1553edo

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1553edo

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Prime factorization	1553 (prime)
Step size	0.772698¢
Fifth	908\1553 (701.61¢)
Semitones (A1:m2)	144:119 (111.3¢ : 91.95¢)
Dual sharp fifth	909\1553 (702.382¢)
Dual flat fifth	908\1553 (701.61¢)
Dual major 2nd	264\1553 (203.992¢)
Consistency limit	5
Distinct consistency limit	5

1553 equal divisions of the octave (abbreviated 1553edo or 1553ed2), also called 1553-tone equal temperament (1553tet) or 1553 equal temperament (1553et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1553 equal parts of about 0.773 ¢ each. Each step represents a frequency ratio of 2^1/1553, or the 1553rd root of 2.

Theory

1553edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.13 subgroup, where it notably tempers out 4096/4095 and 140625/140608.

Odd harmonics

Approximation of odd harmonics in 1553edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-0.345	+0.035	+0.137	+0.082	-0.384	+0.168	-0.310	+0.132	-0.024	-0.208	-0.071
	Relative (%)	-44.7	+4.6	+17.8	+10.6	-49.7	+21.7	-40.1	+17.0	-3.1	-26.9	-9.2
Steps (reduced)		2461 (908)	3606 (500)	4360 (1254)	4923 (264)	5372 (713)	5747 (1088)	6067 (1408)	6348 (136)	6597 (385)	6821 (609)	7025 (813)

Subsets and supersets

1553edo is the 245th prime edo. 3106edo, which doubles it, provides a good correction to the harmonic 3.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.9	[4923 -1553⟩	[⟨1553 4923]]	-0.0130	0.0130	1.68
2.9.5	[93 -33 5⟩, [-36 -26 51⟩	[⟨1553 4923 3606]]	-0.0137	0.0106	1.38
2.9.5.7	[-5 5 5 -8⟩, [2 -10 14 -1⟩, [37 1 -4 -11⟩	[⟨1553 4923 3606 4360]]	-0.0225	0.0178	2.31
2.9.5.7.13	4096/4095, 140625/140608, 28829034/28824005, [4 10 -9 0 -4⟩	[⟨1553 4923 3606 4360 5372]]	-0.0271	0.0184	2.38

Music

Francium

• Stumbling Over Mystery (2023)