321edo

← 320edo

321edo

322edo →

Prime factorization	3 × 107
Step size	3.73832 ¢
Fifth	188\321 (702.804 ¢)
Semitones (A1:m2)	32:23 (119.6 ¢ : 85.98 ¢)
Consistency limit	3
Distinct consistency limit	3

321 equal divisions of the octave (abbreviated 321edo or 321ed2), also called 321-tone equal temperament (321tet) or 321 equal temperament (321et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 321 equal parts of about 3.74 ¢ each. Each step represents a frequency ratio of 2^1/321, or the 321st root of 2.

Theory

321edo is inconsistent in the 5-odd-limit. The patent val tempers out 2401/2400, 5120/5103 and 10976/10935 in the 7-limit, supporting hemififths. In the 11-limit it tempers out 385/384 and 1375/1372, and in the 13-limit 325/324, 352/351, 847/845, 2080/2079 and 4096/4095, providing the optimal patent val for 11- and 13-limit akea temperament.

Prime harmonics

Approximation of prime harmonics in 321edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.85	-1.27	-0.60	-1.79	+0.59	-0.28	+1.55	-0.24	-1.54	-1.11
	Relative (%)	+0.0	+22.7	-33.9	-16.1	-47.8	+15.9	-7.6	+41.5	-6.3	-41.2	-29.7
Steps (reduced)		321 (0)	509 (188)	745 (103)	901 (259)	1110 (147)	1188 (225)	1312 (28)	1364 (80)	1452 (168)	1559 (275)	1590 (306)

Interval table

Main article: Table of 321edo intervals

Scales

JUMBLE's Blastoff scale (9L 8s)

• 25\321
• 37\321
• 62\321
• 74\321
• 99\321
• 111\321
• 136\321
• 148\321
• 173\321
• 185\321
• 210\321
• 222\321
• 247\321
• 259\321
• 284\321
• 296\321
• 321\321

This scale may be thought of as a tuning of the secundly (or lower limit secund) temperament.

Music

JUMBLE

• Sun Through The Window (2023) – ambient
• BLASTOFF! (2023) – synthwave, 9L 8s scale
• Infinity (2023)
• Greige City (2023) – synthwave, 9L 8s scale
• So Long, San Diego (2024)
• Ten Day Forecast (2024) – ambient

No Clue Music

• Frozen Star (2024)