34 equal temperament

In musical theory, 34 equal temperament, also referred to as 34-tet, 34-edo or 34-et, is the tempered tuning derived by dividing the octave into 34 equal-sized steps (equal frequency ratios).  Play  Each step represents a frequency ratio of 2^1/34, or 35.29 cents  Play .

History

Unlike divisions of the octave into 19, 31 or 53 steps, which can be considered as being derived from ancient Greek intervals (the greater and lesser diesis and the syntonic comma), division into 34 steps did not arise 'naturally' out of older music theory, although Cyriakus Schneegass proposed a meantone system with 34 divisions based in effect on half a chromatic semitone (the difference between a major third and a minor third, 25/24 or 70.67 cents). Wider interest in the tuning was not seen until modern times, when the computer made possible a systematic search of all possible equal temperaments. While Barbour discusses it, ^[1] the first recognition of its potential importance appears to be in an article published in 1979 by the Dutch theorist Dirk de Klerk. The luthier Larry Hanson had an electric guitar refretted from 12 to 34 and persuaded well-known American guitarist Neil Haverstick to take it up.

As compared with 31-et, 34-et reduces the combined mistuning from the theoretically ideal just thirds, fifths and sixths from 11.9 to 7.9 cents. Its fifths and sixths are markedly better, and its thirds only slightly further from the theoretical ideal of the 5/4 ratio. Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B, thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9. This can be regarded either as a resource or as a problem, making modulation in the contemporary Western sense more complex. As the number of divisions of the octave is even, the exact halving of the octave (600 cents) appears, as in 12-et. Unlike 31-et, 34 does not give an approximation to the harmonic seventh, ratio 7/4.

Scale diagram

The following are 15 of the 34 notes in the scale:

The remaining notes can easily be added.

Interval size

The following table outlines some of the intervals of this tuning system and their match to various ratios in the harmonic series.

References

• J. Murray Barbour, Tuning and Temperament, Michigan State College Press, 1951.

External links

• Dirk de Klerk. "Equal Temperament", Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150.
• Stickman: Neil Haverstick - Neil Haverstick is a composer and guitarist who uses microtonal tunings, especially 19, 31 and 34 tone equal temperament.

Microtonal music People Easley Blackwood, Jr. Heinz Bohlen Wendy Carlos Julián Carrillo Mildred Couper Ivor Darreg John Eaton Adriaan Fokker Alois Hába Lou Harrison Christiaan Huygens Ben Johnston Joel Mandelbaum Joe Maneri Harry Partch Ezra Sims Nicola Vicentino Elaine Walker Ivan Wyschnegradsky Tunings & scales Non-octave- repeating scales Alpha scale Beta scale Gamma scale Delta scale Lambda scale (Bohlen–Pierce scale) Equal temperament 15 17 19 22 24 31 34 41 53 72 Just intonation Harry Partch's 43-tone scale Concepts & techniques Limit Otonality and Utonality Semitone Sonido 13 Xenharmonic music Groups & publications Boston Microtonal Society Genesis of a Music Huygens-Fokker Foundation Tonalsoft Xenharmonic Bulletin Compositions Beauty in the Beast quarter tone pieces just pieces Mother Sonata for Microtonal Piano Suite for Microtonal Piano Twelve Microtonal Etudes for Electronic Music Media Other Enharmonic keyboard Generalized keyboard Modernism (music)

Musical tunings Measurement Pitch Cent Millioctave Savart Interval Interval ratio Pitch class Consonance and dissonance List of musical intervals List of intervals in 5-limit just intonation List of meantone intervals Microtone Just intonation Euler–Fokker genus Harmonic Scale Harry Partch's 43-tone scale Hexany Limit 5-limit 7-limit List of compositions Numerary nexus Otonality Ptolemy's intense diatonic scale Pythagorean tuning Scale of harmonics Tonality diamond Tonality flux Temperaments Equal 6-tone 12-tone 15-tone 17-tone 19-tone 22-tone 24-tone (pieces) 31-tone 34-tone 41-tone 53-tone 72-tone Linear Meantone (quarter-comma, septimal) Schismatic Miracle Magic Syntonic Irregular Well temperament/Temperament ordinaire (Kirnberger, Werckmeister, Young) Traditional non-Western Chinese musicology Shí-èr-lǜ Dastgah Maqam Arabian maqam Makam Mugham Muqam Octoechos Pelog Raga (Carnatic rāga) Slendro Tetrachord Non-octave 833 cents scale A12 scale Alpha scale Beta scale Gamma scale Delta scale Lambda scale (Bohlen-Pierce)