146edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''146edo''' divides the octave into 146 equal parts of 8.219178 [[cent]]s each. It has an accurate major third, only 0.012344 cents compressed from just [[5/4]] interval. 146edo is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]]. However, it also provides the optimal patent val for the 11-limit [[Semicomma family|newspeak temperament]]. It tempers out the [[semicomma]], 2109375/2097152 and 129140163/125000000 in the 5-limit; 225/224, 1728/1715, and 100442349/97656250 in the 7-limit; 441/440, 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; 1001/1000, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.
{{EDO intro}} 146edo has an accurate major third, only 0.012344 cents compressed from just [[5/4]] interval. 146edo is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]].  


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
However, it also provides the [[optimal patent val]] for the 11-limit [[newspeak]] temperament. It tempers out the 2109375/2097152 ([[semicomma]]), and {{monzo| -6 17 -9 }} in the 5-limit; [[225/224]], [[1728/1715]], and 100442349/97656250 in the 7-limit; [[441/440]], 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; [[1001/1000]], [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit.
 
=== Odd harmonics ===
{{Harmonics in equal|146}}
 
=== Subsets and supersets ===
Since 146 factors into {{factorization|146}}, 146edo contains [[2edo]] and [[73edo]] as its subsets.

Revision as of 15:00, 18 May 2024

← 145edo 146edo 147edo →
Prime factorization 2 × 73
Step size 8.21918 ¢ 
Fifth 85\146 (698.63 ¢)
Semitones (A1:m2) 11:13 (90.41 ¢ : 106.8 ¢)
Dual sharp fifth 86\146 (706.849 ¢) (→ 43\73)
Dual flat fifth 85\146 (698.63 ¢)
Dual major 2nd 25\146 (205.479 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro 146edo has an accurate major third, only 0.012344 cents compressed from just 5/4 interval. 146edo is the denominator of a convergent to log25, after 3, 28 and 59, and before 643.

However, it also provides the optimal patent val for the 11-limit newspeak temperament. It tempers out the 2109375/2097152 (semicomma), and [-6 17 -9⟩ in the 5-limit; 225/224, 1728/1715, and 100442349/97656250 in the 7-limit; 441/440, 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; 1001/1000, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 146edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.32 -0.01 +1.04 +1.57 -0.63 -2.17 -3.34 +1.89 -1.62 -2.29 -3.62
Relative (%) -40.5 -0.2 +12.6 +19.1 -7.7 -26.4 -40.6 +23.0 -19.7 -27.8 -44.0
Steps
(reduced) 		231
(85) 		339
(47) 		410
(118) 		463
(25) 		505
(67) 		540
(102) 		570
(132) 		597
(13) 		620
(36) 		641
(57) 		660
(76)

Subsets and supersets

Since 146 factors into 2 × 73, 146edo contains 2edo and 73edo as its subsets.

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