218edo: Difference between revisions

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Corrected synopsis. To talk of a deviation limit of 5.5 cents is pointless in this edo.
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'''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).
{{Infobox ET}}
{{ED 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:


; [[3-limit]]: 1/1
; [[5-limit]]: 20000/19683, 1220703125/1207959552


; [[5-limit]]: 20000/19683
; [[7-limit]]: 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640


; [[7-limit]]: 4000/3969 65625/65536 245/243 2401/2400 60025/59049
; [[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,


; [[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


; [[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 ===
{{Harmonics in equal|218}}


[[Category:Stub]]
=== Subsets and supersets ===
[[Category:Edo]]
Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring.

Latest revision as of 14:17, 20 February 2025

← 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

218 equal divisions of the octave (abbreviated 218edo or 218ed2), also called 218-tone equal temperament (218tet) or 218 equal temperament (218et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 218 equal parts of about 5.5 ¢ each. Each step represents a frequency ratio of 21/218, or the 218th root of 2.

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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