41edo: Difference between revisions
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== Notation == | == Notation == | ||
Since 41edo has a perfect fifth which is split exactly in half, semisharps and semiflats (as in [[Diatonic notation|neutral diatonic notation]]) can be used to notate it. A useful addition is ups and downs, which naturally reflect 41edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub".) | Since 41edo has a perfect fifth which is split exactly in half, semisharps and semiflats (as in [[Diatonic notation|neutral diatonic notation]]) can be used to notate it. A useful addition is ups and downs, which naturally reflect 41edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub"; preferable to Unque and also indicative of the flat tuning of 5/4 in the temperament.) | ||
== Practice == | == Practice == | ||
Revision as of 01:42, 29 May 2026
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
JI approximation
41edo is most accurately a 2.3.5.7.11 tuning, though it also has an acceptable if sharp 13th harmonic, notably widening the difference between the arto (10:13:15) and tendo (1/10:1/13:1/15) triads such that they become simple Slendric divisions of the fifth. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 7-6-4-7-6-7-4. However, it also features a MOS diatonic of 7-7-3-7-7-7-3.
| 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) | |
| Quality | Subminor | Farminor | Nearminor | Neutral | Nearmajor | Farmajor | Supermajor |
|---|---|---|---|---|---|---|---|
| Cents | 263 | 293 | 322 | 351 | 381 | 410 | 439 |
| Just interpretation | 7/6 | 13/11 | 6/5 | 11/9 | 5/4 | 14/11 | 9/7 |
| Steps | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Thirds available in the diatonic scale generated by stacking the perfect fifth are bolded.
Edostep interpretations
One step of 41edo represents the following ratios of 2.3.5.7.11.13.19:
- 81/80 (between 5/4 and 81/64)
- 49/48 (between 7/6 and 8/7)
- 64/63 (between 7/4 and 16/9)
- 65/64 (between 16/13 and 5/4)
- 66/65 (between 13/11 and 6/5)
- 55/54 (between 27/20 and 11/8)
- 56/55 (between 5/4 and 14/11)
- 57/56 (between 7/6 and 19/16)
- 91/90 (between 9/7 and 13/10)
- 96/95 (between 19/16 and 6/5)
- 99/98 (between 14/11 and 9/7)
- 121/120 (between 12/11 and 11/10)
- 169/168 (between 14/13 and 13/12)
Two steps of 41edo represent the following ratios:
- 28/27 (between 9/8 and 7/6)
- 33/32 (between 4/3 and 11/8)
- 36/35 (between 7/4 and 9/5)
- 39/38 (between 19/16 and 39/16, or 19/15 and 13/10)
Derivation
41edo is derived by tempering out neither 81/80 nor 64/63, but rather setting them equal, and using the same step to separate 5/4 from a neutral third and 8/7 from 7/6.
Chords
41edo has three different flavors of minor and major intervals as well as neutral intervals. Its subminor and supermajor intervals approximate simpler septimal ratios such as 7/4 and 9/7, while its supraminor ("nearminor") and submajor ("nearmajor") 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 a result, 41edo has nine qualities of tertian, fifth-bounded triad: tendo, supermajor, novamajor, nearmajor, neutral, nearminor, novaminor, subminor. However, 41edo lacks true interordinal intervals (to reach a tuning with both neutrals and interordinals, 58edo must be used), so as for latal fourth-bounded triads, there are only four qualities.
Scales
41edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being an augmented second and 5/4 a diminished fourth. Notably, 41edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-1-3-3-1-3-3-3-1-3-3-1-3-3-1, in which the classical (~5/4) major and classical (~6/5) minor thirds span the same number of scale steps, giving the scale familiar chord qualities.
11-limit penslen is available in 41edo (because of Slendric and 385/384 tempering): 2-1-5-1-2-5-1-2-5-1-2-1-5-2-1-5.
Regular temperaments
41edo shares Schismic (and its extension Garibaldi, and thus Marvel and Hemifamity) with 29edo, Slendric (and its extension Miracle) with 31edo, Tetracot with 34edo, and Magic with 22edo. 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.
Notation
Since 41edo has a perfect fifth which is split exactly in half, semisharps and semiflats (as in neutral diatonic notation) can be used to notate it. A useful addition is ups and downs, which naturally reflect 41edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub"; preferable to Unque and also indicative of the flat tuning of 5/4 in the temperament.)
Practice
41edo is used by the musician and conlanger Lamplight as a standard tuning for their Shasavic theory of music.
Kite's work
There is a large amount of evidence suggesting that Kite's body of work is implicitly designed with 41edo in mind.
Most notably, the Kite Guitar itself uses 41edo, and is advertised as a general-purpose just-intonation-esque instrument, notably treating 41edo implicitly as the solution to compromising high-accuracy JI with equal temperament. The Kite Guitar manages the high density of notes by only having frets for every other note of 41edo, and tuning the strings in a way that allows one string to fill in the gaps of an adjacent string.
On another note, the color notation system seems specifically well-designed for 41edo, as a tuning of tetracot with high accuracy in the 2.3.5.7.11 subgroup. Colors can unambiguously be assigned to 41edo interval qualities without any of the extra overhead that comes with managing color notation for just intonation or larger edos, but while feeling like it uses the notation to its full potential. The qualities in order, from subminor to supermajor, are zo (7/), minor wa (3), gu (/5), (i)la (11), yo (5/), major wa (3), and ru (/7). This also serves to explain the unusual nature of "la" as "lavender", with the 11/ and /11 colors conflated due to the fact that in 41edo, rastmic tempering is assumed.
Additionally, ups and downs seems to work particularly well for the combination of undecimal garibaldi, slendric, and rastmic tempering, which is a characteristic of specifically 41edo, due to allowing the arrow to refer to a large number of different structural commas that are in the size range.
