198edo: Difference between revisions

(2 intermediate revisions by one other user not shown)

Line 7:

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 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.

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.

Line 17:

=== Prime harmonics ===

=== 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 ===

Line 32:

Line 35:

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

Line 40:

Line 43:

| 2.3.5.7.11

| 2401/2400, 3025/3024, 3136/3125, 4375/4374

| {{~~mapping~~| 198 314 460 556 685 }}

| {{Mapping| 198 314 460 556 685 }}

| −0.344

| 0.291

Line 47:

Line 50:

| 2.3.5.7.11.13

| 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

| 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]].

Line 60:

Line 77:

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

Line 127:

Line 144:

|-

| 2

| 52\198 (47\198)

| 52\198 (47\198)

| 315.15 (284.85)

| 315.15 (284.85)

| 6/5 (33/28)

| 6/5 (33/28)

| [[Semiparakleismic]]

|-

| 2

| 58\198 (41\198)

| 58\198 (41\198)

| 351.52 (248.48)

| 351.52 (248.48)

| 49/40 (15/13)

| 49/40 (15/13)

| [[Semihemi]]

|-

| 2

| 67\198 (32\198)

| 67\198 (32\198)

| 406.06 (193.94)

| 406.06 (193.94)

| 495/392 (28/25)

| 495/392 (28/25)

| [[Semihemiwürschmidt]]

|-

| 2

| 74\198 (25\198)

| 74\198 (25\198)

| 448.48 (151.51)

| 448.48 (151.51)

| 35/27 (12/11)

| 35/27 (12/11)

| [[Neusec]]

|-

Line 157:

Line 174:

|-

| 3

| 41\198 (25\198)

| 41\198 (25\198)

| 248.48 (151.51)

| 248.48 (151.51)

| 15/13 (12/11)

| 15/13 (12/11)

| [[Hemimist]]

|-

| 6

| 82\198 (16\198)

| 82\198 (16\198)

| 496.97 (96.97)

| 496.97 (96.97)

| 4/3 (200/189)

| 4/3 (200/189)

| [[Semimist]]

|-

| 18

| 52\198 (3\198)

| 52\198 (3\198)

| 315.15 (18.18)

| 315.15 (18.18)

| 6/5 (99/98)

| 6/5 (99/98)

| [[Hemiennealimmal]]

|-

| 22

| 82\198 (1\198)

| 82\198 (1\198)

| 496.97 (6.06)

| 496.97 (6.06)

| 4/3 (385/384)

| 4/3 (385/384)

| [[Icosidillic]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

<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]]

@@ Line 7: / Line 7: @@
 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 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.
+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.
@@ Line 17: / Line 17: @@
 === 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 ===
@@ Line 32: / Line 35: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 40: / Line 43: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 3136/3125, 4375/4374
-| {{mapping| 198 314 460 556 685 }}
+| {{Mapping| 198 314 460 556 685 }}
 | −0.344
 | 0.291
@@ Line 47: / Line 50: @@
 | 2.3.5.7.11.13
 | 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
 | 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]].
@@ Line 60: / Line 77: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 127: / Line 144: @@
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
@@ Line 157: / Line 174: @@
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<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]]