Ratiotonic Temperament: A Proposal for the Organization of Tones in Art Music
as an Alternative to Twelve-Tone Equal Temperament

Peter Lucas Hulen

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In order to accommodate a more or less equal range for all eight contonations and the total network of tones they share, a unified temperament must be developed. In order to fit a suitable range of pitches onto a commercially-available electronic instrument capable of the necessary fine detuning, the tones of all eight contonations must be combined and arranged in ascending order (a custom-made electronic instrument constructed according to this system remains to be created). Many instruments have a range of ten and one-half octaves divided into twelve tones each (128 notes in all). Figure 5 illustrates how a hypothetical instrument with a range of C-2 to G8 (128 notes) could be detuned to accommodate such a range of pitches (in fact the Yamaha TX802 FM tone generator, for example, has a range from C-2 to G8, and it will detune to within 0.5859375 cents [1/1024 of an octave] of a specific value in whole cents, so this chart could be adapted to detune that instrument according to the overall system). The black noteheads indicate the default notes of the hypothetical instrument, while the white noteheads aligned with them indicate the pitches to which they would be detuned (plus or minus so many cents as indicated). The letter names indicate the primary tone(s) of the contonation(s) containing the given note. The ranges of the contonations as given in the "Table of Contonations" in the Appendix correspond to this 128-note range of pitches shown in Figure 5.

Figure 5

Figure 5


The table in figure 6 gives frequency values in hertz for the range of tones in Figure 5. Each line of the table represents a successive tone in the total range. The note name and deviation in cents from equal temperament is given in one or more of the eight columns on the left side; the frequency in hertz of the given tone is in the rightmost column. Each column represents the successive tones in a contonation. A note name listed two or more times on a given line represents a point of modulation between the two or more contonations (columns) in which it is listed.

Figure 6

View Figure 6  |

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Peter Lucas Hulen is Byron K. Trippet Assistant Professor of Music at Wabash College in Crawfordsville, Indiana, USA.

© 2005 Peter Hulen


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