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→Approximation to JI: The proposed update graphic highlights 29edo’s strengths in practical harmony. The previous graphic’s full 15-odd-limit coverage was visually dense and made these strengths harder to identify. A more focused selection improves readability and better represents the interval relationships most useful for composing and playing in 29edo.
 
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Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29edo represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the [[7:11:13|1-11/7-13/7 (7:11:13)]] chord, the [[The Archipelago|barbados]] triad [[10:13:15|1-13/10-3/2 (10:13:15)]], the minor barbados triad [[26:30:39|1-15/13-3/2 (26:30:39)]], the [[22:28:33|1-14/11-3/2 (22:28:33)]] triad, the [[22:26:33|1-13/11-3/2 (22:26:33)]] triad, and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].  
Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29edo represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the [[7:11:13|1-11/7-13/7 (7:11:13)]] chord, the [[The Archipelago|barbados]] triad [[10:13:15|1-13/10-3/2 (10:13:15)]], the minor barbados triad [[26:30:39|1-15/13-3/2 (26:30:39)]], the [[22:28:33|1-14/11-3/2 (22:28:33)]] triad, the [[22:26:33|1-13/11-3/2 (22:26:33)]] triad, and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].  


29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.
29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195, 364/363, and 847/845 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.


Due to 29edo's tone-efficient mapping of 2.3.7/5.11/5.13/5, it makes sense to collapse this subgroup to 29edo. One may then expand the subgroup to the full 13-limit, adding an independent generator to reach primes 5, 7, 11, and 13 in one generator. This is [[mystery]] temperament, which has very low [[badness]] despite so many periods per octave. The 58-note MOS gives scope for harmony, with 29 15-odd-limit otonal chords and 29 utonal chords.
Due to 29edo's tone-efficient mapping of 2.3.7/5.11/5.13/5, it makes sense to collapse this subgroup to 29edo. One may then expand the subgroup to the full 13-limit, adding an independent generator to reach primes 5, 7, 11, and 13 in one generator. This is [[mystery]] temperament, which has very low [[badness]] despite so many periods per octave. The 58-note MOS gives scope for harmony, with 29 15-odd-limit otonal chords and 29 utonal chords.
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| 1
| 1
| 41.379
| 41.379
| [[33/32]],  [[40/39]],  [[45/44]], [[81/80]]
| [[33/32]],  [[40/39]],  [[45/44]],<br>[[81/80]], [[64/63]]
| negative diminished 2nd,<br>double diminished 3rd
| negative diminished 2nd,<br>double diminished 3rd
| ^1, vm2
| ^1, vm2
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| 28
| 28
| 1158.621
| 1158.621
| [[64/33]], [[39/20]], [[88/45]] [[160/81]]
| [[64/33]], [[39/20]], [[88/45]],<br>[[160/81]], [[63/32]]
| diminished 9th,<br>double augmented 6th
| diminished 9th,<br>double augmented 6th
| ^M7, v8
| ^M7, v8
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== Approximation to JI ==
== Approximation to JI ==
[[File:29ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|15-odd-limit intervals approximated in 29edo]]
=== Selected 13-limit intervals ===
[[File:29ed2-002.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|29edo’s strengths in the 13-limit, using an odd-limit of 45 and a 5.7-cent error threshold. Pythagorean thirds and 5-odd-limit intervals are also included for reference, even where they exceed this threshold.]]


=== Interval mappings ===
=== Interval mappings ===
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* [[Blackdye]] / [[syntonic dipentatonic]]: 1 4 3 4 1 4 3 4 1 4
* [[Blackdye]] / [[syntonic dipentatonic]]: 1 4 3 4 1 4 3 4 1 4
* [[Blackville]] / [[Marvel dipentatonic]]: 2 3 4 3 2 3 4 3 2 3]
* [[Blackville]] / [[Marvel dipentatonic]]: 2 3 4 3 2 3 4 3 2 3]
* Baobab{{idio}} (approximated from [[30afdo]]): 8 4 5 4 4 4
== Chords ==
29edo offers an excellent approximation of the 10:11:13:14:15 chord, whose intervals above the root are 11/10, 13/10, 7/5, and 3/2. Mapped to 0–4–11–14–17, this chord can be repeated indefinitely at successive fifths. Its utonal counterpart, 1/(15:14:13:11:10), has intervals of 15/14, 15/13, 15/11, and 3/2 above the root and is mapped to 0–3–6–13–17.
The [[12L 5s]] MOS scale, in the rotation 2 2 1 2 2 2 1 2 2 1 2 2 1 2 2 2 1, provides a useful framework for exploring this chord though successive transpositions by a fifth. It accommodates four successive statements of the 10:11:13:14 core, each transposed up a tempered fifth (3/2) from the preceding one.


== Instruments ==
== Instruments ==
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* [https://www.youtube.com/shorts/NYN8EBllJkE ''A Cyber's World''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2023)
* [https://www.youtube.com/shorts/NYN8EBllJkE ''A Cyber's World''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2023)
* [https://www.youtube.com/watch?v=JOqnRPIOb5o ''Dialtone''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2024)
* [https://www.youtube.com/watch?v=JOqnRPIOb5o ''Dialtone''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2024)
; {{W|Erroll Garner}}
* [https://www.youtube.com/shorts/dqEXdB28hZo ''Misty''] (1954) – microtonal cover in 29edo by [[Bryan Deister]] (2026)


; {{W|Bart Howard}}
; {{W|Bart Howard}}
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* [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)
* [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)
* [https://www.youtube.com/shorts/PuaNvxX11II ''an idea in 29edo''] (2026)
* [https://www.youtube.com/shorts/PuaNvxX11II ''an idea in 29edo''] (2026)
* [https://www.youtube.com/shorts/DNIBJtiuPKU ''29edo improv''] (2026-07-04)
* [https://www.youtube.com/shorts/u8TmaFaEC74 ''29edo improv''] (2026-08-30)


; [[duckapus]]
; [[duckapus]]
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