509edo: 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<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 807 -509 }}
| {{monzo| 807 -509 }}
| {{mapping| 509 807 }}
| {{mapping| 509 807 }}
| -0.1890
| &minus;0.1890
| 0.1889
| 0.1889
| 8.01
| 8.01
Line 34: Line 26:
| {{monzo| 9 -13 5 }}, {{monzo| 93 -3 -38 }}
| {{monzo| 9 -13 5 }}, {{monzo| 93 -3 -38 }}
| {{mapping| 509 807 1182 }}
| {{mapping| 509 807 1182 }}
| -0.1729
| &minus;0.1729
| 0.1559
| 0.1559
| 6.61
| 6.61
Line 41: Line 33:
| 2401/2400, 1600000/1594323, 29360128/29296875
| 2401/2400, 1600000/1594323, 29360128/29296875
| {{mapping| 509 807 1182 1429 }}
| {{mapping| 509 807 1182 1429 }}
| -0.1415
| &minus;0.1415
| 0.1456
| 0.1456
| 6.18
| 6.18
Line 48: Line 40:
| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
| {{mapping| 509 807 1182 1429 1761 }}
| {{mapping| 509 807 1182 1429 1761 }}
| -0.1335
| &minus;0.1335
| 0.1312
| 0.1312
| 5.57
| 5.57
Line 55: Line 47:
| 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625
| 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625
| {{mapping| 509 807 1182 1429 1761 1884 }}
| {{mapping| 509 807 1182 1429 1761 1884 }}
| -0.1618
| &minus;0.1618
| 0.1354
| 0.1354
| 5.74
| 5.74
Line 62: Line 54:
| 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913
| 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913
| {{mapping| 509 807 1182 1429 1761 1884 2081 }}
| {{mapping| 509 807 1182 1429 1761 1884 2081 }}
| -0.1784
| &minus;0.1784
| 0.1318
| 0.1318
| 5.59
| 5.59
|}
{{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 99: Line 85:
| 243/200
| 243/200
| [[Amity]]
| [[Amity]]
|}
{{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}}


== Music ==
== Music ==

Revision as of 03:15, 16 November 2024

← 508edo 509edo 510edo →
Prime factorization 509 (prime)
Step size 2.35756 ¢ 
Fifth 298\509 (702.554 ¢)
Semitones (A1:m2) 50:37 (117.9 ¢ : 87.23 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

509edo has a sharp tendency in lower harmonics. The 13-limit optimal tuning of this temperament is consistent to the 15-integer-limit, so one might want to keep the octave compression tight.

The equal temperament tempers out 1600000/1594323 (amity comma) in the 5-limit; 2401/2400 and 29360128/29296875 in the 7-limit; and 3025/3024, 5632/5625, 41503/41472, 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and 1953125/1948617 in the 11-limit. It provides the optimal patent val for petrtri, the 2.11/5.13/5 subgroup temperament tempering out 2200/2197.

Odd harmonics

Approximation of odd harmonics in 509edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.60 +0.33 +0.13 -1.16 +0.35 +1.12 +0.93 +1.13 -0.46 +0.73 -1.16
Relative (%) +25.4 +13.9 +5.6 -49.2 +14.9 +47.6 +39.3 +48.1 -19.5 +31.0 -49.3
Steps
(reduced) 		807
(298) 		1182
(164) 		1429
(411) 		1613
(86) 		1761
(234) 		1884
(357) 		1989
(462) 		2081
(45) 		2162
(126) 		2236
(200) 		2302
(266)

Subsets and supersets

509edo is the 97th prime edo.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [807 -509⟩ | [⟨509 807]] | −0.1890 | 0.1889 | 8.01 |- | 2.3.5 | [9 -13 5⟩, [93 -3 -38⟩ | [⟨509 807 1182]] | −0.1729 | 0.1559 | 6.61 |- | 2.3.5.7 | 2401/2400, 1600000/1594323, 29360128/29296875 | [⟨509 807 1182 1429]] | −0.1415 | 0.1456 | 6.18 |- | 2.3.5.7.11 | 2401/2400, 3025/3024, 5632/5625, 1600000/1594323 | [⟨509 807 1182 1429 1761]] | −0.1335 | 0.1312 | 5.57 |- | 2.3.5.7.11.13 | 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625 | [⟨509 807 1182 1429 1761 1884]] | −0.1618 | 0.1354 | 5.74 |- | 2.3.5.7.11.13.17 | 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913 | [⟨509 807 1182 1429 1761 1884 2081]] | −0.1784 | 0.1318 | 5.59 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 18\509 | 42.44 | 40/39 | Humorous |- | 1 | 36\509 | 84.87 | 21/20 | Amicable |- | 1 | 115\509 | 271.12 | 1024/875 | Quasiorwell |- | 1 | 144\509 | 339.49 | 243/200 | Amity Template:Rank-2 end Template:Orf

Music

Francium
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