1ed28.7c

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1ed31444/30927

2ed31444/30927 →

Prime factorization	n/a
Step size	28.7014¢
Octave	42\1ed31444/30927 (1205.46¢)
Twelfth	66\1ed31444/30927 (1894.29¢)
Consistency limit	3
Distinct consistency limit	1
Special properties	Highly composite

Arithmetic pitch sequence of 720jot (APS720jot), also known as 1 equal division of 720jot (1ed720jot) or 720-jot equal temperament (720jot-et), is an equal-step scale with steps of 720 jots (~28.7015 cents), or each 720th step of 30103edo.

Although it is not exactly 28.7 cents, it is close enough that it can reasonably be referred to as 1 equal division of 28.7 cents (1ed28.7c). This title is used for ease of navigation and comparison with other arithmetic pitch sequences on the wiki.

Theory

APS720jot divides 2/1 into ~41.8097 equal parts, making it close to 42edo, but with impure octaves and better approximations of harmonics 3/1 and 5/1.

It is a kind of opposite twin to the scale APS715jot, as they improve 42edo’s JI approximation by about the same amount, but in opposite directions (those harmonics which are slightly sharp in one are slightly flat in the other).

42ed257/128’s step size is very close to that of 42ed257/128 and 189zpi.

See Table of stretched 42edo tunings for more.

Harmonics

APS720jot has less than 40% relative error on every single prime harmonic from 2 up to 47, making it a suitable full 47-limit tuning.

No other stretched or compressed version of 42edo has that property.

Harmonics in APS720jot

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53
Error	Absolute (¢)	+5.5	-7.7	-2.3	-10.8	+10.4	+8.2	+3.0	+11.3	-3.7	-3.2	-3.8	+5.6	+0.1	+3.7	-6.8	-13.9
	Relative (%)	+19.0	-26.7	-7.9	-37.5	+36.2	+28.6	+10.4	+39.5	-12.9	-11.1	-13.4	+19.4	+0.2	+12.9	-23.6	-48.3
Steps		42	66	97	117	145	155	171	178	189	203	207	218	224	227	232	239

42edo for comparison:

Harmonics in 42edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53
Error	Absolute (¢)	+0.0	+12.3	+13.7	+2.6	-8.5	-12.0	+9.3	-11.8	+0.3	-1.0	-2.2	+5.8	-0.5	+2.8	-8.4	+12.2
	Relative (%)	+0.0	+43.2	+47.9	+9.1	-29.6	-41.8	+32.7	-41.3	+1.0	-3.5	-7.6	+20.3	-1.7	+9.7	-29.3	+42.7
Steps (reduced)		42 (0)	67 (25)	98 (14)	118 (34)	145 (19)	155 (29)	172 (4)	178 (10)	190 (22)	204 (36)	208 (40)	219 (9)	225 (15)	228 (18)	233 (23)	241 (31)

Notation

APS720jot can use most notation systems designed for 42edo. See 42edo#Notation.

Scala file

Tuning file for anything that supports Scala. Made with Scale Workshop.

! aps720jot.scl
! Created using Scale Workshop 3.0.1
!
! https://scaleworkshop.plainsound.org/scale/M7mh0KX7e
!
Arithmetic pitch sequence of 720jot
 42
!
 28.701432
 57.402864
 86.104296
 114.805728
 143.507160
 172.208592
 200.910024
 229.611456
 258.312888
 287.014320
 315.715752
 344.417184
 373.118616
 401.820048
 430.521480
 459.222912
 487.924344
 516.625776
 545.327208
 574.028640
 602.730072
 631.431504
 660.132936
 688.834368
 717.535800
 746.237232
 774.938664
 803.640096
 832.341528
 861.042960
 889.744392
 918.445824
 947.147256
 975.848688
 1004.550120
 1033.251552
 1061.952984
 1090.654416
 1119.355848
 1148.057280
 1176.758712
 1205.460144

Scales

To generate these in Scale Workshop, you can first generate them in 42edo, then apply a stretch of 1.00455012.

MOS scales

• Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
• Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
• Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3
• Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2
• Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2

Subsets of MOS scales

(Names used are idiosyncratic.)

• Eugene/Tritikleismic[9]
  • Groovy aeolian pentatonic: 11 6 8 3 14
  • Otonal mixolydian pentatonic: 14 3 8 11 6
  • Pseudo-equipentatonic: 11 6 8 6 11
  • Septimal melodic minor pentatonic: 8 3 14 14 3
  • Septimal Picardy pentatonic: 8 6 11 3 14
  • Undecimal lydian-aeolian pentatonic: 8 14 3 11 6
  • Yokai pentatonic: 3 14 8 3 14

Tonality diamonds

These take advantage of APS720jot’s improved approximations of the full 47-limit compared to 42edo, especially its improved 3/1 and 5/1.

• Diamond7: 8 1 2 2 4 3 2 2 4 3 2 2 8
• Diamond9: 6 1 1 1 2 2 2 2 3 2 2 3 1 3 1 2 1 1 6

Combination product sets

These take advantage of APS720jot’s improved approximations of the full 47-limit compared to 42edo, especially its improved 3/1 and 5/1.

• Pentany(4 of 1,3,5,7,11): 6 13 8 10 5
• Pentany(4 of 5,17,23,29,37): 10 14 8 5 5
• Pentany(4 of 5,17,23,29,41): 10 14 7 2 9
• Pentany(4 of 5,17,23,31,41): 7 3 8 6 18
• Hexany(2 of 1,3,5,7): 10 3 10 8 3 8
• Hexany(2 of 5,17,23,31): 10 6 3 5 10 8 '

Instruments

• Lumatone
  • APS720jot can use most types of Lumatone mapping for 42edo

Music

• Four Short Experiments in Octave Stretched 42edo (Dec 2024) - Budjarn Lambeth

See also

• Gallery of arithmetic pitch sequences