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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The octave is [[Octave stretching|stretched]] by about 0.1166 [[cents]] and the step size is about 22.6437 cents. It is consistent to the 10-[[integer-limit]].
{{ED intro}}
 
Lookalikes: [[53edo]], [[84edt]]


== Theory ==
== Theory ==
31edf provides excellent approximations for the classic [[5-limit]] just chords and scales, such as the Ptolemy-Zarlino "[[just major]]" scale.
31edf is almost identical to [[53edo]], but with the 3/2 rather than the [[2/1]] being [[just]]. The octave is [[octave stretching|stretched]] by about 0.1166 [[cents]]. Like 53edo, 31edf is consistent to the [[integer limit|10-integer-limit]].
 
{| class="wikitable"
|-
! Interval
! Ratio
! Size
! Difference
|-
| Perfect octave
| 2/1
| style="text-align: center;" | 31
| +0.12 cents
|-
| major third
| 5/4
| style="text-align: center;" | 17
| −1.37 cents
|-
| minor third
| 6/5
| style="text-align: center;" | 14
| +1.37 cents
|-
| major tone
| 9/8
| style="text-align: center;" | 9
| −0.12 cents
|-
| minor tone
| 10/9
| style="text-align: center;" | 8
| −1.25 cents
|-
| diat. semitone
| 16/15
| style="text-align: center;" | 5
| +1.49 cents
|}
 
One notable property of 31edf is that, like 53edo, it offers good approximations for both pure and [[Pythagorean tuning|Pythagorean]] major thirds.
 
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! Like 53edo, 31edf is practically equal to an extended Pythagorean.
 
The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.


=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|31|3|2|intervals=prime}}
{{Harmonics in equal|31|3|2|intervals=integer}}
{{Harmonics in equal|31|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31edf (continued)}}


[[Category:Edf]]
== See also ==
[[Category:Edonoi]]
* [[53edo]] – relative edo
* [[84edt]] – relative edt
* [[137ed6]] – relative ed6

Revision as of 10:42, 24 March 2025

← 30edf 31edf 32edf →
Prime factorization 31 (prime)
Step size 22.6437 ¢ 
Octave 53\31edf (1200.12 ¢)
(convergent)
Twelfth 84\31edf (1902.07 ¢)
(convergent)
Consistency limit 10
Distinct consistency limit 10

31 equal divisions of the perfect fifth (abbreviated 31edf or 31ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 31 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of (3/2)1/31, or the 31st root of 3/2.

Theory

31edf is almost identical to 53edo, but with the 3/2 rather than the 2/1 being just. The octave is stretched by about 0.1166 cents. Like 53edo, 31edf is consistent to the 10-integer-limit.

Harmonics

Approximation of harmonics in 31edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.12 +0.12 +0.23 -1.14 +0.23 +5.09 +0.35 +0.23 -1.02 -7.52 +0.35
Relative (%) +0.5 +0.5 +1.0 -5.0 +1.0 +22.5 +1.5 +1.0 -4.5 -33.2 +1.5
Steps
(reduced) 		53
(22) 		84
(22) 		106
(13) 		123
(30) 		137
(13) 		149
(25) 		159
(4) 		168
(13) 		176
(21) 		183
(28) 		190
(4)
Approximation of harmonics in 31edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.36 +5.20 -1.02 +0.47 +8.73 +0.35 -2.68 -0.90 +5.20 -7.40 +6.22 +0.47
Relative (%) -10.4 +23.0 -4.5 +2.1 +38.6 +1.5 -11.8 -4.0 +23.0 -32.7 +27.5 +2.1
Steps
(reduced) 		196
(10) 		202
(16) 		207
(21) 		212
(26) 		217
(0) 		221
(4) 		225
(8) 		229
(12) 		233
(16) 		236
(19) 		240
(23) 		243
(26)

See also

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