218edo: Difference between revisions

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{{Infobox ET}}
'''218edo''', having a step size of 5.50458715596 [[cent]]s, contains very accurate ratios, such as [[7/4]], [[9/7]], [[8/7]], [[9/8]], [[10/9]], [[11/10]] and [[17/16]] which are approximated within 0.55¢ deviation (10% the step size).
{{EDO intro}}


The following table shows the nearest matches for the interval, not the matches from the [[patent val]]. '''Bold''' numbers are off within less than 0.1 (10%) of the step size.
218edo is in[[consistent]] to the [[5-odd-limit]], with [[harmonic]] [[3/1|3]] falling about halfway between its steps. However, it contains very accurate ratios, such as [[7/4]], [[9/7]], [[9/8]], [[10/9]], [[11/10]], [[17/16]], and [[19/16]], which are approximated within 0.55-cent deviation (10% the step size). The suggested [[subgroup]]s are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.
 
{| class="wikitable"
|-
! Interval fraction
| [[3/2]]
| [[4/3]]
| [[5/4]]
| [[8/5]]
| [[5/3]]
| [[6/5]]
| '''[[7/4]]'''
| '''[[8/7]]'''
| [[10/9]]
| [[9/5]]
| '''[[9/8]]'''
| '''[[16/9]]'''
|-
! Steps in 218edo
| 128
| 90
| 70
| 148
| 161
| 57
| '''176'''
| '''42'''
| 33
| 185
| '''37'''
| '''181'''
|}
 
Suggested [[subgroup]]s: 2.9.7.17 and 2.9.5.7.11.17.
 
Also explore [[436edo]].


Commas using the [[13-limit]] patent val:
Commas using the [[13-limit]] patent val:
Line 49: Line 14:
; [[13-limit]]: 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512
; [[13-limit]]: 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== Odd harmonics ===
{{Harmonics in equal|218}}
 
=== Subsets and supersets ===
Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring.

Revision as of 07:27, 2 April 2024

← 217edo 218edo 219edo →
Prime factorization 2 × 109
Step size 5.50459 ¢ 
Fifth 128\218 (704.587 ¢) (→ 64\109)
Semitones (A1:m2) 24:14 (132.1 ¢ : 77.06 ¢)
Dual sharp fifth 128\218 (704.587 ¢) (→ 64\109)
Dual flat fifth 127\218 (699.083 ¢)
Dual major 2nd 37\218 (203.67 ¢)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

218edo is inconsistent to the 5-odd-limit, with harmonic 3 falling about halfway between its steps. However, it contains very accurate ratios, such as 7/4, 9/7, 9/8, 10/9, 11/10, 17/16, and 19/16, which are approximated within 0.55-cent deviation (10% the step size). The suggested subgroups are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.

Commas using the 13-limit patent val:

5-limit
20000/19683, 1220703125/1207959552
7-limit
4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640
11-limit
4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576,
13-limit
28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512

Odd harmonics

Approximation of odd harmonics in 218edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.63 -0.99 -0.02 -0.24 -0.86 +1.67 +1.64 -0.37 -0.27 +2.61 -0.75
Relative (%) +47.8 -18.0 -0.3 -4.4 -15.6 +30.4 +29.8 -6.7 -4.8 +47.5 -13.7
Steps
(reduced) 		346
(128) 		506
(70) 		612
(176) 		691
(37) 		754
(100) 		807
(153) 		852
(198) 		891
(19) 		926
(54) 		958
(86) 		986
(114)

Subsets and supersets

Since 218 factors into 2 × 109, 218edo contains 2edo and 109edo as its subsets. 436edo, which doubles it, is worth exploring.

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