34edo

The structure of the 5-limit in 34et, visualized.

34edo is the equal tuning system which splits the octave into 34 equal steps, of about (1200/34) ~= 35.3 cents each. It is a 5-limit and 2.3.5.13 system with a number of melodically intuitive structures.

34edo is the first hemipythagorean edo after 24edo.

Derivation

From doubling 17edo

One can observe that 17edo's step is a nearly perfectly tuned 25/24, and also that 5/4 and 6/5 are almost exactly halfway in-between notes of 17edo. Thus, 17edo can be doubled to improve the tunings of 5-limit intervals.

From the usage of Pythagorean diatonic semitones as classical chromatic semitones

Forcing 17edo's near-just 25/24, which is a Pythagorean diatonic semitone, to surround a neutral third and function as a chromatic semitone, requires offsetting the chain of fifths by a perfect semioctave, effectively allowing one to 'swap' the tunings of diatonic and chromatic semitones. This results in the 34edo tuning of Diaschismic.

From the DKW step sizes

DKW theory suggests that the core step sizes of the 5-limit are S3 = 9/8, S4 = 16/15, and S5 = 25/24. It can be observed that 16/15 stacks twice to approximate 9/8, and that 25/24 stacks 3 times to approximate 9/8. Tempering these equivalences together results in 34edo. Because the latter (kleismic) equalizes 24:25:26:27, and the former (diaschismic) equalizes 15:16:17:18, 34edo can be seen as a 2.3.5.13.17 system. 34edo can, thus, be broken up as 6-3-2-3-6-3-2-3-6, with 6 representing 9/8, 3 representing 16/15, and 2 representing 25/24. (In the Delkian system, the 9/8 intervals are further split into two 16/15s, and in the Roklotian, the 9/8 intervals are split into three 25/24s.)

Theory

Edostep interpretations

34edo's edostep, the sextula, has the following interpretations in the 2.3.5.13.17 subgroup, and equal divisions thereof:

• 81/80, the difference between the fifth-generated major third and the classical major third
• 128/125, the difference between the 5-limit enharmonic intervals
• 40/39, the difference between 13/10 and 4/3 and between 15/13 and 9/8; also between 6/5 and 16/13, among others
• 65/64, the difference between 8/5 and 13/8
• One half of 25/24, the difference between 5/4 and 6/5, such that the two are inflected from the neutral third by a sextula
• One sixth of 9/8 (from whence derives the name sextula)

TODO: add to list

JI approximation

34edo is straddle-7, straddle-11, and straddle-19, but has good accuracy on the 2.3.5.13.17.23 subgroup. 34edo inherits 17edo's mosdiatonic scale, 6-6-2-6-6-6-2, with the "optimally tuned" leading tone approximating 25/24. It also supports the zarlino scale, but because it does not support Porcupine, the zarlino scale requires 2 sets of accidentals to notate, making it awkward to use as the basis of notation. (The best option is to use 5-sharp and 5-flat accidentals from 17edo's diatonic, as if you are notating 5-limit JI, which may in 34edo be represented as ups and downs.)

Approximation of prime harmonics in 34edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+3.9	+1.9	-15.9	+13.4	+6.5	+0.9	-15.2	+7.0	-6.0	-15.6
	Relative (%)	0.0	+11.1	+5.4	-45.0	+37.9	+18.5	+2.6	-43.0	+19.9	-17.1	-44.3
Steps (reduced)		34 (0)	54 (20)	79 (11)	95 (27)	118 (16)	126 (24)	139 (3)	144 (8)	154 (18)	165 (29)	168 (32)

Thirds in 34edo

Quality	Inframinor	Farminor	Nearminor	Neutral	Nearmajor	Farmajor	Ultramajor
Cents	247	282	318	353	388	424	459
Just interpretation	15/13	20/17	6/5	16/13	5/4	32/25	13/10

MOS diatonic thirds are bolded.

Chords

34edo supports arto and tendo theory with its inframinor and ultramajor thirds. Being a Diaschismic edo, it has a series of tetrads wherein the third and seventh are separated by 600 cents, but due to not supporting Pajara, these do not approximate simple 7-limit chords.

Scales

34edo contains 17edo's diatonic scale and alongside it a zarlino scale. Other scales it includes are:

• the blackdye scale, with steps 1-5-3-5-1-5-3-5-1-5 (sLmLsLmLsL)
• the 5-odd-limit tonality diamond 9-2-3-6-3-2-9 (LsmMmsL)
  • the 5-limit DKW signature, 6-3-2-3-6-3-2-3-6 (LmsmLmsmL)
• Diaschismic[12], with steps 3-3-3-3-3-2-3-3-3-3-3-2 (LLLLLsLLLLLs)
  • the Delkian scale, a MODMOS of diaschismic[12] with steps 3-3-3-2-3-3-3-3-3-3-2-3 (LLLsLLLLLLsL)
• the hemipythagorean decatonic, with steps 3-4-3-4-3-3-4-3-4-3 (sLsLssLsLs)
  • the Sixanian scale, a MODMOS of the above with steps 3-4-3-4-3-3-3-4-3-4 (sLsLsssLsL)
• the Roklotian scale, 2-2-2-3-2-3-2-2-2-3-2-3-2-2-2 (sssLsLsssLsLsss)
• the MOS pentatonic, pythagorean[5], 6-8-6-8-6 (sLsLs)
• the equable pentatonic, Semaphore[5], 7-7-6-7-7 (LLsLL)
• the vertical pentatonic, 5-9-6-5-9 (sLmsL)

Additional regular temperaments

Alongside Kleismic (shared with 15edo and 19edo), and Diaschismic (shared with 12edo), 34edo supports the following temperaments:

• Tetracot (splitting 3/2 into four 10/9s), shared with 41edo and 27edo
• Gammic (setting 25/24 to a tenth of 3/2), shared with 103edo and 171edo

Notation

Notation

34edo may use ups and downs notation. It may also use the diaschismic notation detailed on 22edo's page. Because it splits the perfect fifth exactly in half, it may use neutral diatonic notation for a 17edo subset, relying on ups and downs to notate the rest. Note that 5/4 is simultaneously an upneutral 3rd and a downmajor 3rd; while downmajor may seem like the obvious choice, 34edo's emphasis on dividing intervals and its tendency towards the 10-form as a sharp hemipythagorean edo makes it somewhat logical to compare the 5-limit thirds to the neutral third instead, especially if it is understood that the semisharp represents a 25/24 semitone or one third of a whole tone.

Multiples

68edo

68edo is the double of 34edo, and improves its mapping of 7 much as 34edo improves 17edo's mapping of 5. This improves the mappings of 11 and 19 as well, making 68edo function as a general 19-limit system. While its 13 is inaccurate in relative error, it is 34edo's kleismic tuning of 13 and shares all the structural properties thereof, which justifies it. Due to its sharp fifth, the septimal comma 64/63 is mapped to one step while 81/80 is two steps, thus mapping the aberschisma to a negative step, and making 68edo a Bidic temperament.

The new 7/4 supports Sensamagic, doubling 9/7 to reach 5/3, and 2.5.7 Didacus, splitting 5/4 into two wholetones that stack 5 times to reach 7/4. Additionally, the new 11/8 makes 14/11 equal to 81/64, supporting Pentacircle (and various gentle/neogothic temperaments).

To notate 68edo, there is a rather neat scheme wherein semisharps and semiflats represent 17edo steps, ups and downs represent 34edo steps, and lifts and drops represent 68edo steps.

Approximation of prime harmonics in 68edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+3.9	+1.9	+1.8	-4.3	+6.5	+0.9	+2.5	+7.0	-6.0	+2.0
	Relative (%)	0.0	+22.3	+10.9	+10.0	-24.1	+37.0	+5.3	+14.1	+39.8	-34.3	+11.5
Steps (reduced)		68 (0)	108 (40)	158 (22)	191 (55)	235 (31)	252 (48)	278 (6)	289 (17)	308 (36)	330 (58)	337 (65)

306edo

306 is the decominator of a continued fraction convergent to log₂(3/2), and as such 306edo has a nearly perfectly accurate 3/2 representation. Its step is the difference between 34edo's 3/2 and the near-just one. It also has a 7/4 accurate to within 0.2 cents.

Approximation of prime harmonics in 306edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+0.0	+1.9	-0.2	+1.6	-1.3	+0.9	+0.5	-0.8	+1.8	+0.1
	Relative (%)	0.0	+0.1	+49.0	-5.1	+41.4	-33.5	+23.6	+13.4	-21.0	+45.8	+1.6
Steps (reduced)		306 (0)	485 (179)	711 (99)	859 (247)	1059 (141)	1132 (214)	1251 (27)	1300 (76)	1384 (160)	1487 (263)	1516 (292)

612edo

612edo doubles 306edo, adding the perfect fifth from 12edo and a nearly perfect 5/4. Its main utility is as a fine-grained interval size measurement system for the 11-limit, wherein 3/2 is 358 steps and 5/4 is 197 steps, as its step size is almost exactly a (consistently represented) schisma. The 12edo perfect fifth is 357 steps, 34edo's is 360 steps. Thus, 34edo is about twice as inaccurate as 12edo in its tuning of 3/2.

Approximation of prime harmonics in 612edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+0.0	-0.0	-0.2	-0.3	+0.6	+0.9	+0.5	-0.8	-0.2	+0.1
	Relative (%)	0.0	+0.3	-2.0	-10.1	-17.2	+33.1	+47.3	+26.8	-42.0	-8.4	+3.2
Steps (reduced)		612 (0)	970 (358)	1421 (197)	1718 (494)	2117 (281)	2265 (429)	2502 (54)	2600 (152)	2768 (320)	2973 (525)	3032 (584)