229edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 363 -229 }}
| {{monzo| 363 -229 }}
| {{mapping| 229 363 }}
| {{mapping| 229 363 }}
| -0.072
| &minus;0.072
| 0.072
| 0.072
| 1.38
| 1.38
Line 38: Line 30:
| 393216/390625, {{monzo| 39 -29 3 }}
| 393216/390625, {{monzo| 39 -29 3 }}
| {{mapping| 229 363 532 }}
| {{mapping| 229 363 532 }}
| -0.258
| &minus;0.258
| 0.269
| 0.269
| 5.13
| 5.13
Line 45: Line 37:
| 2401/2400, 3136/3125, 14348907/14336000
| 2401/2400, 3136/3125, 14348907/14336000
| {{mapping| 229 363 532 643 }}
| {{mapping| 229 363 532 643 }}
| -0.247
| &minus;0.247
| 0.233
| 0.233
| 4.46
| 4.46
Line 52: Line 44:
| 2401/2400, 3025/3024, 3136/3125, 8019/8000
| 2401/2400, 3025/3024, 3136/3125, 8019/8000
| {{mapping| 229 363 532 643 792 }}
| {{mapping| 229 363 532 643 792 }}
| -0.134
| &minus;0.134
| 0.308
| 0.308
| 5.87
| 5.87
Line 59: Line 51:
| 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125
| 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125
| {{mapping| 229 363 532 643 792 936 }}
| {{mapping| 229 363 532 643 792 936 }}
| -0.106
| &minus;0.106
| 0.288
| 0.288
| 5.50
| 5.50
Line 66: Line 58:
| 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400
| 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400
| {{mapping| 229 363 532 643 792 936 973 }}
| {{mapping| 229 363 532 643 792 936 973 }}
| -0.130
| &minus;0.130
| 0.273
| 0.273
| 5.22
| 5.22
Line 73: Line 65:
| 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215
| 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215
| {{mapping| 229 363 532 643 792 936 973 1036 }}
| {{mapping| 229 363 532 643 792 936 973 1036 }}
| -0.129
| &minus;0.129
| 0.256
| 0.256
| 4.88
| 4.88
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11.13
| 2.3.5.7.11.13
| style="border-top: double;" | 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125
| 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125
| style="border-top: double;" | {{mapping| 229 363 532 643 792 847 }} (229)
| {{mapping| 229 363 532 643 792 847 }} (229)
| style="border-top: double;" | -0.017
| &minus;0.017
| style="border-top: double;" | 0.384
| 0.384
| style="border-top: double;" | 7.32
| 7.32
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11.13
| 2.3.5.7.11.13
| style="border-top: double;" | 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024
| 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024
| style="border-top: double;" | {{mapping| 229 363 532 643 792 848 }} (229f)
| {{mapping| 229 363 532 643 792 848 }} (229f)
| style="border-top: double;" | -0.253
| &minus;0.253
| style="border-top: double;" | 0.387
| 0.387
| style="border-top: double;" | 7.39
| 7.39
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 148: Line 134:
| 18/13
| 18/13
| [[Trident]] (229)
| [[Trident]] (229)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Hemiwürschmidt]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]

Revision as of 05:27, 16 November 2024

← 228edo 229edo 230edo →
Prime factorization 229 (prime)
Step size 5.24017 ¢ 
Fifth 134\229 (702.183 ¢)
Semitones (A1:m2) 22:17 (115.3 ¢ : 89.08 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

While not highly accurate for its size, 229edo is the point where a few important temperaments meet, and is distinctly consistent in the 11-odd-limit. The equal temperament tempers out 393216/390625 (würschmidt comma) and [39 -29 3⟩ (tricot comma) in the 5-limit; 2401/2400, 3136/3125, 6144/6125, and 14348907/14336000 in the 7-limit; 3025/3024, 3388/3375, 8019/8000, 14641/14580 and 15488/15435 in the 11-limit, notably supporting hemiwürschmidt, newt, and trident.

It extends less well to the 13-limit. Using the patent val ⟨229 363 532 643 792 847], it tempers out 351/350, 1573/1568, 2080/2079, and 4096/4095. Using the alternative 229f val ⟨229 363 532 643 792 848], it tempers out 352/351, 729/728, 1001/1000, and 1716/1715.

Higher harmonics like 17, 19, and 23 are well approximated, so it shows great potential in the no-13 23-limit. It tempers out 561/560, 1089/1088, and 1701/1700 in the 17-limit; 476/475, 1216/1215, 1445/1444, and 1540/1539 in the 19-limit; and 484/483, 576/575 and 736/735 in the 23-limit.

The 229b val supports a septimal meantone close to the CTE tuning.

Prime harmonics

Approximation of prime harmonics in 229edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.23 +1.46 +0.61 -1.10 -2.10 -0.15 +1.18 +0.55 -2.50 +2.56
Relative (%) +0.0 +4.4 +27.8 +11.6 -21.0 -40.1 -2.9 +22.5 +10.4 -47.8 +48.9
Steps
(reduced) 		229
(0) 		363
(134) 		532
(74) 		643
(185) 		792
(105) 		847
(160) 		936
(20) 		973
(57) 		1036
(120) 		1112
(196) 		1135
(219)

Subsets and supersets

229edo is the 50th prime edo.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [363 -229⟩ | [⟨229 363]] | −0.072 | 0.072 | 1.38 |- | 2.3.5 | 393216/390625, [39 -29 3⟩ | [⟨229 363 532]] | −0.258 | 0.269 | 5.13 |- | 2.3.5.7 | 2401/2400, 3136/3125, 14348907/14336000 | [⟨229 363 532 643]] | −0.247 | 0.233 | 4.46 |- | 2.3.5.7.11 | 2401/2400, 3025/3024, 3136/3125, 8019/8000 | [⟨229 363 532 643 792]] | −0.134 | 0.308 | 5.87 |- | 2.3.5.7.11.17 | 561/560, 1089/1088, 1701/1700, 2401/2400, 3136/3125 | [⟨229 363 532 643 792 936]] | −0.106 | 0.288 | 5.50 |- | 2.3.5.7.11.17.19 | 476/475, 561/560, 1089/1088, 1216/1215, 1445/1444, 2401/2400 | [⟨229 363 532 643 792 936 973]] | −0.130 | 0.273 | 5.22 |- | 2.3.5.7.11.17.19.23 | 476/475, 484/483, 561/560, 576/575, 736/735, 1089/1088, 1216/1215 | [⟨229 363 532 643 792 936 973 1036]] | −0.129 | 0.256 | 4.88 |- style="border-top: double;" | 2.3.5.7.11.13 | 351/350, 1573/1568, 2080/2079, 2197/2187, 3136/3125 | [⟨229 363 532 643 792 847]] (229) | −0.017 | 0.384 | 7.32 |- style="border-top: double;" | 2.3.5.7.11.13 | 352/351, 729/728, 1001/1000, 1716/1715, 3025/3024 | [⟨229 363 532 643 792 848]] (229f) | −0.253 | 0.387 | 7.39 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 16\229 | 83.84 | 16807/16000 | Sextilimeans |- | 1 | 19\229 | 99.56 | 18/17 | Quintagar / quinsandra (229) / quinsandric (229) |- | 1 | 37\229 | 193.87 | 28/25 | Didacus / hemiwürschmidt |- | 1 | 67\229 | 351.09 | 49/40 | Newt (229) |- | 1 | 74\229 | 387.77 | 5/4 | Würschmidt (5-limit) |- | 1 | 95\229 | 497.82 | 4/3 | Gary |- | 1 | 75\229 | 503.06 | 147/110 | Quadrawürschmidt |- | 1 | 108\229 | 565.94 | 18/13 | Trident (229) Template:Rank-2 end Template:Orf

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