41edo
41edo, or 41 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. It is known for its relatively good approximation of 11-limit just intonation.
Theory
41edo has three different flavors of minor and major intervals as well as neutral intervals. Its subminor and supermajor intervals approximate simpler septal ratios such as 7/4 and 9/7, while its superminor and submajor intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its plain major and plain minor intervals approximate classic 3-limit ratios as well as ones in the 2.3.7/5, 2.3.19, and 2.3.19/5 just intonation subgroups (such as 63/40, 19/12, and 19/15, respectively). Its neutral intervals approximate 2.3.11 and, less accurately, 2.3.13.
41edo is used by the musician and conlanger Lamplight as a standard tuning for their Shasavic theory of music.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +0.5 | -5.8 | -3.0 | +4.8 | +8.3 | +12.1 | -4.8 | -13.6 | -5.2 | -3.6 |
| Relative (%) | 0.0 | +1.7 | -19.9 | -10.2 | +16.3 | +28.2 | +41.4 | -16.5 | -46.6 | -17.7 | -12.2 | |
| Steps
(reduced) |
41
(0) |
65
(24) |
95
(13) |
115
(33) |
142
(19) |
152
(29) |
168
(4) |
174
(10) |
185
(21) |
199
(35) |
203
(39) | |
Notation
Since 41edo has a perfect fifth which is split exactly in half, Stein-Zimmerman accidentals can be used alongside standard accidentals to notate it. These accidentals represent half-sharps, half-flats, sesquisharps, and sesquiflats.
Ups and downs notation is a great fit for 41edo. An up arrow represents 1 step up in pitch, and a down arrow represents 1 step down in pitch. These arrows represent comma-sized deviations from the central chain of fifths so that, for example, C to ↓E represents a tempered 5/4 ratio, and C to ↓B♭ represents a tempered 7/4.
Regular temperaments
41edo supports a number of notable linear temperaments including schismic, slendric, tetracot, miracle, and magic. Magic is especially important here as it forms the fret layout and main string tuning for the Kite Guitar.
It also contains a slightly-stretched version of equal Bohlen-Pierce tuning (where the perfect twelfth of 3/1 is split into 13 equal parts) via every fifth step. If used in a linear temperament as the generator, this temperament is called bohpier.
Planar temperaments it supports include marvel and aberschismic (also known as hemifamity).
