2901533edo: Difference between revisions

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m why remove unbiased original research in favour of wrong information? the note at the top states it's "a topic of primarily mathematical interest", and its tuning properties w.r.t very large odd-limits are unparalleled afaik
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m readd info that was missing; note that distinct consistency
 
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{{ED intro}}
{{ED intro}}


Except for 8 barely in[[consistent]] interval pairs, 2901533edo is consistent in the 137-prime-limited no-247's 255-odd-limit (a total of 4067 interval pairs), with primes 151, 157, 163, 173, 181, 197 and 211 being includeable to that odd limit for a tiny penalty of only 3 more barely-inconsistent interval pairs (and for a total of 4830). Including odd 247 adds 8 more inconsistent interval pairs and 90 more consistent interval pairs for a total of 4928 interval pairs (of which 19 interval pairs are inconsistent). Because of its unusual [[consistency]] at its size range, it could be a candidate for "miracle edo" (not [[miracle]], the temperament) after [[311edo]], although this is not entirely certain or clear because a deep exhaustive search of comprehensive odd-limit performance has not been done up until this point, but it is at least significant that it holds a significant amount of records for [[odd limit]] [[consistency]] as detailed on the page for [[minimal consistent edos]]. Furthermore, it is consistent up to the [[odd prime sum limit|25-OPSL]], and is [[Consistency #Consistency to distance d|consistent to distance 4]] in the 16-OPSL.
As noted in the page on [[minimal consistent EDOs]], 2901533edo is the smallest edo to be [[consistent]] to every odd-limit from 79 to 131. Due to its extremely small step size, distinct consistency is a given, and its tuning properties are in fact a lot more exceptional: Except for 8 barely in[[consistent]] interval pairs, 2901533edo is consistent in the 137-prime-limited no-247's 255-odd-limit (a total of 4067 interval pairs), with primes 151, 157, 163, 173, 181, 197 and 211 being includable to that odd limit for a tiny penalty of only 3 more barely-inconsistent interval pairs (and for a total of 4830). Including odd 247 adds 8 more inconsistent interval pairs and 90 more consistent interval pairs for a total of 4928 interval pairs (of which 19 interval pairs are inconsistent). Because of its unusual [[consistency]] at its size range, it could be a candidate for "miracle edo" (not [[miracle]], the temperament) after [[311edo]], although this is not entirely certain or clear because a deep exhaustive search of comprehensive odd-limit performance has not been done up until this point, but it is at least significant that it holds a significant amount of records for [[odd limit]] [[consistency]] as detailed on the page for [[minimal consistent edos]]. Furthermore, it is consistent up to the [[odd prime sum limit|25-OPSL]], and is [[Consistency #Consistency to distance d|consistent to distance 4]] in the 16-OPSL.


=== Prime harmonics ===
=== Prime harmonics ===

Latest revision as of 03:38, 24 March 2026

This page presents a topic of primarily mathematical interest.

While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

← 2901532edo 2901533edo 2901534edo →
Prime factorization 433 × 6701
Step size 0.000413574 ¢ 
Fifth 1697288\2901533 (701.955 ¢)
Semitones (A1:m2) 274884:218159 (113.7 ¢ : 90.22 ¢)
Consistency limit 131
Distinct consistency limit 131

2901533 equal divisions of the octave (abbreviated 2901533edo or 2901533ed2), also called 2901533-tone equal temperament (2901533tet) or 2901533 equal temperament (2901533et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2901533 equal parts of about 0.000414 ¢ each. Each step represents a frequency ratio of 21/2901533, or the 2901533rd root of 2.

As noted in the page on minimal consistent EDOs, 2901533edo is the smallest edo to be consistent to every odd-limit from 79 to 131. Due to its extremely small step size, distinct consistency is a given, and its tuning properties are in fact a lot more exceptional: Except for 8 barely inconsistent interval pairs, 2901533edo is consistent in the 137-prime-limited no-247's 255-odd-limit (a total of 4067 interval pairs), with primes 151, 157, 163, 173, 181, 197 and 211 being includable to that odd limit for a tiny penalty of only 3 more barely-inconsistent interval pairs (and for a total of 4830). Including odd 247 adds 8 more inconsistent interval pairs and 90 more consistent interval pairs for a total of 4928 interval pairs (of which 19 interval pairs are inconsistent). Because of its unusual consistency at its size range, it could be a candidate for "miracle edo" (not miracle, the temperament) after 311edo, although this is not entirely certain or clear because a deep exhaustive search of comprehensive odd-limit performance has not been done up until this point, but it is at least significant that it holds a significant amount of records for odd limit consistency as detailed on the page for minimal consistent edos. Furthermore, it is consistent up to the 25-OPSL, and is consistent to distance 4 in the 16-OPSL.

Prime harmonics

Approximation of prime harmonics in 2901533edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.000000 +0.000000 +0.000004 +0.000021 -0.000001 +0.000018 -0.000132 +0.000057 -0.000121 -0.000071 -0.000034 +0.000061
Relative (%) +0.0 +0.0 +0.9 +5.1 -0.3 +4.3 -32.0 +13.8 -29.3 -17.1 -8.3 +14.8
Steps
(reduced) 		2901533
(0) 		4598821
(1697288) 		6737151
(934085) 		8145633
(2342567) 		10037655
(1333056) 		10736948
(2032349) 		11859908
(253776) 		12325502
(719370) 		13125264
(1519132) 		14095592
(2489460) 		14374764
(2768632) 		15115401
(607736)
Approximation of prime harmonics in 2901533edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) +0.000025 -0.000104 +0.000060 -0.000091 +0.000027 -0.000041 +0.000014 -0.000086 -0.000092 +0.000056 -0.000118 -0.000103
Relative (%) +5.9 -25.3 +14.5 -22.0 +6.5 -9.9 +3.3 -20.9 -22.2 +13.4 -28.6 -24.9
Steps
(reduced) 		15545114
(1037449) 		15744486
(1236821) 		16116823
(1609158) 		16619750
(2112085) 		17068683
(2561018) 		17208230
(2700565) 		17600958
(191760) 		17843694
(434496) 		17959980
(550782) 		18290628
(881430) 		18497387
(1088189) 		18789554
(1380356)
Approximation of prime harmonics in 2901533edo (continued)
Harmonic 97 101 103 107 109 113 127 131 137 139 149 151
Error Absolute (¢) +0.000038 -0.000140 +0.000027 +0.000029 -0.000070 -0.000135 -0.000101 +0.000024 -0.000134 -0.000185 +0.000126 -0.000090
Relative (%) +9.1 -33.8 +6.6 +7.1 -16.8 -32.5 -24.5 +5.8 -32.4 -44.8 +30.5 -21.9
Steps
(reduced) 		19149865
(1740667) 		19319020
(1909822) 		19401102
(1991904) 		19560589
(2151391) 		19638110
(2228912) 		19788974
(2379776) 		20277899
(2868701) 		20407709
(96978) 		20595174
(284443) 		20655842
(345111) 		20946656
(635925) 		21002470
(691739)
Approximation of prime harmonics in 2901533edo (continued)
Harmonic 157 163 167 173 179 181 191 193 197 199 211 223
Error Absolute (¢) +0.000072 -0.000102 +0.000138 -0.000060 -0.000183 -0.000139 +0.000083 +0.000147 +0.000049 -0.000143 -0.000038 -0.000202
Relative (%) +17.5 -24.8 +33.3 -14.5 -44.4 -33.6 +20.0 +35.5 +11.8 -34.5 -9.2 -48.9
Steps
(reduced) 		21165583
(854852) 		21322577
(1011846) 		21424062
(1113331) 		21571819
(1261088) 		21714538
(1403807) 		21761050
(1450319) 		21986160
(1675429) 		22029765
(1719034) 		22115635
(1804904) 		22157918
(1847187) 		22403024
(2092293) 		22634568
(2323837)

Subsets and supersets

2901533 = 433 × 6701, so 2901533edo contains 433edo and 6701edo as subsets.

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