198edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
| 2.3.5.7.11
| 2401/2400, 3025/3024, 3136/3125, 4375/4374
| 2401/2400, 3025/3024, 3136/3125, 4375/4374
| {{mapping| 198 314 460 556 685 }}
| {{mapping| 198 314 460 556 685 }}
| -0.344
| &minus;0.344
| 0.291
| 0.291
| 4.80
| 4.80
Line 45: Line 37:
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| {{mapping| 198 314 460 556 685 733 }}
| {{mapping| 198 314 460 556 685 733 }}
| -0.372
| &minus;0.372
| 0.273
| 0.273
| 4.50
| 4.50
|}
{{comma basis end}}
* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]].  
* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]].  


Line 54: Line 46:
Note: temperaments supported by 99et are not included.  
Note: temperaments supported by 99et are not included.  


{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 123: Line 109:
|-
|-
| 2
| 2
| 52\198<br>(47\198)
| 52\198<br />(47\198)
| 315.15<br>(284.85)
| 315.15<br />(284.85)
| 6/5<br>(33/28)
| 6/5<br />(33/28)
| [[Semiparakleismic]]
| [[Semiparakleismic]]
|-
|-
| 2
| 2
| 58\198<br>(41\198)
| 58\198<br />(41\198)
| 351.52<br>(248.48)
| 351.52<br />(248.48)
| 49/40<br>(15/13)
| 49/40<br />(15/13)
| [[Semihemi]]
| [[Semihemi]]
|-
|-
| 2
| 2
| 67\198<br>(32\198)
| 67\198<br />(32\198)
| 406.06<br>(193.94)
| 406.06<br />(193.94)
| 495/392<br>(28/25)
| 495/392<br />(28/25)
| [[Semihemiwürschmidt]]
| [[Semihemiwürschmidt]]
|-
|-
| 2
| 2
| 74\198<br>(25\198)
| 74\198<br />(25\198)
| 448.48<br>(151.51)
| 448.48<br />(151.51)
| 35/27<br>(12/11)
| 35/27<br />(12/11)
| [[Neusec]]
| [[Neusec]]
|-
|-
Line 153: Line 139:
|-
|-
| 3
| 3
| 41\198<br>(25\198)
| 41\198<br />(25\198)
| 248.48<br>(151.51)
| 248.48<br />(151.51)
| 15/13<br>(12/11)
| 15/13<br />(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
| 6
| 6
| 82\198<br>(16\198)
| 82\198<br />(16\198)
| 496.97<br>(96.97)
| 496.97<br />(96.97)
| 4/3<br>(200/189)
| 4/3<br />(200/189)
| [[Semimist]]
| [[Semimist]]
|-
|-
| 18
| 18
| 52\198<br>(3\198)
| 52\198<br />(3\198)
| 315.15<br>(18.18)
| 315.15<br />(18.18)
| 6/5<br>(99/98)
| 6/5<br />(99/98)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 22
| 22
| 82\198<br>(1\198)
| 82\198<br />(1\198)
| 496.97<br>(6.06)
| 496.97<br />(6.06)
| 4/3<br>(385/384)
| 4/3<br />(385/384)
| [[Icosidillic]]
| [[Icosidillic]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:99edo]]
[[Category:99edo]]
[[Category:Major minthmic]]
[[Category:Major minthmic]]
[[Category:Namaka]]
[[Category:Namaka]]

Revision as of 05:35, 16 November 2024

← 197edo 198edo 199edo →
Prime factorization 2 × 32 × 11
Step size 6.06061 ¢ 
Fifth 116\198 (703.03 ¢) (→ 58\99)
Semitones (A1:m2) 20:14 (121.2 ¢ : 84.85 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

Theory

198edo is distinctly consistent through the 15-odd-limit with harmonics of 3 through 13 all tuned sharp. It is enfactored 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, 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.

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 chords including cuthbert chords, sinbadmic chords, and petrmic chords in the 13-odd-limit, in addition to island chords in the 15-odd-limit.

Notably, it is the last edo to map 64/63 and 81/80 to the same step consistently.

The 198b val supports 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

Approximation of prime harmonics in 198edo
Harmonic 2 3 5 7 11 13 17 19 23 29 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
Steps
(reduced) 		198
(0) 		314
(116) 		460
(64) 		556
(160) 		685
(91) 		733
(139) 		809
(17) 		841
(49) 		896
(104) 		962
(170) 		981
(189)

Subsets and supersets

Since 198 factors into 2 × 32 × 11, 198edo has subset edos 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.

A step of 198edo is exactly 50 purdals or 62 primas.

Intervals

Main article: Table of 198edo intervals

Regular temperament properties

Template:Comma basis begin |- | 2.3.5.7.11 | 2401/2400, 3025/3024, 3136/3125, 4375/4374 | [⟨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 | [⟨198 314 460 556 685 733]] | −0.372 | 0.273 | 4.50 Template:Comma basis end

  • 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past 190 and followed by 224.

Rank-2 temperaments

Note: temperaments supported by 99et are not included.

Template:Rank-2 begin |- | 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
(47\198) | 315.15
(284.85) | 6/5
(33/28) | Semiparakleismic |- | 2 | 58\198
(41\198) | 351.52
(248.48) | 49/40
(15/13) | Semihemi |- | 2 | 67\198
(32\198) | 406.06
(193.94) | 495/392
(28/25) | Semihemiwürschmidt |- | 2 | 74\198
(25\198) | 448.48
(151.51) | 35/27
(12/11) | Neusec |- | 3 | 5\198 | 30.30 | 55/54 | Hemichromat |- | 3 | 41\198
(25\198) | 248.48
(151.51) | 15/13
(12/11) | Hemimist |- | 6 | 82\198
(16\198) | 496.97
(96.97) | 4/3
(200/189) | Semimist |- | 18 | 52\198
(3\198) | 315.15
(18.18) | 6/5
(99/98) | Hemiennealimmal |- | 22 | 82\198
(1\198) | 496.97
(6.06) | 4/3
(385/384) | Icosidillic Template:Rank-2 end Template:Orf

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