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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|35}}
{{ED intro}}


== Theory ==
== Theory ==
As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71¢ and 5edo's wide fifth of 720¢. Because it includes 7edo, 35edo tunes the 29th harmonic with +1 cent of error. 35edo can also represent the 2.3.5.7.11.17 [[Just_intonation_subgroups|subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among whitewood tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings), and if you ignore [[22edo]]'s more in-tune versions of 35edo MOS's and consistent representation of both subgroups. 35edo has the optimal patent val for [[greenwood]] and [[secund]] temperaments, as well as 11-limit [[muggles]], and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a diatonic scale (unless 7edo is considered a diatonic scale).
As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{Cent}} and 5edo's wide fifth of 720{{C}}. Since it has two approximations of the perfect fifth which are close to equally off, 35edo is a classic example of a [[dual-fifth]] system. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{C}} of error.
 
35edo can also represent the 2.3.5.7.11.17 [[subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of the higher primes ([[7/5]] and [[17/11]] stand out, having less than 1 cent error). Therefore among [[whitewood]] tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings).
 
35edo has the optimal [[patent val]] for [[greenwood]] and [[secund]] temperaments, as well as 11-limit [[muggles]], and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a [[diatonic scale]] (unless 7edo is considered a diatonic scale).


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|35}}
{{Harmonics in equal|35}}
=== Dual-fifth harmony ===
35edo has two viable mappings of the [[3/2|perfect fifth]], one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned.
Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).
Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. There are two mappings of the perfect fifth, with some chords preferring the flat fifth, while other chords prefer the sharp fifth.
For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).
The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo that is a subset of the 9-odd-limit otonal or utonal pentad (up to octave equivalence) can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315|1/(9:7:6:5:4)]], that being 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.
Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15|1/(6:5:4)]], [[6:7:9]], and [[14:18:21|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28|1/(8:7:6)]], [[9:10:12]], and [[15:18:20|1/(12:10:9)]], are also tuned fairly close to DR.
==== Caveats of dual-fifth ====
However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.
Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several [[mos scale]], such as the traditional [[diatonic]] scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an [[equalized]] version of the scale, while the 21\35 fifth generates a [[collapsed]] version of the scale. Since 35edo does not have a diatonic scale, [[chain-of-fifths notation]] also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or [[Hobbled scale|hobbling]] a [[34edo]] or [[36edo]] diatonic scale.
35edo is only one of many dual-fifth systems, with others including [[18edo]], [[23edo]], [[25edo]], [[28edo]], [[30edo]], [[37edo]], and [[40edo]], each with their own unique properties.
=== Subsets and supersets ===
Since 35 factors as 5 × 7, its nontrivial subsets are [[5edo]] and [[7edo]]. Its double [[70edo]] corrects the perfect fifth, as well as the [[13/1|13th harmonic]], though the [[5/1|5th]] and [[7/1|7th]] harmonics become relatively inaccurate. The quadruple of 35edo, which is [[140edo]], additionally corrects the mappings of primes 5 and 7, and makes for an excellent [[17-limit]] system and beyond.
== Intervals ==
(Bolded ratio indicates that the ratio is most accurately tuned by the given 35edo interval.)
{| class="wikitable center-1 right-2"
|-
! #
! Cents value
! Ratios in the<br>2.5.7.11.17 subgroup
! Ratios with<br>flat 3
! Ratios with<br>sharp 3
! Ratios with<br>best 9
|-
| 0
| 0.000
| '''1/1'''
|
|
|
|-
| 1
| 34.286
| '''50/49''', '''121/119''', 33/32
| '''36/35'''
| 25/24
| '''64/63''', '''81/80'''
|-
| 2
| 68.571
| 128/125
| '''25/24'''
| 81/80
|
|-
| 3
| 102.857
| '''17/16'''
| '''15/14'''
| '''16/15'''
| '''18/17'''
|-
| 4
| 137.143
|
| '''12/11''', 16/15
|
|
|-
| 5
| 171.429
| '''11/10'''
|
| 12/11
| '''10/9'''
|-
| 6
| 205.714
|
|
|
| '''9/8'''
|-
| 7
| 240.000
| '''8/7'''
|
| 7/6
|
|-
| 8
| 274.286
| '''20/17'''
| '''7/6'''
|
|
|-
| 9
| 308.571
|
| '''6/5'''
|
|
|-
| 10
| 342.857
| '''17/14'''
|
| 6/5
| '''11/9'''
|-
| 11
| 377.143
| '''5/4'''
|
|
|
|-
| 12
| 411.429
| '''14/11'''
|
|
|
|-
| 13
| 445.714
| '''22/17''', 32/25
|
|
| '''9/7'''
|-
| 14
| 480.000
|
|
| 4/3, '''21/16'''
|
|-
| 15
| 514.286
|
| '''4/3'''
|
|
|-
| 16
| 548.571
| '''11/8'''
|
|
|
|-
| 17
| 582.857
| '''7/5'''
| '''24/17'''
| 17/12
|
|-
| 18
| 617.143
| '''10/7'''
| '''17/12'''
| 24/17
|
|-
| 19
| 651.429
| '''16/11'''
|
|
|
|-
| 20
| 685.714
|
| '''3/2'''
|
|
|-
| 21
| 720.000
|
|
| 3/2, '''32/21'''
|
|-
| 22
| 754.286
| '''17/11''', 25/16
|
|
| '''14/9'''
|-
| 23
| 788.571
| '''11/7'''
|
|
|
|-
| 24
| 822.857
| '''8/5'''
|
|
|
|-
| 25
| 857.143
| '''28/17'''
|
| 5/3
| '''18/11'''
|-
| 26
| 891.429
|
| '''5/3'''
|
|
|-
| 27
| 925.714
| '''17/10'''
| '''12/7'''
|
|
|-
| 28
| 960.000
| '''7/4'''
|
|
|
|-
| 29
| 994.286
|
|
|
| '''16/9'''
|-
| 30
| 1028.571
| '''20/11'''
|
|
| '''9/5'''
|-
| 31
| 1062.857
|
| '''11/6''', 15/8
|
|
|-
| 32
| 1097.143
| '''32/17'''
| '''28/15'''
| '''15/8'''
| '''17/9'''
|-
| 33
| 1131.429
|
|
|
|
|-
| 34
| 1165.714
|
|
|
|
|-
| 35
| 1200.000
| 2/1
|
|
|
|}


== Notation ==
== Notation ==
Line 273: Line 565:
===Sagittal notation===
===Sagittal notation===
====Best fifth notation====
====Best fifth notation====
This notation uses the same sagittal sequence as EDOs [[30edo#Sagittal notation|30b]] and [[40edo#Sagittal notation|40]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[40edo#Sagittal notation|40]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].


<imagemap>
<imagemap>
Line 284: Line 576:
</imagemap>
</imagemap>


====Second best fifth notation====
====Second-best fifth notation====
This notation uses the same sagittal sequence as [[42edo#Sagittal notation|42-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].
This notation uses the same sagittal sequence as [[42edo#Sagittal notation|42-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].


Line 317: Line 609:
0-9-20-29 = C vE G vB = Cv7 = C down seven
0-9-20-29 = C vE G vB = Cv7 = C down seven


For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions|Ups and downs notation - Chords and Chord Progressions]].


== JI Intervals ==
== Approximation to JI ==
(Bolded ratio indicates that the ratio is most accurately tuned by the given 35edo interval.)
 
{{Q-odd-limit intervals|35}}


== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable"
{| class="wikitable"
|-
|-
| Degrees
! Periods<br>per 8ve
| Cents value
! Generator
| Ratios in 2.5.7.11.17 subgroup
! Temperaments with<br>flat 3/2 (patent val)
| Ratios with flat 3
! Temperaments with sharp 3/2 (35b val)
| Ratios with sharp 3
! [[Mos scale]]s
| Ratios with best 9
|-
| 0
| 0.000
| '''1/1'''
| (see comma table)
|
|
|-
|-
| 1
| 1
| 34.286
| 1\35
| '''50/49''', '''121/119''', 33/32
| '''36/35'''
| 25/24
| '''81/80'''
|-
| 2
| 68.571
| 128/125
| '''25/24'''
| 81/80
|
|-
| 3
| 102.857
| '''17/16'''
| '''15/14'''
| '''16/15'''
| '''18/17'''
|-
| 4
| 137.143
|  
|  
| '''12/11''', 16/15
|  
|  
|  
|  
|-
|-
| 5
| 1
|171.429
| 2\35
| '''11/10'''
|
| 12/11
| '''10/9'''
|-
| 6
| 205.714
|  
|  
|  
|  
|  
| [[1L 16s]], [[17L 1s]]
| '''9/8'''
|-
|-
| 7
| 1
| 240
| 3\35
| '''8/7'''
|
| 7/6
|  
|  
| [[Ripple]]
| [[1L 10s]], [[11L 1s]], [[12L 11s]]
|-
|-
| 8
| 1
| 274.286
| 4\35
| '''20/17'''
| [[Secund]]
| '''7/6'''
|  
|  
|  
| [[1L 7s]], [[8L 1s]], [[9L 8s]], [[9L 17s]]
|-
|-
| 9
| 1
| 308.571
| 6\35
|  
| colspan="2" | [[Baldy]] (messed-up){{idiosyncratic}}
| '''6/5'''
| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[6L 11s]], [[6L 17s]], [[6L 23s]]
|  
|  
|-
|-
| 10
| 1
|342.857
| 8\35
| '''17/14'''
|  
|  
| 6/5
| [[Orwell]] (messed-up){{idiosyncratic}}
| '''11/9'''
| [[1L 3s]], [[4L 1s]], [[4L 5s]], [[9L 4s]], [[13L 9s]]
|-
|-
| 11
| 1
| 377.143
| 9\35
| '''5/4'''
| [[Myna]]
|  
|  
|  
|  
| [[1L 3s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], …, [[4L 27s]]
|-
|-
| 12
| 1
| 411.429
| 11\35
| '''14/11'''
| [[Muggles]]
|
|  
|  
|  
| [[3L 1s]], [[3L 4s]], [[3L 7s]] [[3L 10s]], [[3L 13s]], [[16L 3s]]
|-
|-
| 13
| 1
| 445.714
| 12\35
| '''22/17''', 32/25
|  
|  
|  
| [[Roman]]
| '''9/7'''
| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[3L 14s]], [[3L 17s]], [[3L 20s]], …, [[3L 29s]]
|-
|-
| 14
| 1
| 480
| 13\35
|  
| colspan="2" | Inconsistent 2.5/3.9/7 [[sensi]]/[[subgroup_temperaments#Sentry|sentry]]
|
| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[8L 3s]], [[8L 11s]], [[8L 19s]]
| 4/3, '''21/16'''
|  
|-
|-
| 15
| 1
|514.286
| 16\35
|
| '''4/3'''
|  
|  
|  
|  
| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[11L 2s]], [[11L 13s]]
|-
|-
| 16
| 1
| 548.571
| 17\35
| '''11/8'''
|  
|  
|  
|  
|  
| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]], [[2L 17s]], [[2L 19s]], …, [[2L 31s]]
|-
|-
| 17
| 5
| 582.857
| 1\35
| '''7/5'''
| '''24/17'''
| 17/12
|  
|  
| [[Blackwood]] (favoring 7/6)
| [[5L 5s]], [[5L 10s]], [[5L 15s]], [[5L 20s]], [[5L 25s]]
|-
|-
| 18
| 5
| 617.143
| 2\35
| '''10/7'''
| '''17/12'''
| 24/17
|  
|  
| [[Blackwood]] (favoring 6/5 and 20/17)
| [[5L 5s]], [[5L 10s]], [[15L 5s]]
|-
|-
| 19
| 5
| 651.429
| 3\35
| '''16/11'''
|
|  
|  
|  
| [[Blackwood]] (favoring 5/4 and 17/14)
| [[5L 5s]], [[10L 5s]], [[10L 15s]]
|-
|-
| 20
| 7
|685.714
| 1\35
|  
| [[Whitewood]] / [[redwood]]
| '''3/2'''
|
|  
|  
| [[7L 7s]], [[7L 14s]], [[7L 21s]]
|-
|-
| 21
| 7
| 720
| 2\35
|
| [[Greenwood]]
|
| 3/2, '''32/21'''
|
|-
| 22
| 754.286
| '''17/11''', 25/16
|
|
| '''14/9'''
|-
| 23
| 788.571
| '''11/7'''
|
|
|
|-
| 24
| 822.857
| '''8/5'''
|
|
|
|-
| 25
|857.143
| '''28/17'''
|
| 5/3
| '''18/11'''
|-
| 26
| 891.429
|
| '''5/3'''
|  
|  
|
| [[7L 7s]], [[14L 7s]]
|-
| 27
| 925.714
| '''17/10'''
| '''12/7'''
|
|
|-
| 28
| 960
| '''7/4'''
|
|
|
|-
| 29
| 994.286
|
|
|
| '''16/9'''
|-
| 30
|1028.571
| '''20/11'''
|
|
| '''9/5'''
|-
| 31
| 1062.857
|
| '''11/6''', 15/8
|
|
|-
| 32
| 1097.143
| '''32/17'''
| '''28/15'''
| '''15/8'''
| '''17/9'''
|-
| 33
| 1131.429
|
|
|
|
|-
| 34
| 1165.714
|
|
|
|
|-
|3
|1200
|
|
|
|
|}
 
{{15-odd-limit|35}}
 
== Rank-2 temperaments ==
{| class="wikitable"
|-
! | Periods
 
per octave
! | Generator
! | Temperaments with
 
flat 3/2 (patent val)
! | <span style="display: block; text-align: center;">Temperaments with sharp 3/2 (35b val)</span>
! | [[Mos]] Scales
|-
| | 1
| | 1\35
| |
| |
| |
|-
| | 1
| | 2\35
| |
| |
| | [[1L 16s]], [[17L 1s]]
|-
| | 1
| | 3\35
| |
| | [[Ripple|Ripple]]
| | [[1L 10s]], [[11L 1s]], [[12L 11s]]
|-
| | 1
| | 4\35
| | [[Greenwoodmic_temperaments#Secund|Secund]]
| |
| | [[1L 7s]], [[8L 1s]], [[9L 8s]], [[9L 17s]]
|-
| | 1
| | 6\35
| colspan="2" | Messed-up [[Subgroup temperaments#Baldy|Baldy]]
| | [[1L 4s]], [[5L 1s]], [[6L 5s]], [[6L 11s]], [[6L 17s]], [[6L 23s]]
|-
| | 1
| | 8\35
| |
| | Messed-up [[Orwell]]
| | [[1L 3s]], [[4L 1s]], [[4L 5s]], [[9L 4s]], [[13L 9s]]
|-
| | 1
| | 9\35
| | [[Myna|Myna]]
| |
| | [[1L 3s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], etc ... [[4L 27s]]
|-
| | 1
| | 11\35
| | [[Muggles]]
| |
| | [[3L 1s]], [[3L 4s]], [[3L 7s]] [[3L 10s]], [[3L 13s]], [[16L 3s]]
|-
| | 1
| | 12\35
| |
| | [[Avicennmic_temperaments#Roman|Roman]]
| | [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[3L 14s]], [[3L 17s]], [[3L 20s]], etc ... [[3L 29s]]
|-
| | 1
| | 13\35
| colspan="2" | Inconsistent 2.9'/7.5/3 [[Sensi]]
| | [[2L 1s]], [[3L 2s]], [[3L 5s]], [[8L 3s]], [[8L 11s]], [[8L 19s]]
|-
| | 1
| | 16\35
| |
| |
| | [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[11L 2s]], [[11L 13s]]
|-
| | 1
| | 17\35
| |
| |
| | [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]], [[2L 17s]], [[2L 19s]] etc ... [[2L 31s]]
|-
| | 5
| | 1\35
| |
| | [[Blackwood]] (favoring 7/6)
| | [[5L 5s]], [[5L 10s]], [[5L 15s]], [[5L 20s]], [[5L 25s]]
|-
| | 5
| | 2\35
| |
| | [[Blackwood]] (favoring 6/5 and 20/17)
| | [[5L 5s]], [[5L 10s]], [[15L 5s]]
|-
| | 5
| | 3\35
| |
| | [[Blackwood]] (favoring 5/4 and 17/14)
| | [[5L 5s]], [[10L 5s]], [[10L 15s]]
|-
| | 7
| | 1\35
| | [[Apotome_family|Whitewood]]/[[Apotome_family#Redwood|Redwood]]
| |
| | [[7L 7s]], [[7L 14s]], [[7L 21s]]
|-
| | 7
| | 2\35
| | [[Greenwoodmic_temperaments#Greenwood|Greenwood]]
| |
| | [[7L 7s]], [[14L 7s]]
|}
|}


== Scales ==
=== Commas ===
A good beginning for start to play 35-EDO is with the Sub-diatonic scale, that is a [[MOS]] of 3L2s: 9 4 9 9 4.
35et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 35 55 81 98 121 130 }}.)
 
== Commas ==
35 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 35 55 81 98 121 130 }}.)


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime <br> limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
Line 720: Line 743:
| 113.69
| 113.69
| Lawa
| Lawa
| Apotome, Whitewood comma
| Whitewood comma, apotome, Pythagorean chroma
|-
|-
| 5
| 5
Line 741: Line 764:
| 29.61
| 29.61
| Laquinyo
| Laquinyo
| Small diesis, Magic comma
| Magic comma, small diesis
|-
|-
| 7
| 7
Line 755: Line 778:
| 44.13
| 44.13
| Laquinzo
| Laquinzo
| Cloudy
| Cloudy comma
|-
|-
| 7
| 7
Line 762: Line 785:
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicenna
| Avicennma
|-
|-
| 7
| 7
Line 769: Line 792:
| 13.79
| 13.79
| Zotrigu
| Zotrigu
| Starling comma, Septimal semicomma
| Septimal semicomma, starling comma
|-
|-
| 11
| 11
Line 786: Line 809:
|}
|}
<references/>
<references/>
== Octave stretch or compression ==
35edo's [[prime]]s 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from [[octave stretching]]. Some stretched-octave 35edo tunings (least to most stretched) include [[149zpi]], [[equal tuning|116ed10]], [[ed7|98ed7]], [[ed5|81ed5]], [[ed12|125ed12]] or [[ed6|90ed6]].
== Scales ==
=== Polymicrotonal scales ===
; 12-tone 7edo&5edo
The ''12-tone 7edo&5edo scale'' is designed to be mapped to the key of C on a conventional piano keyboard, with [[7edo]] on the white keys, and [[5edo]] on black:
* 5 2 3 4 1 5 1 4 3 2 5 0
; 24-tone blackwood&greenwood
You can have two pianos/keyboards, one 68.6 [[cents]] sharp of the other, both tuned to the 12-tone 7edo&5edo scale. The combined black keys across both keyboards will be ''[[blackwood]][10]'' and the white keys will be ''[[greenwood]][14]''.
* 3 2 0 2 1 2 2 1 1 1 3 1 1 1 2 2 1 2 0 2 3 0 2 0
; 20-tone blackwood&greenwood
Removing the duplicates from the previous scale (perhaps for use on other instruments beside keyboard) gives this ''20-tone scale,'' which includes both blackwood[10] and greenwood[14] as subsets.
* 3 2 2 1 2 2 1 1 1 3 1 1 1 2 2 1 2 2 3 2
=== MOS scales ===
Of the [[MOS scale]]s available in 35edo, the [[muggles]] scales most closely approximate [[just intonation]].
; MOS scales
* [[Greenwood]][7]/[[whitewood]][7]: 5 5 5 5 5 5 5 (''a.k.a. [[7edo]]; an [[equiheptatonic]] scale'')
* [[Greenwood]][14]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2
* [[Greenwood]][21]: 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
* [[Muggles]][5] (a.k.a. sub-diatonic): 9 4 9 9 4
* [[Muggles]][13]: 2 2 5 2 2 2 5 2 2 2 5 2 2
* [[Muggles]][16]: 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2
* [[Muggles]][19]: 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2
* [[Ripple]][12]: 3 3 3 3 3 3 3 3 2 3 3 3
* [[Ripple]][23]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1
* [[Secund]][17]: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
* [[Whitewood]][14]: 1 4 1 4 1 4 1 4 1 4 1 4 1 4
* [[Whitewood]][21]: 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1
* [[Blackwood]][5]: 7 7 7 7 7 (''a.k.a. [[5edo]]; an [[equipentatonic]] scale; [[slendro]]-like; works with all three blackwood tunings'')
* [[Blackwood|5/4-blackwood]][10]: 4 3 4 3 4 3 4 3 4 3
* [[Blackwood|5/4-blackwood]][15]: 3 1 3 3 1 3 3 1 3 3 1 3 3 1 3
* [[Blackwood|5/4-blackwood]][25]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
* [[Blackwood|6/5-blackwood]][10]: 2 5 2 5 2 5 2 5 2 5
* [[Blackwood|6/5-blackwood]][15]: 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2
* [[Blackwood|6/5-blackwood]][20]: 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2
* 2L 9s (4:3) [11]: 3 3 4 3 3 3 3 3 4 3 3 --- A scale doing great job tempering the 2.9.11.17 subgroup near JI.
=== Ripple scales ===
; Ripple[23]
The ''[[ripple]][23]'' [[MOS scale]] makes maximum use of 35edo's dual-fifth nature, with both its sizes of fifth and fourth occurring frequently throughout the whole scale:
* Symmetrical mode (has the most consonances): 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
* Mode that includes the clear pond{{idio}} modmos: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1
{| class="wikitable mw-collapsible mw-collapsed"
|+Ripple[23] subsets approximated from [[96edo]]
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
* Flattened major: 6 5 4 5 6 6 3
* Sharpened minor: 6 3 6 5 4 6 5
* Sharpened harmonic minor: 6 3 6 6 3 9 2
* Flattened major pentatonic: 5 6 9 6 9
* Sharpened minor pentatonic: 9 6 5 10 5
* Evened minor hexatonic: 5 4 6 5 9 6
* Roughened augmented: 10 2 9 2 10 2
* Evened dominant pentatonic: 6 6 8 9 6
* Sharpened Dorian: 6 3 6 6 6 3 5
* Flattened Ionian pentatonic: 11 4 5 12 3
* Sharpened Dorian harmonic: 6 3 9 3 6 3 5
* Evened Mixolydian pentatonic: 11 4 6 8 6
* Roughened Phrygian dominant: 2 10 2 6 3 6 6
* Evened Phrygian dominant hexatonic: 3 8 4 6 8 6
* Sharpened Phrygian pentatonic: 3 6 12 3 11
* Sharpened blues Aeolian hexatonic: 9 6 3 2 3 12
* Flattened blues Aeolian pentatonic I: 8 6 6 3 12
* Sharpened blues Aeolian pentatonic II: 9 12 2 6 6
* Roughened blues Dorian heptatonic: 9 6 3 2 7 2 6
* Sharpened blues Dorian hexatonic: 9 6 6 5 4 5
* Roughened blues Dorian pentatonic: 9 11 7 2 6
* Roughened blues pentachordal: 6 3 5 4 2 15
* Sharpened minor harmonic pentatonic I: 6 3 12 12 2
* Sharpened minor harm. pent. II: 9 6 6 12 2
* Evened hirajoshi: 6 3 11 4 11
* Sharpened hirajoshi: 6 3 12 3 11
* Roughened hirajoshi: 6 2 13 2 12
* Evened akebono I: 6 3 11 6 9
* Sharpened akebono I: 6 3 12 5 9
* Roughened akebono I: 7 1 13 6 8
* Roughened Javanese pentachordal: 2 7 9 2 15
* Roughened cosmic: 14 6 2 7 6
* Roughened cosmic II: 6 2 7 5 15
* [[Lost spirit]]: 9 6 2 3 7 3 5
* Moonbeam: 6 3 11 12 3
* Palace: 5 4 6 5 5 4 6
* Underpass: 9 11 7 3 5
|}
; Clear pond{{idio}}
The ''clear pond scale''{{idio}}, a [[modmos]] of ripple[12], tries to sound close to the familiar [[12edo]]:
* 3 3 3 2 3 3 3 4 2 3 3 3
{| class="wikitable mw-collapsible mw-collapsed"
|+Clear pond subsets
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
* Lydian: 6 5 6 3 6 6 3
* Major: 6 5 3 6 6 6 3
* Mixolydian: 6 5 3 6 6 3 6
* Dorian: 6 3 5 6 6 3 6
* Minor: 6 3 5 6 4 5 6
* Phrygian: 3 6 5 6 4 5 6
* Locrian: 3 6 5 3 7 5 6
* Harmonic minor: 6 3 5 6 4 8 3
* Melodic minor: 6 3 5 6 6 6 3
* Major pentatonic: 6 8 6 6 9
* Minor pentatonic: 9 5 6 9 6
* Minor blues: 9 5 3 3 9 6
* Minor blues heptatonic: 9 5 3 3 6 3 6
* Akebono I: 6 3 11 6 9
|}
=== Secund scales ===
; Secund[17]
The ''secund[17]'' MOS scale includes a motley mix of quirky, quite [[xenharmonic]] subsets, suited for exploring those consonances very different to any found in [[12edo]].
* 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
{| class="wikitable mw-collapsible mw-collapsed"
|+Secund[17] subsets
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
*[[Antipental blues]]: 8 7 1 4 8 7
* Antipental blues maj 6th: 8 7 1 4 7 1 7
* Antipental blues neutral 7th: 8 7 1 4 8 3 4
* Antipental blues maj 7th: 8 7 1 4 8 4 3
* Antipental blues harmonic: 8 7 1 4 3 9 3
* [[Pelog]]-like heptatonic: 3 5 7 5 3 8 4 (''Phrygian-like'')
* Pelog-like pentatonic: 3 5 12 3 12
* Secund chance ([[modmos]] of secund[8]): 4 7 4 1 4 4 7 4
* Secund-tempered rotated [[5afdo]]: 7 4 9 8 7
* Secund-tempered [[6afdo]]: 8 7 5 7 4 4
* Undecimal Mixolydian: 7 4 4 5 7 1 7
* Undecimal minor hexatonic: 7 1 7 5 8 7
* Undecimal quasi-equipentatonic: 7 8 5 8 7
* 12 from secund[17]: 7 1 3 4 1 4 3 4 1 3 1 3
|}
=== Blackwood scales ===
; The three blackwood temperaments
There are actually three versions of the ''[[blackwood]] temperament'' available in 35edo. One optimises the subminor third [[7/6]], one optimises the minor third [[6/5]], the other optimises the major third [[5/4]]. Try them each and see which one you prefer:
* [[Blackwood|5/4-blackwood]][15]: 3 1 3 3 1 3 3 1 3 3 1 3 3 1 3
* [[Blackwood|5/4-blackwood]][25]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
* [[Blackwood|6/5-blackwood]][15]: 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2
* [[Blackwood|6/5-blackwood]][20]: 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2
* [[Blackwood|7/6-blackwood]][15]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
* [[Blackwood|7/6-blackwood]][20]: 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1
{| class="wikitable mw-collapsible mw-collapsed"
|+6/5-blackwood[20] subsets
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
*Blackwood meta-Hirajoshi: 2 3 4 2 5 7 2 12
** ''Blackwood pseudo-Akebono neutral: 5 9 7 2 12''
** ''Blackwood pseudo-Akebono supermajor: 7 7 7 2 12''
** ''Blackwood pseudo-Hirajoshi: 2 12 7 2 12''
** ''Blackwood pseudo-[[pelog]]: 5 4 12 2 12''
* Blackwood meta-partial: 4 3 2 2 3 7 7 7
** ''Blackwood-tempered [[5afdo]]: 7 4 10 7 7''
** ''Mechanical (from [[16afdo]]): 9 2 10 7 7''
** ''Starship (from [[68ifdo]]'', see [[ifdo]]''): 4 7 3 7 7 7''
** ''Volcanic (from [[16afdo]]): 4 7 10 7 7''
* Meta-monsoon: 7 4 3 2 5 9 5
** ''Monsoon (from [[47zpi]]): 7 7 7 9 5''
** ''Monsoon otonal: 7 9 5 9 5''
** ''Monsoon major: 11 5 5 9 5''
* Blackwood neutral nonatonic: 4 7 3 2 5 4 5 2 3
* Blackwood undecimal harmonic: 4 8 4 5 4 5 5
* Dungeon (from [[30afdo]]): 11 3 7 2 12
* Moonbeam (from [[16afdo]]): 7 2 12 12 2
* Underpass (from [[10afdo]]): 9 12 5 4 5
* 12 from 6/5-blackwood[20]: 4 3 2 2 3 7 2 3 2 2 3 2
|}
=== Other scales ===
* Amulet{{idiosyncratic}}, approximated from [[magic]] in [[25edo]]: 3 1 3 3 1 3 4 3 3 1 3 4 3
* Fourfourths{{idio}} ([[modmos]] of 7/6-blackwood[20]): 3 1 1 2 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4
* Near-just rotated [[5afdo]]: 6 5 9 8 7
* Near-just [[6afdo]]: 8 7 5 6 5 4


== Instruments ==
== Instruments ==
=== Lumatone ===
35edo can be played on the [[Lumatone]]. See [[Lumatone mapping for 35edo]]
35edo can be played on the [[Lumatone]]. See [[Lumatone mapping for 35edo]]
=== Skip fretting ===
'''Skip fretting system 35 3 8''' is a [[skip fretting]] system for [[35edo]]. All examples on this page are for 7-string [[guitar]].
; Prime harmonics
1/1: string 2 open
2/1: string 3 fret 9 and string 6 fret 1
3/2: string 3 fret 4 and string 4 fret 13
5/4: string 3 fret 1, string 4 fret 10, and string 7 fret 2
7/4: string 4 fret 4
11/8: string 1 fret 8, string 4 open, and string 5 fret 9
13/8: string 1 fret 11, string 4 fret 3, and string 5 fret 12
17/16: string 2 fret 1 and string 3 fret 10


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Frederick Chopin}}
* [https://www.youtube.com/watch?v=1odAmqiQaz0 ''CHOPIN Waltz op 64 #2''] (1847) – rendered in 35-edo with alternating sharp and flat fifths by [[Claudi Meneghin]] (2025)
; {{W|Carlo Gesualdo}}
* [https://www.youtube.com/watch?v=idUG-x8kT3o&t=305 ''Dolcissima mia vita''] – in three comparative tunings including 35edo (5:05–10:05), rendered by [[Chris Vaisvil]] (2025)
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/6X1-692axAk ''35edo''] (2025)
* ''Whistling Like An Oberon - 35edo'' (2026)
** [https://www.youtube.com/shorts/rTkr2YHDvZM <nowiki>[short 1]</nowiki>]
** [https://www.youtube.com/shorts/AvIGI8TG9_8 <nowiki>[short 2]</nowiki>]
** [https://m.youtube.com/watch?v=zPRYktfbJj8 <nowiki>[full piece]</nowiki>]
* [https://www.youtube.com/watch?v=x8doWEgXMCY ''35edo improv''] (2026)
; [[dotuXil]]
; [[dotuXil]]
* [https://dotuxil.bandcamp.com/track/icebound-gallery-of-refractions-2 "Icebound Gallery of Refractions"] from [https://dotuxil.bandcamp.com/album/collected-refractions ''Collected Refractions''] (2024)
* [https://www.youtube.com/watch?v=61ssLv9H6rk "Icebound Gallery of Refractions"] from [https://dotuxil.bandcamp.com/album/collected-refractions ''Collected Refractions''] (2024)


; [[E8 Heterotic]]
; [[E8 Heterotic]]
* [https://youtu.be/07-wj6BaTOw ''G2 Manifold''] (2020) – uses a combination of 5edo and 7edo, which can be classified as a 35edo subset.
* [https://youtu.be/07-wj6BaTOw ''G2 Manifold''] (2020) – uses a combination of 5edo and 7edo, which can be classified as a 35edo subset.
; [[Francium]]
* "What Kind Of Things" from ''TOTMC 2025'' (2025) – [https://francium223.bandcamp.com/track/what-kind-of-things Bandcamp] | [https://www.youtube.com/watch?v=WaRm0dlUqQU YouTube]
; [[groundfault]]
* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 27edo


; [[JUMBLE]]
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=2qpsI26JfjY ''Penguins...?''] (2024)
* [https://www.youtube.com/watch?v=2qpsI26JfjY ''Penguins...?''] (2024)
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=ZPXaMTdTSgw ''Lighting the Jack-o'-lanterns''] (2025, uses meta-monsoon scale{{idio}} from 6/5-[[Blackwood]][20])


; [[Chuckles McGee]]
; [[Chuckles McGee]]
* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 Self-Destructing Mechanical Forest] (in Secund[9])
* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 ''Self-Destructing Mechanical Forest''] – in Secund[9], 35edo tuning


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude &amp; Fugue, "The Bijingle"''] (2014)
* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude & Fugue, "The Bijingle"''] (2014)
* [https://www.youtube.com/watch?v=JPie2YDwA8I ''MicroFugue on Happy Birthday for Baroque Ensemble''] (2023)
* [https://www.youtube.com/watch?v=JPie2YDwA8I ''MicroFugue on Happy Birthday for Baroque Ensemble''] (2023)
* [https://www.youtube.com/shorts/c9rCrQwF1HI ''NEOBAROQUE CANON, 3-in-1 without Bass in 35-edo for Baroque Consort: Oboe, Recorder, Violin''] (2025)


; [[No Clue Music]]
; [[No Clue Music]]
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