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

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 and tendo 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.

Approximation of prime harmonics in 41edo

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)

Chords

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 supraminor ("pentaminor") and submajor ("pentamajor") 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, pentamajor, neutral, pentaminor, novaminor, subminor. However, 41edo lacks true interseptimal intervals (to reach a tuning with both neutrals and interseptimals, 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, which features a mosdegree representing both the 5-limit major and minor thirds, and thus has familiar chord qualities.

Regular temperaments

41edo shares schismic (and its extension garibaldi, and thus marvel and hemifamity) with 29edo, slendric (and its extension miracle) with 31edo, and tetracot with 34edo, and magic with 22edo. Magic is especially important here as it forms the fret layout and main string tuning for the most popular 41edo skip-fretting system.

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".)

Practice

41edo is used by the musician and conlanger Lamplight as a standard tuning for their Shasavic theory of music.