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41 Tone Equal TemperamentDeutsch

Introduction | Commas | Temperaments | Intervals | Instruments | Harmonic Scale | Nonoctave Temperaments | Music | Links

## Introduction

The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 cents, an interval close in size to 64/63, the septimal comma. 41-ET can be seen as a tuning of theGaribaldi temperament^{[1]},^{[2]},^{[3]}theMagic temperament^{[4]}and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after 29edo) whose perfect fifth is closer to just intonation than that of 12-ET, and is the seventh zeta integral edo after 31; it is not, however, a zeta gap edo. This has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp. Various 13-limit magic extensions are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.41edo is consistent in the 15 odd limit. In fact,

allof its intervals between 100 and 1100 cents in size are 15-odd-limit consonances. (In comparison, 31edo is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).41-ET forms the foundation of the H-System, which uses the scale degrees of 41-ET as the basic 13-limit intervals requiring fine tuning +/- 1 average JND from the 41-ET circle in 205edo.

41edo is the 13th prime edo, following 37edo and coming before 43edo.

## Commas

41 EDO tempers out the following commas using its patent val, < 41 65 95 115 142 152 168 174 185 199 203 |.## Temperaments

List of edo-distinct 41et rank two temperaments## Intervals

Ratios in the 11-limit

downs

notation

solfege

syllable

33/32

octacot

Ab

Ab^

64/33

41edo chord names using ups and downs:

0-10-20 = D F Ab = Ddim = "D dim"

0-10-21 = D F Ab^ = Ddim(^5) = "D dim up-five"

0-10-22 = D F Avv = Dm(vv5) = "D minor double-down five", or possibly Ddim(^^5)

0-10-23 = D F Av = Dm(v5) = "D minor down-five"

0-10-24 = D F A = Dm = "D minor"

0-11-24 = D F^ A = D.^m = "D upminor"

0-12-24 = D F^^ A = D.~ = "D mid"

0-13-24 = D F#v A = D.v = "D downmajor" or "D dot down"

0-14-24 = D F# A = D = "D" or "D major"

0-14-25 = D F# A^ = D(^5) = "D up-five"

0-14-26 = D F# A^^ = D(^^5) = "D double-up-five", or possibly Daug(vv5)

0-14-27 = D F# A#v = Daug(v5) = "D aug down-five"

0-14-28 = D F# A# is Daug = "D aug"

etc.

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 41edo (ordered by absolute error).Interval, complementError (abs., in cents)## Instruments

41-EDO Electric guitar, by Gregory Sanchez.41-EDO Classical guitar, by Ron Sword.A possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.

## Harmonic Scale

41edo is the first edo to do some justice to Mode 8 of the harmonic series, which Dante Rosati calls the "Diatonic Harmonic Series Scale," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)

7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.

6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).

5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).

4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).

The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.

## Nonoctave Temperaments

Taking every third degree of 41edo produces a scale extremely close to 88cET or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered Bohlen-Pierce Scale (or the 13th root of 3). See chart:## Notation

A red-note/blue-note system, similar to the one proposed for 36edo, is one option for notating 41edo. (This is separate from and not compatible with Kite's color notation.) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:

A, red A, blue Bb, Bb, A#, red A#, blue B, B, red B, blue C, C, red C, blue Db, Db, C#, red C#, blue D, D, red D, blue Eb, Eb, D#, red D#, blue E, E, red E, blue F, F, red F, blue Gb, Gb, F#, red F#, blue G, G, red G, blue Ab, Ab, G#, red G#, blue A, A.

Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.

The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.

## Music

EveningHorizon play by Cameron Bobro## Links