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'''198 equal temperament''' divides the octave into 198 parts of 6.061 cents each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== 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]].
198edo is [[enfactoring|enfactored]] in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good [[11-limit|11-]] and [[13-limit]] system. It is [[consistency|distinctly consistent]] through the [[15-odd-limit]], and demonstrates a sharp tendency, with [[harmonic]]s 3 through 13 all tuned sharp.  


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]]. It factors into 2 × 3<sup>2</sup> × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
Like 99, it [[tempering out|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].  


=== Prime intervals ===
It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered chord]]s including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[island chords]] in the [[15-odd-limit]].
{{Primes in edo|198|columns=11}}
 
Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.
 
Extending it beyond the 13-limit can be tricky, as the approximated [[17/1|harmonic 17]] is almost 1/3-edostep flat of just, which does not blend well with the sharp tendency from the lower harmonics. The 198g val in turn gives you an alternative that is more than 2/3-edostep sharp. However, if we skip prime 17 altogether and treat 198edo as a no-17 [[23-limit]] system, it is almost consistent to the no-17 [[23-odd-limit]] with the sole exception of [[19/15]] and its [[octave complement]]. It tempers out [[361/360]] and [[456/455]] in the [[19-limit]], and [[484/483]] and [[576/575]] in the [[23-limit]]. Finally, the harmonics [[29/1|29]] and [[31/1|31]] are quite accurate, though the [[25/1|25]] and [[27/1|27]] are sharp enough to have incurred more inconsistencies.
 
The 198b val [[support]]s a [[septimal meantone]] close to the [[CTE tuning]], although [[229edo]] is even closer, and besides, the 198be val supports an undecimal meantone almost identical to the [[POTE tuning]].
 
=== Prime harmonics ===
{{Harmonics in equal|198}}
 
=== Octave stretch ===
198edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[512ed6]] or [[710ed12]]. This improves the approximated harmonics 3, 5, 7, 13, and 23; the 11 may become less accurate depending on the specific tuning. The 19 also gets worse on compression, so the compression should be very mild if the target is the no-17 23-limit.
 
=== Subsets and supersets ===
Since 198 factors into primes as {{nowrap| 2 × 3<sup>2</sup> × 11 }}, 198edo has subset edos {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.
 
A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s.


== Intervals ==
== Intervals ==
{{main|Table of 198edo intervals}}
{{Main| Table of 198edo intervals }}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 3136/3125, 4375/4374
| {{Mapping| 198 314 460 556 685 }}
| −0.344
| 0.291
| 4.80
|-
| 2.3.5.7.11.13
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| {{Mapping| 198 314 460 556 685 733 }}
| −0.372
| 0.273
| 4.50
|-
| 2.3.5.7.11.13.19
| 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330
| {{Mapping| 198 314 460 556 685 733 841 }}
| −0.301
| 0.307
| 5.07
|-
| 2.3.5.7.11.13.19.23
| 352/351, 361/360, 456/455, 484/483, 576/575, 676/675, 847/845
| {{Mapping| 198 314 460 556 685 733 841 896 }}
| −0.319
| 0.291
| 4.81
|}
* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]].
 
=== Rank-2 temperaments ===
Note: temperaments supported by 99et are not included.
 
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 7\198
| 42.42
| 40/39
| [[Humorous]]
|-
| 1
| 19\198
| 115.15
| 77/72
| [[Semigamera]]
|-
| 1
| 23\198
| 139.39
| 13/12
| [[Quasijerome]]
|-
| 1
| 65\198
| 393.93
| 49/39
| [[Hitch]]
|-
| 1
| 83\198
| 503.03
| 147/110
| [[Quadrawürschmidt]]
|-
| 2
| 14\198
| 84.85
| 21/20
| [[Floral]]
|-
| 2
| 31\198
| 187.87
| 39/35
| [[Semiwitch]]
|-
| 2
| 38\198
| 230.30
| 8/7
| [[Hemigamera]]
|-
| 2
| 40\198
| 242.42
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\198
| 260.61
| 64/55
| [[Hemiamity]]
|-
| 2
| 52\198<br>(47\198)
| 315.15<br>(284.85)
| 6/5<br>(33/28)
| [[Semiparakleismic]]
|-
| 2
| 58\198<br>(41\198)
| 351.52<br>(248.48)
| 49/40<br>(15/13)
| [[Semihemi]]
|-
| 2
| 67\198<br>(32\198)
| 406.06<br>(193.94)
| 495/392<br>(28/25)
| [[Semihemiwürschmidt]]
|-
| 2
| 74\198<br>(25\198)
| 448.48<br>(151.51)
| 35/27<br>(12/11)
| [[Neusec]]
|-
| 3
| 5\198
| 30.30
| 55/54
| [[Hemichromat]]
|-
| 3
| 41\198<br>(25\198)
| 248.48<br>(151.51)
| 15/13<br>(12/11)
| [[Hemimist]]
|-
| 6
| 82\198<br>(16\198)
| 496.97<br>(96.97)
| 4/3<br>(200/189)
| [[Semimist]]
|-
| 18
| 52\198<br>(3\198)
| 315.15<br>(18.18)
| 6/5<br>(99/98)
| [[Hemiennealimmal]]
|-
| 22
| 82\198<br>(1\198)
| 496.97<br>(6.06)
| 4/3<br>(385/384)
| [[Icosidillic]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave]]
[[Category:99edo]]
[[Category:99edo]]
[[Category:198edo]]
[[Category:Major minthmic]]
[[Category:Minthmic]]
[[Category:Namaka]]
[[Category:Misty]]
[[Category:Hemifamity]]
[[Category:Canou]]
[[Category:Semicanousmic]]
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