298edo: Difference between revisions

Line 3:

== Theory ==

298edo is [[consistent]] in the 5-odd-limit~~, where it is [[enfactoring|enfactored~~]], 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 2.3.5.17 but differs on the mapping for ~~harmonics~~ [[7/4|7]], [[11/8|11]], [[13/8|13]]. ~~298edo~~ tempers out the [[rastma]], splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]].

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.

~~The patent val supports~~ 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.

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 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]].

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

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

In higher limits, 298edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 r¢. A comma basis for the 2.5.11.17.23.43.53.59 subgroup is {1376/1375, 3128/3127, 4301/4300, 25075/25069, 38743/38720, 58351/58300, 973360/972961}.

=== ~~Prime~~ harmonics ===

=== Odd harmonics ===

== Regular temperament properties ==

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

! rowspan="2" |Subgroup

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

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

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

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

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

! rowspan="2" |Optimal

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

8ve ~~stretch~~ (¢)

! colspan="2" | Tuning Error

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

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

![[TE simple badness|Relative]] (%)

! [[TE simple badness|Relative]] (%)

|-

|2.3.5.7

| 2.3.5.7

|6144/6125, ~~321489~~/~~320000~~, 3796875/3764768

| 6144/6125, 78732/78125, 3796875/3764768

|[{{~~val~~|298 472 692 837}}]

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

|0.0275

| +0.0275

|0.5022

| 0.5022

|?

| 12.5

|-

|2.3.5.7.11

| 2.3.5.7.11

|243/242, 1375/1372, 6144/6125, 72171/71680

| 243/242, 1375/1372, 6144/6125, 72171/71680

|[{{~~val~~|298 472 692 837 1031}}]

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

|0.0012

| +0.0012

|0.4523

| 0.4523

|?

| 11.2

|-

|2.3.5.7.11

| 2.3.5.7.11

|243/242, 1029/1024, 3136/3125, 9801/9800

| 243/242, 1029/1024, 3136/3125, 9801/9800

|[{{~~val~~|298 472 692 836 1031}}] (298d)

| {{mapping| 298 472 692 836 1031 }} (298d)

|0.2882

| +0.2882

|0.4439

| 0.4439

|?

| 11.0

|-

|2.3.5.7.11.13

| 2.3.5.7.11.13

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

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

|[{{~~val~~|298 472 692 837 1031 1103}}]

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

|

| -0.0478

|

| 0.4271

|?

| 10.6

|-

|2.3.5.7.11.13.17

| 2.3.5.7.11.13.17

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

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

|[{{~~val~~|298 472 692 837 1031 1103 1218}}]

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

|

| 0.3974

|

| 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 ~~represented~~ by ~~149edo~~ are not ~~included~~.

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

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

! Periods per ~~Octave~~

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

|1

| 1

|113\298

| 113\298

|455.033

| 455.033

|13/10

| 13/10

|[[Petrtri]] (2.11/5.13/5)

| [[Petrtri]] (2.11/5.13/5)

|-

| 1

Line 93:

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.

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:Bison]]

[[Category:Emka ~~family~~]]

[[Category:Emka]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is [[consistent]] in the 5-odd-limit, where it is [[enfactoring|enfactored]], 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 2.3.5.17 but differs on the mapping for harmonics [[7/4|7]], [[11/8|11]], [[13/8|13]]. 298edo tempers out the [[rastma]], splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]].
+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.
-The patent val supports 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.
-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 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]].
-The 298cd val, {{val|298 472 '''691''' '''836''' 1031}} supports [[miracle]].
+The 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].
 In higher limits, 298edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 r¢. A comma basis for the 2.5.11.17.23.43.53.59 subgroup is {1376/1375, 3128/3127, 4301/4300, 25075/25069, 38743/38720, 58351/58300, 973360/972961}.
-=== Prime harmonics ===
+=== Odd harmonics ===
 {{Harmonics in equal|298}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |Subgroup
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Mapping]]
-! rowspan="2" |Optimal
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-ve stretch (¢)
+! colspan="2" | Tuning Error
-! colspan="2" |Tuning error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.3.5.7
+| 2.3.5.7
-|6144/6125, 321489/320000, 3796875/3764768
+| 6144/6125, 78732/78125, 3796875/3764768
-|[{{val|298 472 692 837}}]
+| {{mapping| 298 472 692 837 }} (298)
-|0.0275
+| +0.0275
-|0.5022
+| 0.5022
-|?
+| 12.5
 |-
-|2.3.5.7.11
+| 2.3.5.7.11
-|243/242, 1375/1372, 6144/6125, 72171/71680
+| 243/242, 1375/1372, 6144/6125, 72171/71680
-|[{{val|298 472 692 837 1031}}]
+| {{mapping| 298 472 692 837 1031 }} (298)
-|0.0012
+| +0.0012
-|0.4523
+| 0.4523
-|?
+| 11.2
 |-
-|2.3.5.7.11
+| 2.3.5.7.11
-|243/242, 1029/1024, 3136/3125, 9801/9800
+| 243/242, 1029/1024, 3136/3125, 9801/9800
-|[{{val|298 472 692 836 1031}}] (298d)
+| {{mapping| 298 472 692 836 1031 }} (298d)
-|0.2882
+| +0.2882
-|0.4439
+| 0.4439
-|?
+| 11.0
 |-
-|2.3.5.7.11.13
+| 2.3.5.7.11.13
-|243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
+| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
-|[{{val|298 472 692 837 1031 1103}}]
+| {{mapping| 298 472 692 837 1031 1103 }}
-|
+| -0.0478
-|
+| 0.4271
-|?
+| 10.6
 |-
-|2.3.5.7.11.13.17
+| 2.3.5.7.11.13.17
-|243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
+| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
-|[{{val|298 472 692 837 1031 1103 1218}}]
+| {{mapping| 298 472 692 837 1031 1103 1218 }}
-|
+| 0.3974
-|
+| 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 represented by 149edo are not included.
+Note: 5-limit temperaments supported by 149et are not listed.
 {| class="wikitable center-all left-5"
-! Periods<br>per Octave
+! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
-|1
+| 1
-|113\298
+| 113\298
-|455.033
+| 455.033
-|13/10
+| 13/10
-|[[Petrtri]] (2.11/5.13/5)
+| [[Petrtri]] (2.11/5.13/5)
 |-
 | 1
@@ Line 93: / Line 93: @@
 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.
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Bison]]
-[[Category:Emka family]]
+[[Category:Emka]]