298edo: Difference between revisions

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== Theory ==

298edo is [[enfactoring|enfactored]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to ~~149edo – it~~ retains the mapping for [[harmonic]]s [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[17/1|17]] but differs on the mapping for [[7/4|7]], [[11/8|11]], [[13/8|13]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.

298edo is [[enfactoring|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[17/1|17]] but differs on the mapping for [[7/4|7]], [[11/8|11]], [[13/8|13]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]] and is a strong tuning for [[barton]]. In the full 13-limit, 298edo supports an unnamed {{nowrap|77 & 298}} temperament with [[13/8]] as its generator.

Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent ~~val – for~~ example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the {{nowrap|31 & 298d}} variant and the {{nowrap|118 & 298d}} variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

The 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

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

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925

| {{mapping| 298 472 692 837 1031 1103 }} (298)

| -0.0478

| −0.0478

| 0.4271

| 10.6

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| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925

| {{mapping| 298 472 692 837 1031 1103 1218 }} (298)

| -0.0320

| −0.0320

| 0.3974

| 9.87

|}

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

=== Rank-2 temperaments ===

Note: 5-limit temperaments supported by 149et are not listed.

{| class="wikitable center-all left-5"

! Periods per 8ve

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| [[Bison]]

|}

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

== Scales ==

The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is 105 & 298.

The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is {{nowrap|105 & 298}}.

[[Category:Bison]]

[[Category:Emka]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|298}}
+{{ED intro}}
 == Theory ==
-edo is [[enfactoring|enfactored]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo – it retains the mapping for [[harmonic]]s [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[17/1|17]] but differs on the mapping for [[7/4|7]], [[11/8|11]], [[13/8|13]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.
+edo is [[enfactoring|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[17/1|17]] but differs on the mapping for [[7/4|7]], [[11/8|11]], [[13/8|13]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]] and is a strong tuning for [[barton]]. In the full 13-limit, 298edo supports an unnamed {{nowrap|77 &amp; 298}} temperament with [[13/8]] as its generator.
-Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val – for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].
+Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the {{nowrap|31 &amp; 298d}} variant and the {{nowrap|118 &amp; 298d}} variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].
 The 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].
@@ Line 16: / Line 16: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 49: / Line 50: @@
 | 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
 | {{mapping| 298 472 692 837 1031 1103 }} (298)
-| -0.0478
+| −0.0478
 | 0.4271
 | 10.6
@@ Line 56: / Line 57: @@
 | 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
 | {{mapping| 298 472 692 837 1031 1103 1218 }} (298)
-| -0.0320
+| −0.0320
 | 0.3974
 | 9.87
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 === Rank-2 temperaments ===
 Note: 5-limit temperaments supported by 149et are not listed.
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 89: / Line 91: @@
 | [[Bison]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
-The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is 105 & 298.
+The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is {{nowrap|105 &amp; 298}}.
 [[Category:Bison]]
 [[Category:Emka]]