298edo: Difference between revisions

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== Theory ==
== 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}.
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}}
{{Harmonics in equal|298}}


== 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|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>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 ===
=== 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"
{| 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
! 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
| 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.
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:Bison]]
[[Category:Emka family]]
[[Category:Emka]]

Revision as of 08:16, 4 March 2024

← 297edo 298edo 299edo →
Prime factorization 2 × 149
Step size 4.02685 ¢ 
Fifth 174\298 (700.671 ¢) (→ 87\149)
Semitones (A1:m2) 26:24 (104.7 ¢ : 96.64 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

298edo is 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 harmonics 2, 3, 5, and 17 but differs on the mapping for 7, 11, 13. Using the patent val, the equal temperament 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 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, ⟨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, ⟨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}.

Odd harmonics

Approximation of odd harmonics in 298edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.28 +0.26 +1.64 +1.46 +0.36 +1.08 -1.02 -0.26 +0.47 +0.36 -0.09
Relative (%) -31.9 +6.5 +40.8 +36.2 +8.9 +26.9 -25.3 -6.4 +11.8 +8.9 -2.1
Steps
(reduced) 		472
(174) 		692
(96) 		837
(241) 		945
(51) 		1031
(137) 		1103
(209) 		1164
(270) 		1218
(26) 		1266
(74) 		1309
(117) 		1348
(156)

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5.7 6144/6125, 78732/78125, 3796875/3764768 [⟨298 472 692 837]] (298) +0.0275 0.5022 12.5
2.3.5.7.11 243/242, 1375/1372, 6144/6125, 72171/71680 [⟨298 472 692 837 1031]] (298) +0.0012 0.4523 11.2
2.3.5.7.11 243/242, 1029/1024, 3136/3125, 9801/9800 [⟨298 472 692 836 1031]] (298d) +0.2882 0.4439 11.0
2.3.5.7.11.13 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925 [⟨298 472 692 837 1031 1103]] -0.0478 0.4271 10.6
2.3.5.7.11.13.17 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925 [⟨298 472 692 837 1031 1103 1218]] 0.3974 0.3974 9.87

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

Rank-2 temperaments

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

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 113\298 455.033 13/10 Petrtri (2.11/5.13/5)
1 137\298 551.67 11/8 Emka
2 39\298 157.04 35/32 Bison

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.

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