Sixth Tone Guitar Neck - an equal tempered microtonal instrument (more)
Photo and Fretting by Glen Peterson
Equal Temperament |
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These two systems are grouped together because the two tuning processes yield similar results. Equal Tempered systems are based on equal divisions of the octave. Pythagorean systems are based on circles of intervals like the familiar Circle of Fifths. Equal divisions of the octave make possible circles of any interval within the system. Similarly, Circles of intervals which "close" define equal temperaments.
Pythagorean circles with Just intervals are possible but the first note is not necessarily reached again - they tend to become spirals instead of circles. For instance, a Just Intonation Perfect Fifth (3/2) is 702 cents, while a 12TET Perfect Fifth is 700 cents. If you go around the Circle of Fifths with a 12TET fifth, you start and end on the same note, C, G, D, A, E, B, F#, C#, G#, D#, A#, F, C. If you do the same with a Just Intonation Perfect Fifth, each note is 2 cents sharper than the previous. By the time you get back to C you will be a little more than 20 cents sharp; C, G+2, D+4, A+6, E+8, B+10, F#+12, C#+14, G#+16, D#+18, A#+20, F+22, C+24. This difference is called a Pythagorean Comma and is often mentioned in tuning treatises.
Alois Haba, Lou Harisson, Joe Maneri, Charles Ives, Bela Bartok (in a violin concerto but sometimes omitted), and many other composers have worked with equal temperaments other than 12TET with very interesting results. Equal tempered systems are very easy to build fretted string instruments for because unbroken frets can travel straight underneath all the strings.
Comparison of equal temperaments.
Charles Ives, Piano Pieces (Three Quarter-tone Pieces for Two Pianos) - Ives quarter-tone piano pieces are very interesting. He uses several interesting effects such as drifting by a quarter-tone, and even takes a stab at some counterpoint.
Alois Haba, Qrt String 11/12/15/16/Fant/St (I haven't heard this but the one I know well is no longer available. Just enter "Haba" in the search field).