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## Table of Contents

50edodivides the octave into 50 equal parts of precisely 24 cents each. In the 5-limit, it tempers out 81/80, making it a meantone system, and in that capacity has historically has drawn some notice. In "Harmonics or the Philosophy of Musical Sounds" (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the least squares tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While 31edo extends meantone with a 7/4 which is nearly pure, 50 has a flat 7/4 but both 11/8 and 13/8 are nearly pure.50 tempers out 126/125, 225/224 and 3136/3125 in the 7-limit, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the 11-limit and 105/104, 144/143 and 196/195 in the 13-limit, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&50 temperament (Coblack), and provides the optimal patent val for 11 and 13 limit bimeantone. It is also the unique equal temperament tempering out both 81/80 and the vishnuzma, |23 6 -14>, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.

## Relations

The 50edo system is related to 7edo, 12edo, 19edo, 31edo as the next approximation to the "Golden Tone System" (Das Goldene Tonsystem) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "A Theory of Evolving Tonality").## Intervals

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 50edo (ordered by absolute error).Interval, complementError (abs., in cents)## Commas

50 EDO tempers out the following commas. (Note: This assumes the val < 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.## Music

Twinkle canon – 50 edo by Claudi MeneghinFantasia Catalana by Claudi Meneghin

Fugue on the Dragnet theme by Claudi Meneghin

the late little xmas album by Cam Taylor

Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor

Long improvisation 2 in 50EDO by Cam Taylor

Chord sequence for Difference tones in 50EDO by Cam Taylor

Enharmonic Modulations in 50EDO by Cam Taylor

Harmonic Clusters on 50EDO Harpsichord by Cam Taylor

Fragment in Fifty by Cam Taylor

## Additional reading

Robert Smith's book onlineMore information about Robert Smith's temperament

50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor

iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor