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218edo: Difference between revisions

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128 and 90 are the patent val. Patent val works by combining nearest match for primes, and neither 127 or 91 are nearest. 37 is not patent val of 9 because 128+128-218=38.
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'''218edo''', having a step size of 5.50458715596 cents, contains very accurate ratios, such as 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10 and 17/16, and its step size can be approximated as 5.5 cents.
{{Infobox ET}}
{{ED intro}}


'''Bold numbers are off within less than 0.1 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"
Commas using the [[13-limit]] patent val:
|-
| | 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'''
|}


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


Suggested subgroups: 2.9.7.17 and 2.9.5.7.11.17.
; [[7-limit]]: 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640


Also explore [[436edo]].
; [[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,


Commas using the 13-limit patent val:
; [[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


3-limit: 1/1
=== Odd harmonics ===
{{Harmonics in equal|218}}


5-limit: 20000/19683
=== Subsets and supersets ===
 
Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring.
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
 
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:Stub]]
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