198edo: Difference between revisions

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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 2^1/198, 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
Error	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
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

198edo: Difference between revisions

Latest revision as of 18:54, 28 July 2025

Contents

Theory

Prime harmonics

Octave stretch

Subsets and supersets

Intervals

Regular temperament properties

Rank-2 temperaments

Navigation menu

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-07 17:50:10 UTC</tt>.<br>
-: The original revision id was <tt>251717114</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //198 equal temperament// divides the octave into 198 parts of 6.061 cents each. It is contorted in the [[7-limit]], with the same the same tuning as [[99edo]], but makes for a good 11- and 13-limit system. Like 99, it tempers out 2401/2400, 4375/4374, 3136/3135, 5120/5103 and 6144/6125 in the 7-limit; in the [[11-limit]] it tempers 3025/3024 and 9801/9800; and in the [[13-limit]] 352/351, 847/845, 676/675, 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 temperment]]. It is distinctly [[consistent]] through the 15-limit, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.</pre></div>
+== Theory ==
-<h4>Original HTML content:</h4>
+edo 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.
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;198edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;198 equal temperament&lt;/em&gt; divides the octave into 198 parts of 6.061 cents each. It is contorted in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, with the same the same tuning as &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt;, but makes for a good 11- and 13-limit system. Like 99, it tempers out 2401/2400, 4375/4374, 3136/3135, 5120/5103 and 6144/6125 in the 7-limit; in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; it tempers 3025/3024 and 9801/9800; and in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; 352/351, 847/845, 676/675, 1001/1000, 1716/1715 and 2080/2079.&lt;br /&gt;
-&lt;br /&gt;
+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 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 &lt;a class="wiki_link" href="/Misty%20family#Hemimist"&gt;hemimist temperment&lt;/a&gt;. It is distinctly &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through the 15-limit, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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.
+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 ===
+edo 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 ==
+{{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:99edo]]
+[[Category:Major minthmic]]
+[[Category:Namaka]]

198edo: Difference between revisions

Latest revision as of 18:54, 28 July 2025

Theory

Prime harmonics

Octave stretch

Subsets and supersets

Intervals

Regular temperament properties

Rank-2 temperaments

Navigation menu

Search