198edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 3<sup>2</sup> × 11
{{ED intro}}
| Step size = 6.06061¢
| Fifth = 116\198 (703.03¢) (→ [[99edo|58\99]])
| Semitones = 20:14 (121.21¢ : 84.85¢)
| Consistency = 15
}}
The '''198 equal divisions of the octave''' ('''198edo'''), or the '''198(-tone) equal temperament''' ('''198tet''', '''198et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 198 parts of about 6.06 [[cent]]s each.


== Theory ==
== Theory ==
198edo is distinctly [[consistent]] through the [[15-odd-limit]] with harmonics of 3 through 13 all tuned sharp. It 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.  
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.  


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]].  
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]].  


It provides the [[optimal patent val]] for the rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[minthmic chords]], it enables the [[cuthbert triad]], the [[island chords]], the [[sinbadmic chords]], and the [[petrmic triad]].  
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]].  


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


198 factors into 2 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.  
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 ===
=== Prime harmonics ===
{{Primes in edo|198|columns=11}}
{{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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 40: Line 43:
| 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
| [{{val| 198 314 460 556 685 }}]
| {{Mapping| 198 314 460 556 685 }}
| -0.344
| −0.344
| 0.291
| 0.291
| 4.80
| 4.80
Line 47: Line 50:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| [{{val| 198 314 460 556 685 733 }}]
| {{Mapping| 198 314 460 556 685 733 }}
| -0.372
| −0.372
| 0.273
| 0.273
| 4.50
| 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 ===
=== Rank-2 temperaments ===
Line 57: Line 75:


{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 69: Line 88:
| 40/39
| 40/39
| [[Humorous]]
| [[Humorous]]
|-
| 1
| 19\198
| 115.15
| 77/72
| [[Semigamera]]
|-
|-
| 1
| 1
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| 13/12
| 13/12
| [[Quasijerome]]
| [[Quasijerome]]
|-
| 1
| 65\198
| 393.93
| 49/39
| [[Hitch]]
|-
|-
| 1
| 1
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| 21/20
| 21/20
| [[Floral]]
| [[Floral]]
|-
| 2
| 31\198
| 187.87
| 39/35
| [[Semiwitch]]
|-
|-
| 2
| 2
Line 129: Line 166:
| 35/27<br>(12/11)
| 35/27<br>(12/11)
| [[Neusec]]
| [[Neusec]]
|-
| 3
| 5\198
| 30.30
| 55/54
| [[Hemichromat]]
|-
|-
| 3
| 3
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| [[Icosidillic]]
| [[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:Namaka]]

Latest revision as of 18:54, 28 July 2025

← 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

198 equal divisions of the octave (abbreviated 198edo or 198ed2), also called 198-tone equal temperament (198tet) or 198 equal temperament (198et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 198 equal parts of about 6.06 ¢ each. Each step represents a frequency ratio of 21/198, or the 198th root of 2.

Theory

198edo is enfactored in the 7-limit, with the same tuning as 99edo, but makes for a good 11- and 13-limit system. It is distinctly consistent through the 15-odd-limit, and demonstrates a sharp tendency, with harmonics 3 through 13 all tuned sharp.

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.

Extending it beyond the 13-limit can be tricky, as the approximated 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 and 31 are quite accurate, though the 25 and 27 are sharp enough to have incurred more inconsistencies.

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)

Octave stretch

198edo can benefit from slightly 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 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
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
2.3.5.7.11.13.19 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330 [⟨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 [⟨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 190 and followed by 224.

Rank-2 temperaments

Note: temperaments supported by 99et are not included.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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