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[http://en.wikipedia.org/wiki/53_equal_temperament Wikipedia article about 53edo]
[http://en.wikipedia.org/wiki/53_equal_temperament Wikipedia article about 53edo]
=Linear temperaments=
[[List_of_edo-distinct_53et_rank_two_temperaments|List of edo-distinct 53et rank two temperaments]]
=Just Approximation=
53edo provides excellent approximations for the classic 5-limit [[just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
{| class="wikitable"
|-
! | interval
! | ratio
! | size
! | difference
|-
| | perfect fifth
| | 3/2
| style="text-align:center;" | 31
| | −0.07 cents
|-
| | major third
| | 5/4
| style="text-align:center;" | 17
| | −1.40 cents
|-
| | minor third
| | 6/5
| style="text-align:center;" | 14
| | +1.34 cents
|-
| | major tone
| | 9/8
| style="text-align:center;" | 9
| | −0.14 cents
|-
| | minor tone
| | 10/9
| style="text-align:center;" | 8
| | −1.27 cents
|-
| | diat. semitone
| | 16/15
| style="text-align:center;" | 5
| | +1.48 cents
|}
One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO 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 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal_kleisma|septimal kleisma]], 225/224.


=Intervals=
=Intervals=
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 723: Line 674:
| style="text-align:center;" | 9/7, 12/7
| style="text-align:center;" | 9/7, 12/7
|}
|}
All 53edo chords can be named using ups and downs. Here are the zo, gu, ilo, lu, yo and ru triads:
All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.


Here are the zo, gu, ilo, lu, yo and ru triads:
{| class="wikitable"
{| class="wikitable"
|-
|-
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| style="text-align:center;" | C upmajor or C up
| style="text-align:center;" | C upmajor or C up
|}
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
 
= Relationship to 12-edo =
Whereas 12-edo has a circle of twelve 5ths, 53-edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53-edo in a 12-edo-friendly format. Excellent for introducing 53-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
[[File:53-edo spiral.png|702x702px]]


This chart shows 53-edo in a 12-edo-friendly format.
=Just Approximation=
53edo provides excellent approximations for the classic 5-limit [[just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.


[[File:53-edo spiral.png|702x702px]]
{| class="wikitable"
|-
! | interval
! | ratio
! | size
! | difference
|-
| | perfect fifth
| | 3/2
| style="text-align:center;" | 31
| | −0.07 cents
|-
| | major third
| | 5/4
| style="text-align:center;" | 17
| | −1.40 cents
|-
| | minor third
| | 6/5
| style="text-align:center;" | 14
| | +1.34 cents
|-
| | major tone
| | 9/8
| style="text-align:center;" | 9
| | −0.14 cents
|-
| | minor tone
| | 10/9
| style="text-align:center;" | 8
| | −1.27 cents
|-
| | diat. semitone
| | 16/15
| style="text-align:center;" | 5
| | +1.48 cents
|}
 
One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.
 
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO 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 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal_kleisma|septimal kleisma]], 225/224.


==Selected just intervals by error==
==Selected just intervals by error==
The following table shows how [[15-odd-limit|some prominent just intervals]] are represented in 53edo (ordered by absolute error).
The following table shows how [[15-odd-limit|some prominent just intervals]] are represented in 53edo (ordered by absolute error).  Octave-reduced prime harmonics are '''bolded'''.


{| class="wikitable"
{| class="wikitable"
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| | '''Error (abs., in [[cent]]s)'''  
| | '''Error (abs., in [[cent]]s)'''  
|-  
|-  
| style="text-align:center;" | [[4/3]], [[3/2]]  
| style="text-align:center;" | [[4/3]], [[3/2|'''3/2''']]  
| style="text-align:center;" | 0.068  
| style="text-align:center;" | '''0.068'''
|-  
|-  
| style="text-align:center;" | [[9/8]], [[16/9]]  
| style="text-align:center;" | [[9/8]], [[16/9]]  
Line 808: Line 805:
| style="text-align:center;" | 1.384  
| style="text-align:center;" | 1.384  
|-  
|-  
| style="text-align:center;" | [[5/4]], [[8/5]]  
| style="text-align:center;" | '''[[5/4]]''', [[8/5]]  
| style="text-align:center;" | 1.408  
| style="text-align:center;" | '''1.408'''
|-  
|-  
| style="text-align:center;" | [[16/15]], [[15/8]]  
| style="text-align:center;" | [[16/15]], [[15/8]]  
Line 820: Line 817:
| style="text-align:center;" | 2.724  
| style="text-align:center;" | 2.724  
|-  
|-  
| style="text-align:center;" | [[16/13]], [[13/8]]  
| style="text-align:center;" | [[16/13]], [[13/8|'''13/8''']]  
| style="text-align:center;" | 2.792  
| style="text-align:center;" | '''2.792'''
|-  
|-  
| style="text-align:center;" | [[8/7]], [[7/4]]  
| style="text-align:center;" | [[8/7]], '''[[7/4]]'''
| style="text-align:center;" | 4.759  
| style="text-align:center;" | '''4.759'''
|-  
|-  
| style="text-align:center;" | [[7/6]], [[12/7]]  
| style="text-align:center;" | [[7/6]], [[12/7]]  
Line 856: Line 853:
| style="text-align:center;" | 7.854  
| style="text-align:center;" | 7.854  
|-  
|-  
| style="text-align:center;" | [[11/8]], [[16/11]]  
| style="text-align:center;" | '''[[11/8]]''', [[16/11]]  
| style="text-align:center;" | 7.922  
| style="text-align:center;" | '''7.922'''
|-  
|-  
| style="text-align:center;" | [[14/11]], [[11/7]]  
| style="text-align:center;" | [[14/11]], [[11/7]]  
Line 863: Line 860:
|}
|}


=Compositions=
=Linear temperaments=
[[List_of_edo-distinct_53et_rank_two_temperaments|List of edo-distinct 53et rank two temperaments]]
 
=Music=
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo_Khramov|Mykhaylo Khramov]]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo_Khramov|Mykhaylo Khramov]]


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