198edo: Difference between revisions
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'''198 equal temperament''' divides the octave into 198 parts of 6.061 cents each | '''198 equal temperament''' divides the octave into 198 parts of 6.061 cents each. | ||
It is the [[optimal patent val]] for the rank five temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[Misty family #Hemimist|hemimist]], [[Hemifamity family #Namaka|namaka]] and [[Canou family #Semicanou|semicanou]]. It is distinctly [[consistent]] through the 15-limit, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99. | == Theory == | ||
198edo is contorted in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and 6144/6125 in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] and [[14641/14580]]; and in the [[13-limit]] [[352/351]], 676/675, 847/845, 1001/1000, 1716/1715 and 2080/2079. | |||
It is the [[optimal patent val]] for the rank five temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[Misty family #Hemimist|hemimist]], [[Hemifamity family #Namaka|namaka]] and [[Canou family #Semicanou|semicanou]]. It is distinctly [[consistent]] through the [[15-odd-limit]], and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99. | |||
== Intervals == | |||
{{main|Table of 198edo intervals}} | |||
== Just approximation == | == Just approximation == | ||
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[[Category:Edo]] | [[Category:Edo]] | ||
[[Category:99edo]] | [[Category:99edo]] | ||
[[Category:198edo]] | |||
[[Category:Minthmic]] | [[Category:Minthmic]] | ||
[[Category:Misty]] | [[Category:Misty]] | ||
Revision as of 10:35, 10 November 2020
198 equal temperament divides the octave into 198 parts of 6.061 cents each.
Theory
198edo is contorted in the 7-limit, with the same tuning as 99edo, but makes for a good 11- and 13-limit system. Like 99, it tempers out 2401/2400, 4375/4374, 3136/3125, 5120/5103 and 6144/6125 in the 7-limit; in the 11-limit it tempers 3025/3024, 9801/9800 and 14641/14580; and in the 13-limit 352/351, 676/675, 847/845, 1001/1000, 1716/1715 and 2080/2079.
It is the optimal patent val for the rank five temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as hemimist, namaka and semicanou. It is distinctly consistent through the 15-odd-limit, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
Intervals
Just approximation
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | prime 29 | prime 31 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.00 | +1.08 | +1.57 | +0.87 | +0.20 | +1.90 | -1.93 | -0.54 | +2.03 | +0.73 | +0.42 |
| relative (%) | 0.0 | +17.7 | +25.8 | +14.4 | +3.3 | +31.3 | -31.8 | -9.0 | +33.5 | +12.0 | +6.9 | |