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Major rearrangement of interval table
 
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!Degree
!Degree
!Cents
!Cents
!Native Fifths
!Interval
!Additional Categories
!Notation
!Notation
!JI*
!Notes
!Notes
|-
|-
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|0.000
|0.000
|P1
|P1
|
|C
|C
|1/1
|
|
|-
|-
|1
|1
|41.379
|41.379
|A7
|A7 ([[Diesis (interval region)|Diesis]])
|[[Diesis (interval region)|Diesis]]
|B♯
|B♯
|[[45/44]], [[50/49]], [[37/36]]
|Distinct from the octave.  Three major thirds reach this augmented seventh.
|Distinct from the octave.  Three major thirds reach this augmented seventh.
|-
|-
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|82.759
|82.759
|m2
|m2
|
|D♭
|D♭
|[[256/243]], [[22/21]]
|
|
|-
|-
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|124.138
|124.138
|[[Chroma]] (A1)
|[[Chroma]] (A1)
|Supraminor 2nd
|C♯
|C♯
|[[15/14]], [[14/13]], [[29/27]]
|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
|-
|-
|4
|4
|165.517
|165.517
|d3
|[[Porcupine|Quill]] (d3)
|Submajor 2nd
|B𝄪, E𝄫
|B𝄪, E𝄫
|[[11/10]], [[32/29]]
|
|
|-
|-
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|206.897
|206.897
|M2
|M2
|
|D
|D
|[[9/8]]
|
|
|-
|-
|6
|6
|248.276
|248.276
|[[Chthonic]]
|[[Chthonic]] ([[Extraclassical tonality|r3]])
|[[Extraclassical tonality|Arto]] 3rd
|C𝄪, F𝄫
|C𝄪, F𝄫
|[[37/32]]
|New region in between M2 and m3; two of them make a perfect fourth.
|New region in between M2 and m3; two of them make a perfect fourth.
|-
|-
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|289.655
|289.655
|m3
|m3
|
|E♭
|E♭
|[[32/27]], [[13/11]]
|
|
|-
|-
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|331.034
|331.034
|A2
|A2
|Supraminor 3rd
|D♯
|D♯
|[[29/24]]
|Distinct from the minor third, as can be seen in the Harmonic Minor modes.
|Distinct from the minor third, as can be seen in the Harmonic Minor modes.
|-
|-
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|372.414
|372.414
|d4
|d4
|Submajor 3rd
|F♭
|F♭
|[[36/29]]
|May be used as a neutral third, but does not bisect the perfect fifth.
|May be used as a neutral third, but does not bisect the perfect fifth.
|-
|-
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|413.793
|413.793
|M3
|M3
|
|E
|E
|[[81/64]], [[14/11]]
|
|
|-
|-
|11
|11
|455.172
|455.172
|[[Naiadic]]
|[[Naiadic]] ([[Extraclassical tonality|T3]])
|[[Extraclassical tonality|Tendo]] 3rd
|D𝄪, G𝄫
|D𝄪, G𝄫
|[[13/10]], [[48/37]]
|New region in between M3 and P4; two of them make a major sixth.
|New region in between M3 and P4; two of them make a major sixth.
|-
|-
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|496.552
|496.552
|P4
|P4
|
|F
|F
|[[4/3]]
|Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.
|Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.
|-
|-
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|537.931
|537.931
|A3
|A3
|Wolf 4th
|E♯
|E♯
|[[37/27]]
|
|
|-
|-
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|579.310
|579.310
|d5
|d5
|Minor tritone
|G♭
|G♭
|[[7/5]]
|Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
|Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
|-
|-
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|620.690
|620.690
|A4
|A4
|Major tritone
|F♯
|F♯
|[[10/7]]
|Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
|Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
|-
|-
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|662.069
|662.069
|d6
|d6
|Wolf 5th
|E𝄪, A𝄫
|E𝄪, A𝄫
|[[54/37]]
|
|
|-
|-
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|703.448
|703.448
|P5
|P5
|
|G
|G
|[[3/2]]
|Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.
|Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.
|-
|-
|18
|18
|744.828
|744.828
|[[Cocytic]]
|[[Cocytic]] ([[Extraclassical tonality|r6]])
|Arto 6th
|F𝄪
|F𝄪
|[[20/13]], [[37/24]]
|New region between P5 and m6; two of them reduce to a minor third.
|New region between P5 and m6; two of them reduce to a minor third.
|-
|-
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|786.207
|786.207
|m6
|m6
|
|A♭
|A♭
|[[128/81]], [[11/7]]
|
|
|-
|-
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|827.586
|827.586
|A5
|A5
|Supraminor 6th
|G♯
|G♯
|[[29/18]]
|Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
|Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
|-
|-
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|868.966
|868.966
|d7
|d7
|Submajor 6th
|B𝄫
|B𝄫
|[[48/29]]
|Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
|Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
|-
|-
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|910.345
|910.345
|M6
|M6
|
|A
|A
|[[27/16]], [[22/13]]
|
|
|-
|-
|23
|23
|951.724
|951.724
|[[Ouranic]]
|[[Ouranic]] ([[Extraclassical tonality|T6/r7]])
|Tendo 6th/arto 7th
|G𝄪, C𝄫
|G𝄪, C𝄫
|[[64/37]]
|New region between M6 and m7; two of them reduce to a perfect fifth.
|New region between M6 and m7; two of them reduce to a perfect fifth.
|-
|-
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|993.103
|993.103
|m7
|m7
|
|B♭
|B♭
|[[16/9]]
|
|
|-
|-
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|1034.483
|1034.483
|A6
|A6
|Supraminor 7th
|A♯
|A♯
|[[20/11]], [[29/16]]
|[[wikipedia:Augmented_sixth_chord|Augmented Sixth chords]] use this interval, not the typical minor seventh.
|[[wikipedia:Augmented_sixth_chord|Augmented Sixth chords]] use this interval, not the typical minor seventh.
|-
|-
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|1075.862
|1075.862
|d8
|d8
|Submajor 7th
|C♭
|C♭
|[[13/7]], [[54/29]]
|
|
|-
|-
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|1117.241
|1117.241
|M7
|M7
|
|B
|B
|[[243/128]], [[21/11]]
|
|
|-
|-
|28
|28
|1158.621
|1158.621
|d2
|d2 ([[Extraclassical tonality|T7]])
|Tendo 7th
|A𝄪, D𝄫
|A𝄪, D𝄫
|[[88/45]], [[72/37]]
|Distinct from the octave.  Four minor thirds reach this diminished ninth.
|Distinct from the octave.  Four minor thirds reach this diminished ninth.
|-
|-
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|1200.000
|1200.000
|P8
|P8
|
|C
|C
|[[2/1]]
|
|
|}
|}
<small>*Assuming the 2.3.7/5.11/5.13/5.29.37 subgroup; see Harmonic Approximations for details.</small>
As can be seen here, the familiar diatonic categories allow composers to root themselves in established structures, permitting them to fall back onto comprehensible harmony while still allowing for the interordinals and other new colors to be utilized alongside them.
As can be seen here, the familiar diatonic categories allow composers to root themselves in established structures, permitting them to fall back onto comprehensible harmony while still allowing for the interordinals and other new colors to be utilized alongside them.


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One benefit of using extraclassical intervals is that the art and tendo thirds (unlike their diatonic counterparts) do not crowd one another when used in a chord together, being separated by a whole tone rather than a chroma.
One benefit of using extraclassical intervals is that the art and tendo thirds (unlike their diatonic counterparts) do not crowd one another when used in a chord together, being separated by a whole tone rather than a chroma.
=== Harmonic Approximations ===
Critics of 29edo have often said that it fails to approximate most ratios beyond the 3-limit; primes 5, 7, 11, and 13 are all of relatively high error, and as such the tuning does not provide the necessary colors to suffice as an introductory system to tunings beyond 12edo.
Assuming for the sake of the argument that approximating JI is somehow the only thing that one may want of a tuning, and further assuming that one even knows what that entails on their first venture into xenharmonic tunings, 29edo does not fall quite as flat as some would have you believe.  Notably, the aforementioned error on primes 5, 7, 11, and 13 are all in the same direction and of roughly the same amount, which leads their difference tones to be tuned quite well.
{| class="wikitable"
|+13-limit difference tones
! colspan="2" |Interval
!7/5
!11/5
!13/5
!11/7
!13/7
!13/11
|-
! rowspan="2" |Error
!Absolute (¢)
| -3.2
| +0.5
| +1.0
| +3.7
| +4.2
| +0.5
|-
!Relative (%)
| -7.7
| +1.2
| +2.3
| +8.9
| +10.0
| +1.1
|-
! colspan="2" |Steps
(Reduced)
|14
(14)
|33
(4)
|40
(11)
|19
(19)
|26
(26)
|7
(7)
|}
Because, for example, 11/7 can be found at (11/5)/(7/5), we only need to add three fractions to the subgroup; here, I'll use 7/5, 11/5, and 13/5.
Additionally, 4\29 can be interpreted as 32/29, adding prime 29 to the subgroup; this allows the chromatic semitone to be interpreted as 29/27, the supraminor third as 29/24, and the submajor third as 36/29.
Finally, the arto third / semifourth at 6\29 can be interpreted as 37/32, adding prime 37 to the subgroup; this allows the upfourth to be interpreted as 37/27, the tendo third as 48/37, and the diesis as 37/36.
So in total, our accurate subgroup for 29edo is '''2.3.7/5.11/5.13/5.29.37'''.  Not bad for a tuning that supposedly isn't useful beyond the 3-limit.
== Chords of 29edo ==
== Chords of 29edo ==


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=== 4L 3s ===
=== 4L 3s ===
The [[4L 3s]] scale can be thought of as an alteration of the Harmonic Minor scale, which is unique to 29edo.  If we notice that the augmented second is precisely three steps larger than a major second, we can distribute this error amongst the three semitones that occur in the scale, which reduces the scale to a maximum variety of two.  We may also notice that this scale's pattern creates a circle of augmented seconds, which can be used to quantify the brightness of the seven modes.
The [[4L 3s]] scale can be thought of as an alteration of the Harmonic Minor scale, which is unique to 29edo.  If we notice that the augmented second is precisely three steps larger than a major second, we can distribute this error amongst the three semitones that occur in the scale, which reduces the scale to a maximum variety of two.  We may also notice that this scale's pattern creates a circle of upminor thirds, which can be used to quantify the brightness of the seven modes.


See [[User:Unque/Dietic Minor|Dietic Minor]] for a more in-depth discussion of how the Harmonic Minor structure can be treated in 29edo, and how this idea generalizes to other tuning systems.
See [[User:Unque/Dietic Minor|Dietic Minor]] for a more in-depth discussion of how the Harmonic Minor structure can be treated in 29edo, and how this idea generalizes to other tuning systems.
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