42ed257/128

← 41ed257/128

42ed257/128

43ed257/128 →

Prime factorization	2 × 3 × 7
Step size	28.7321¢
Octave	42\42ed257/128 (1206.75¢) (→1\1ed257/128)
Twelfth	66\42ed257/128 (1896.32¢) (→11\7ed257/128)
Consistency limit	3
Distinct consistency limit	1

42ed257/128 is the equal division of the interval 257/128 into forty-two parts of 28.73 cents each, corresponding to ~41.77edo.

It can be approached as a compressed-octaves version of 42edo, which improves most of 42edo’s JI approximations.

Harmonics

42ed257/128 is a kind of opposite twin to the scale 42ed255/128, 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 APS720jot and 189zpi.

See Table of stretched 42edo tunings for more.

Approximation of prime harmonics in 42ed257/128

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+6.7	-5.6	+0.7	-7.2	-13.9	+13.0	+8.2	-11.9	+2.1	+3.0	+2.5
	Relative (%)	+23.5	-19.6	+2.4	-24.9	-48.3	+45.1	+28.7	-41.5	+7.3	+10.6	+8.8
Steps (reduced)		42 (0)	66 (24)	97 (13)	117 (33)	144 (18)	155 (29)	171 (3)	177 (9)	189 (21)	203 (35)	207 (39)

42edo for comparison:

Approximation of prime harmonics in 42edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+12.3	+13.7	+2.6	-8.5	-12.0	+9.3	-11.8	+0.3	-1.0	-2.2
	Relative (%)	+0.0	+43.2	+47.9	+9.1	-29.6	-41.8	+32.7	-41.3	+1.0	-3.5	-7.6
Steps (reduced)		42 (0)	67 (25)	98 (14)	118 (34)	145 (19)	155 (29)	172 (4)	178 (10)	190 (22)	204 (36)	208 (40)

Notation

42ed257/128 can use most notation systems designed for 42edo. See 42edo#Notation.

Scala file

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

! 42ed257over128.scl
! Created using Scale Workshop 3.0.1
!
! https://scaleworkshop.plainsound.org/scale/H7mskiu00
!
42 equal divisions of 257/128
 42
!
 28.732130
 57.464260
 86.196390
 114.928520
 143.660650
 172.392780
 201.124910
 229.857040
 258.589170
 287.321300
 316.053430
 344.785560
 373.517690
 402.249820
 430.981950
 459.714080
 488.446210
 517.178340
 545.910470
 574.642600
 603.374730
 632.106859
 660.838989
 689.571119
 718.303249
 747.035379
 775.767509
 804.499639
 833.231769
 861.963899
 890.696029
 919.428159
 948.160289
 976.892419
 1005.624549
 1034.356679
 1063.088809
 1091.820939
 1120.553069
 1149.285199
 1178.017329
 1206.749459

Scales

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
• Seville/Sevond is not available because it is generated by 42edo’s sharp fifth, and 42ed257/128 is designed to improve 42edo’s flat fifth instead

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 42ed257/128’s improved approximations of the full 7-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

Instruments

• Lumatone
  • 42ed257/128 can use most types of Lumatone mapping for 42edo