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autopatrolled, Sysops
3,990 edits
updated ups/down notation, general cleanup, added to the paragraph about quartertones notation
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Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[Harmonic_Entropy|harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[Harmonic_Entropy|harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.


The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDO steps, just one EDO step beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.
The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDOsteps, just one EDOstep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it's simple enough, and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.


==Intervals==
==Intervals==
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{| class="wikitable"
{| class="wikitable"
|-
|-
| style="text-align:center;" | Degree
! style="text-align:center;" | Degree
| style="text-align:center;" | Cents value
! style="text-align:center;" | Cents
| style="text-align:center;" | Approximate
! colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Ups and Downs Notation]]
 
! style="text-align:center;" | Approximate
Ratios*
Ratios*
| style="text-align:center;" | Solfege
! style="text-align:center;" | Solfege
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" | 0.00
| style="text-align:center;" | 0.00
| style="text-align:center;" | 1/1
| style="text-align:center;" | do
| style="text-align:center;" | P1
| style="text-align:center;" | P1
| style="text-align:center;" | perfect unison
| style="text-align:center;" | perfect unison
| style="text-align:center;" | D
| style="text-align:center;" | D
| style="text-align:center;" | 1/1
| style="text-align:center;" | do
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 1
| style="text-align:center;" | 44.44
| style="text-align:center;" | 44.44
| style="text-align:center;" | 36/35, 49/48, 50/49
| style="text-align:center;" | di
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | up-unison, minor 2nd
| style="text-align:center;" | up-unison, minor 2nd
| style="text-align:center;" | ^D, Eb
| style="text-align:center;" | ^D, Eb
| style="text-align:center;" | 36/35, 49/48, 50/49
| style="text-align:center;" | di
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
| style="text-align:center;" | 88.89
| style="text-align:center;" | 88.89
| style="text-align:center;" | 16/15, 21/20, 25/24
| style="text-align:center;" | ra
| style="text-align:center;" | ^m2
| style="text-align:center;" | ^m2
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | ^Eb
| style="text-align:center;" | ^Eb
| style="text-align:center;" | 16/15, 21/20, 25/24
| style="text-align:center;" | ra
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
| style="text-align:center;" | 133.33
| style="text-align:center;" | 133.33
| style="text-align:center;" | 14/13, 13/12
| style="text-align:center;" | ru
| style="text-align:center;" | ~2
| style="text-align:center;" | ~2
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | vvE
| style="text-align:center;" | vvE
| style="text-align:center;" | 14/13, 13/12
| style="text-align:center;" | ru
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
| style="text-align:center;" | 177.78
| style="text-align:center;" | 177.78
| style="text-align:center;" | 10/9
| style="text-align:center;" | reh
| style="text-align:center;" | vM2
| style="text-align:center;" | vM2
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | vE
| style="text-align:center;" | vE
| style="text-align:center;" | 10/9
| style="text-align:center;" | reh
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 5
| style="text-align:center;" | 222.22
| style="text-align:center;" | 222.22
| style="text-align:center;" | 8/7, 9/8
| style="text-align:center;" | re
| style="text-align:center;" | M2
| style="text-align:center;" | M2
| style="text-align:center;" | major 2nd
| style="text-align:center;" | major 2nd
| style="text-align:center;" | E
| style="text-align:center;" | E
| style="text-align:center;" | 8/7, 9/8
| style="text-align:center;" | re
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
| style="text-align:center;" | 266.67
| style="text-align:center;" | 266.67
| style="text-align:center;" | 7/6
| style="text-align:center;" | ma
| style="text-align:center;" | m3
| style="text-align:center;" | m3
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | F
| style="text-align:center;" | F
| style="text-align:center;" | 7/6
| style="text-align:center;" | ma
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
| style="text-align:center;" | 311.11
| style="text-align:center;" | 311.11
| style="text-align:center;" | 6/5
| style="text-align:center;" | me
| style="text-align:center;" | ^m3
| style="text-align:center;" | ^m3
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | ^F
| style="text-align:center;" | ^F
| style="text-align:center;" | 6/5
| style="text-align:center;" | me
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
| style="text-align:center;" | 355.56
| style="text-align:center;" | 355.56
| style="text-align:center;" | 16/13
| style="text-align:center;" | mu
| style="text-align:center;" | ~3
| style="text-align:center;" | ~3
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | ^^F
| style="text-align:center;" | ^^F
| style="text-align:center;" | 16/13
| style="text-align:center;" | mu
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 9
| style="text-align:center;" | 400
| style="text-align:center;" | 400
| style="text-align:center;" | 5/4
| style="text-align:center;" | mi
| style="text-align:center;" | vM3
| style="text-align:center;" | vM3
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | vF#
| style="text-align:center;" | vF#
| style="text-align:center;" | 5/4
| style="text-align:center;" | mi
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
| style="text-align:center;" | 444.44
| style="text-align:center;" | 444.44
| style="text-align:center;" | 9/7, 13/10
| style="text-align:center;" | mo
| style="text-align:center;" | M3
| style="text-align:center;" | M3
| style="text-align:center;" | major 3rd
| style="text-align:center;" | major 3rd
| style="text-align:center;" | F#
| style="text-align:center;" | F#
| style="text-align:center;" | 9/7, 13/10
| style="text-align:center;" | mo
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:center;" | 488.89
| style="text-align:center;" | 488.89
| style="text-align:center;" | 4/3
| style="text-align:center;" | fa
| style="text-align:center;" | P4
| style="text-align:center;" | P4
| style="text-align:center;" | perfect 4th
| style="text-align:center;" | perfect 4th
| style="text-align:center;" | G
| style="text-align:center;" | G
| style="text-align:center;" | 4/3
| style="text-align:center;" | fa
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
| style="text-align:center;" | 533.33
| style="text-align:center;" | 533.33
| style="text-align:center;" | 49/36, 48/35
| style="text-align:center;" | fih
| style="text-align:center;" | ^4
| style="text-align:center;" | ^4
| style="text-align:center;" | up 4th
| style="text-align:center;" | up 4th
| style="text-align:center;" | ^G
| style="text-align:center;" | ^G
| style="text-align:center;" | 49/36, 48/35
| style="text-align:center;" | fih
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
| style="text-align:center;" | 577.78
| style="text-align:center;" | 577.78
| style="text-align:center;" | ~4, vd5
| style="text-align:center;" | mid 4th, updim 5th
| style="text-align:center;" | ^^G, ^Ab
| style="text-align:center;" | 7/5, 18/13
| style="text-align:center;" | 7/5, 18/13
| style="text-align:center;" | fi
| style="text-align:center;" | fi
| style="text-align:center;" | ~4
| style="text-align:center;" | mid 4th
| style="text-align:center;" | ^^G
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
| style="text-align:center;" | 622.22
| style="text-align:center;" | 622.22
| style="text-align:center;" | vA4, ~5
| style="text-align:center;" | downaug 4th, mid 5th
| style="text-align:center;" | vG#, vvA
| style="text-align:center;" | 10/7, 13/9
| style="text-align:center;" | 10/7, 13/9
| style="text-align:center;" | se
| style="text-align:center;" | se
| style="text-align:center;" | ~5
| style="text-align:center;" | mid 5th
| style="text-align:center;" | vvA
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
| style="text-align:center;" | 666.67
| style="text-align:center;" | 666.67
| style="text-align:center;" | 72/49, 35/24
| style="text-align:center;" | sih
| style="text-align:center;" | v5
| style="text-align:center;" | v5
| style="text-align:center;" | down fifth
| style="text-align:center;" | down fifth
| style="text-align:center;" | vA
| style="text-align:center;" | vA
| style="text-align:center;" | 72/49, 35/24
| style="text-align:center;" | sih
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
| style="text-align:center;" | 711.11
| style="text-align:center;" | 711.11
| style="text-align:center;" | 3/2
| style="text-align:center;" | so/sol
| style="text-align:center;" | P5
| style="text-align:center;" | P5
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | A
| style="text-align:center;" | A
| style="text-align:center;" | 3/2
| style="text-align:center;" | so/sol
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 17
| style="text-align:center;" | 755.56
| style="text-align:center;" | 755.56
| style="text-align:center;" | 14/9, 20/13
| style="text-align:center;" | lo
| style="text-align:center;" | m6
| style="text-align:center;" | m6
| style="text-align:center;" | minor 6th
| style="text-align:center;" | minor 6th
| style="text-align:center;" | Bb
| style="text-align:center;" | Bb
| style="text-align:center;" | 14/9, 20/13
| style="text-align:center;" | lo
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
| style="text-align:center;" | 800
| style="text-align:center;" | 800
| style="text-align:center;" | 8/5
| style="text-align:center;" | le
| style="text-align:center;" | ^m6
| style="text-align:center;" | ^m6
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | ^Bb
| style="text-align:center;" | ^Bb
| style="text-align:center;" | 8/5
| style="text-align:center;" | le
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 19
| style="text-align:center;" | 844.44
| style="text-align:center;" | 844.44
| style="text-align:center;" | 13/8
| style="text-align:center;" | lu
| style="text-align:center;" | ~6
| style="text-align:center;" | ~6
| style="text-align:center;" | mid 6th
| style="text-align:center;" | mid 6th
| style="text-align:center;" | vvB
| style="text-align:center;" | vvB
| style="text-align:center;" | 13/8
| style="text-align:center;" | lu
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
| style="text-align:center;" | 888.89
| style="text-align:center;" | 888.89
| style="text-align:center;" | 5/3
| style="text-align:center;" | la
| style="text-align:center;" | vM6
| style="text-align:center;" | vM6
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | vB
| style="text-align:center;" | vB
| style="text-align:center;" | 5/3
| style="text-align:center;" | la
|-
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 21
| style="text-align:center;" | 933.33
| style="text-align:center;" | 933.33
| style="text-align:center;" | 12/7
| style="text-align:center;" | li
| style="text-align:center;" | M6
| style="text-align:center;" | M6
| style="text-align:center;" | major 6th
| style="text-align:center;" | major 6th
| style="text-align:center;" | B
| style="text-align:center;" | B
| style="text-align:center;" | 12/7
| style="text-align:center;" | li
|-
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 22
| style="text-align:center;" | 977.78
| style="text-align:center;" | 977.78
| style="text-align:center;" | 7/4, 16/9
| style="text-align:center;" | ta
| style="text-align:center;" | m7
| style="text-align:center;" | m7
| style="text-align:center;" | minor 7th
| style="text-align:center;" | minor 7th
| style="text-align:center;" | C
| style="text-align:center;" | C
| style="text-align:center;" | 7/4, 16/9
| style="text-align:center;" | ta
|-
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 23
| style="text-align:center;" | 1022.22
| style="text-align:center;" | 1022.22
| style="text-align:center;" | 9/5
| style="text-align:center;" | te
| style="text-align:center;" | ^m7
| style="text-align:center;" | ^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | ^C
| style="text-align:center;" | ^C
| style="text-align:center;" | 9/5
| style="text-align:center;" | te
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 24
| style="text-align:center;" | 1066.67
| style="text-align:center;" | 1066.67
| style="text-align:center;" | 13/7, 24/13
| style="text-align:center;" | tu
| style="text-align:center;" | ~7
| style="text-align:center;" | ~7
| style="text-align:center;" | mid 7th
| style="text-align:center;" | mid 7th
| style="text-align:center;" | ^^C
| style="text-align:center;" | ^^C
| style="text-align:center;" | 13/7, 24/13
| style="text-align:center;" | tu
|-
|-
| style="text-align:center;" | 25
| style="text-align:center;" | 25
| style="text-align:center;" | 1111.11
| style="text-align:center;" | 1111.11
| style="text-align:center;" | 40/21
| style="text-align:center;" | ti
| style="text-align:center;" | vM7
| style="text-align:center;" | vM7
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | vC#
| style="text-align:center;" | vC#
| style="text-align:center;" | 40/21
| style="text-align:center;" | ti
|-
|-
| style="text-align:center;" | 26
| style="text-align:center;" | 26
| style="text-align:center;" | 1155.56
| style="text-align:center;" | 1155.56
| style="text-align:center;" | 35/18, 96/49, 49/25
| style="text-align:center;" | da
| style="text-align:center;" | M7
| style="text-align:center;" | M7
| style="text-align:center;" | major 7th
| style="text-align:center;" | major 7th
| style="text-align:center;" | C#
| style="text-align:center;" | C#
| style="text-align:center;" | 35/18, 96/49, 49/25
| style="text-align:center;" | da
|-
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 27
| style="text-align:center;" | 1200
| style="text-align:center;" | 1200
| style="text-align:center;" | 2/1
| style="text-align:center;" | do
| style="text-align:center;" | P8
| style="text-align:center;" | P8
| style="text-align:center;" | 8ve
| style="text-align:center;" | 8ve
| style="text-align:center;" | D
| style="text-align:center;" | D
| style="text-align:center;" | 2/1
| style="text-align:center;" | do
|}
|}
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
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