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#redirect [[Diaschismic family #Diaschismic]]
{{Infobox regtemp
| Title = Diaschismic
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
| Comma basis = [[2048/2025]] (5-limit); <br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)
| Edo join 1 = 46 | Edo join 2 = 58
| Mapping = 2; 1 -2 -8 -12 -15 1
| Generators = 3/2 | Generators tuning = 703.9 | Optimization method = CWE
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| Pergen = (P8/2, P5)
| Color name = Saguguti
| Odd limit 1 = 5 | Mistuning 1 = 3.259 | Complexity 1 = 12
| Odd limit 2 = 17-limit 21 | Mistuning 2 = ??? | Complexity 2 = 46
}}
'''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]], ideally tuned slightly sharp, or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]] = 9/8 ÷ (16/15)<sup>2</sup>, is [[tempering out|tempered out]], and a half-octave represents [[45/32]], and subtracting a whole tone from that gives [[5/4]]. Since the whole tone has been split in two equal halves of [[~]]16/15, it makes sense to equate that semitone with [[17/16]] and [[18/17]], by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]], thus giving the [[2.3.5.17 subgroup|2.3.5.17-subgroup]] version of diaschismic, sometimes known as ''srutal archagall''.


The canonical [[extension]] to the [[7-limit]] reaches [[7/4]] at -8 fifths (a diminished fourth) [[octave-reduced]] plus a semioctave, tempering out the starling comma, [[126/125]], as well as the aberschisma, [[5120/5103]]. This equates the [[64/63|septimal comma]] with the [[81/80|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. This extension works best between [[46edo]] and [[58edo]].
In this tuning range, a stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), or two comma steps, is close in size to a [[quartertone]], and that interval plus a half-octave can be used to represent [[16/11]]. Three more fifths on top of 16/11 gives [[16/13]] when octave-reduced, and thus [[13/11]] is equated with the Pythagorean minor third [[32/27]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp. Adding the mapping of 17/16 to the semitone, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. While this provides much more harmonic resources, the 2.3.5.17-subgroup version is simpler and has a more flexible tuning range.
An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is again mapped to the semitone. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler (and arguably more elegant) representation of the 7-limit.
See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic|#Septimal diaschismic]] for technical data.
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
{| class="wikitable center-1 right-2 right-4"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate ratios
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
| 600.0
| 17/12, 24/17
|-
| 1
| 703.9
| '''3/2'''
| 103.9
| '''16/15''', '''17/16''', 18/17
|-
| 2
| 207.7
| '''9/8'''
| 807.7
| '''8/5'''
|-
| 3
| 911.6
| 17/10, 22/13
| 311.6
| 6/5
|-
| 4
| 415.4
| 14/11
| 1015.4
| 9/5
|-
| 5
| 1119.5
| 21/11, 40/21, 48/25
| 519.5
| 27/20
|-
| 6
| 623.1
| 10/7
| 23.1
| 56/55, 64/63, 81/80
|-
| 7
| 127.0
| 14/13, 15/14
| 727.0
| '''32/21'''
|-
| 8
| 830.8
| 21/13, 34/21
| 230.8
| '''8/7'''
|-
| 9
| 334.7
| 17/14, 40/33
| 934.7
| 12/7
|-
| 10
| 1038.5
| 20/11
| 438.5
| 9/7
|-
| 11
| 542.4
| 15/11
| 1142.4
| 27/14, 64/33
|-
| 12
| 46.2
| 36/35, 40/39, 45/44, 50/49
| 646.2
| '''16/11'''
|-
| 13
| 750.1
| 17/11, 20/13
| 150.1
| 12/11
|-
| 14
| 253.9
| 15/13
| 853.9
| 18/11
|-
| 15
| 957.8
| 45/26, 68/39
| 357.8
| '''16/13'''
|-
| 16
| 461.6
| 17/13
| 1061.6
| 24/13
|-
| 17
| 1165.5
| 51/26, 96/49, 108/55
| 565.5
| 18/13
|}
<nowiki/>* In 17-limit CWE tuning, octave-reduced
=== As a detemperament of 12et ===
[[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]
Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14 steps. 58 is the largest number of tones for a [[mos]] where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region. In general, ratios closer to intervals of 12edo, such as [[17/16]], are simpler in terms of generator steps, while ratios further from 12edo intervals, such as [[11/9]], are more complex. The 13th harmonic is just beyond the specified generator range, so it does not appear in the diagram.
Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise, though it requires using the sharp, second-best approximation of [[5/4]] (aka the 104c [[val]]).
{{Clear}}
== Chords and harmony ==
{{See also| Chords of diaschismic }}
Diaschismic finds the 5-limit triads very simply, with [[4:5:6|1–5/4–3/2]] (≈0–392–704{{c}}) and [[10:12:15|1–6/5–3/2]] (≈0–312–704{{c}}) each occuring six times in the 12-note [[10L 2s]] mos scale. In optimal tunings, the [[3/2]] perfect fifth and [[5/4]] major third are both tuned slightly sharp, giving major triads an overall bright sound. Unlike in [[meantone]], the syntonic comma [[81/80]] is not tempered out, meaning the [[9/8]] and [[10/9]] whole tones are distinguished. Due to the sharp fifth, the [[Pythagorean tuning|Pythagorean]] major and minor triads are [[neogothic major and minor|neogothic]] in quality, representing [[22:28:33|1–14/11–3/2]] (≈0–416–704{{c}}) and [[22:26:33|1–13/11–3/2]] (≈0–288–704{{c}}) respectively. It can thus be seen as an expansion on the harmony of [[12edo]], which adds new flavors and distinctions.
Prime 17 also plays a role: For example, the interval [[17/8]], which is one octave above [[17/16]], can be used as a minor ninth in chords such as 1–5/4–3/2–17/8 (≈0–392–704{{c}}). Additionally, the semioctave represents [[17/12]]~[[24/17]], and can act as a [[tritone]] in [[diminished triad]]s such as [[85:102:120|1–6/5–24/17]] (≈0–312–600{{c}}) and its melodic inverse [[17:20:24|1–20/17–24/17]] (≈0–288–600{{c}}). The sharp ~9/8 whole tone is equated with a flat ~[[17/15]], and [[20/17]] is equated with [[32/27]]~[[13/11]]. Additionally, the [[dominant seventh chord]] can be seen as representing [[68:85:102:120|1–5/4–3/2–30/17]] (≈0–392–704–992{{c}}), thus becoming [[dyadic chord|dyadically consonant]] in the [[17-odd-limit]], meaning any interval between two notes has an interpretation with an [[odd limit]] of 17 or less. Its {{W|negative harmony}} version is [[10:12:15:17|1–6/5–3/2–17/10]] (≈0–312–704–912{{c}}), a {{W|minor sixth chord}} with a relatively simple otonal signature of 10:12:15:17. The [[2.3.5.17 subgroup|2.3.5.17-subgroup]] is optimized with a somewhat sharper fifth, around 705 cents, close to [[34edo]] or [[80edo]].
== Scales ==
{{Todo|complete section}}
=== 10-note (proper) ===
{{Main| 2L&nbsp;8s }}
=== 12-note (proper) ===
{{Main| 10L&nbsp;2s }}
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 705.8655{{c}}
| CSEE: ~3/2 = 705.5568{{c}}
| POEE: ~3/2 = 704.9311{{c}}
|-
! Tenney
| CTE: ~3/2 = 705.1363{{c}}
| CWE: ~3/2 = 704.9585{{c}}
| POTE: ~3/2 = 704.8982{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 704.8398{{c}}
| CSBE: ~3/2 = 704.7309{{c}}
| POBE: ~3/2 = 704.7977{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 704.0394{{c}}
| CSEE: ~3/2 = 703.8161{{c}}
| POEE: ~3/2 = 703.3785{{c}}
|-
! Tenney
| CTE: ~3/2 = 704.0493{{c}}
| CWE: ~3/2 = 703.7738{{c}}
| POTE: ~3/2 = 703.6809{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 704.0589{{c}}
| CSBE: ~3/2 = 703.7520{{c}}
| POBE: ~3/2 = 703.7438{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~3/2 = 704.0090{{c}}
| CSEE: ~3/2 = 703.9204{{c}}
| POEE: ~3/2 = 703.9138{{c}}
|-
! Tenney
| CTE: ~3/2 = 704.0164{{c}}
| CWE: ~3/2 = 703.8520{{c}}
| POTE: ~3/2 = 703.8121{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 704.0285{{c}}
| CSBE: ~3/2 = 703.7782{{c}}
| POBE: ~3/2 = 703.7642{{c}}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 17/9
| 698.955
|
|-
| [[12edo|7\12]]
|
| 700.000
| 12f val, lower bound of 7-, 9- and 11-odd-limit diamond monotone
|-
|
| 3/2
| 701.955
|
|-
|
| 15/14
| 702.778
|
|-
| [[70edo|41\70]]
|
| 702.857
| 70ef val
|-
|
| 7/5
| 702.915
|
|-
|
| 21/20
| 703.107
|
|-
|
| 15/11
| 703.359
|
|-
|
| 15/13
| 703.410
|
|-
| [[58edo|34\58]]
|
| 703.448
| Lower bound of 13-, 15-, 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 11/10
| 703.500
|
|-
|
| 9/7
| 703.508
|
|-
|
| 13/10
| 703.522
|
|-
|
| 13/11
| 703.597
|
|-
|
| 7/6
| 703.681
|
|-
| [[162edo|95\162]]
|
| 703.704
| 162cef val
|-
|
| 13/9
| 703.728
|
|-
|
| 11/9
| 703.757
|
|-
|
| 21/13
| 703.782
|
|-
|
| 49/48
| 703.783
|
|-
|
| 13/12
| 703.839
|
|-
| [[104edo|61\104]]
|
| 703.846
| 104c val
|-
|
| 21/11
| 703.893
|
|-
|
| 11/6
| 703.895
|
|-
|
| 7/4
| 703.897
|
|-
|
| 13/8
| 703.965
|
|-
| [[150edo|88\150]]
|
| 704.000
| 150c val
|-
|
| 17/14
| 704.014
|
|-
|
| 17/13
| 704.027
|
|-
|
| 13/7
| 704.043
|
|-
|
| 11/8
| 704.057
|
|-
|
| 36/35
| 704.064
|
|-
|
| 17/11
| 704.126
|
|-
|
| 21/16
| 704.174
|
|-
|
| 21/17
| 704.272
|
|-
| [[46edo|27\46]]
|
| 704.348
| Upper bound of 11-, 13-, 15-, 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 11/7
| 704.377
|
|-
|
| 9/5
| 704.399
|
|-
|
| 17/16
| 704.955
|
|-
|
| 5/3
| 705.214
|
|-
|
| 25/24
| 705.866
|
|-
| [[34edo|20\34]]
|
| 705.882
| 34ef val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|
| 17/10
| 706.214
|
|-
|
| 5/4
| 706.843
|
|-
|
| 17/15
| 708.343
|
|-
|
| 15/8
| 711.731
|
|}
<nowiki/>* Besides the octave
[[Category:Diaschismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Diaschismic family]]
[[Category:Diaschismic family]]
{{IoT}}
[[Category:Starling temperaments]]
[[Category:Aberschismic temperaments]]

Latest revision as of 03:02, 21 June 2026

Diaschismic
Subgroups 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
Comma basis 2048/2025 (5-limit);
126/125, 2048/2025 (7-limit);
126/125, 136/135, 176/175,
196/195, 256/255 (17-limit)
Reduced mapping ⟨2; 1 -2 -8 -12 -15 1]
ET join 46 & 58
Generators (CWE) ~3/2 = 703.9 ¢
MOS scales 2L 8s, 10L 2s, 12L 10s
Ploidacot diploid monocot
Pergen (P8/2, P5)
Color name Saguguti
Minimax error 5-odd-limit: 3.259 ¢;
17-limit 21-odd-limit: ??? ¢
Target scale size 5-odd-limit: 12 notes;
17-limit 21-odd-limit: 46 notes

Diaschismic, sometimes known as srutal in the 5-limit, is a half-octave temperament generated by a perfect fifth, ideally tuned slightly sharp, or that minus a half-octave period, which is a semitone representing 16/15. Two of these semitones give a whole tone of 9/8, so the diaschisma, 2048/2025 = 9/8 ÷ (16/15)2, is tempered out, and a half-octave represents 45/32, and subtracting a whole tone from that gives 5/4. Since the whole tone has been split in two equal halves of ~16/15, it makes sense to equate that semitone with 17/16 and 18/17, by tempering out 256/255 (S16), 289/288 (S17), and their product 136/135, thus giving the 2.3.5.17-subgroup version of diaschismic, sometimes known as srutal archagall.

The canonical extension to the 7-limit reaches 7/4 at -8 fifths (a diminished fourth) octave-reduced plus a semioctave, tempering out the starling comma, 126/125, as well as the aberschisma, 5120/5103. This equates the septimal comma with the syntonic comma and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. This extension works best between 46edo and 58edo.

In this tuning range, a stack of twelve perfect fifths octave reduced (a diesis), or two comma steps, is close in size to a quartertone, and that interval plus a half-octave can be used to represent 16/11. Three more fifths on top of 16/11 gives 16/13 when octave-reduced, and thus 13/11 is equated with the Pythagorean minor third 32/27. The mappings of primes 11 and 13 can also be characterized by parapyth, where the major third at +4 fifths represents 14/11, and the minor third at -3 fifths represents 13/11, which makes sense as the fifth is tuned slightly sharp. Adding the mapping of 17/16 to the semitone, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, 176/175, 196/195, and 256/255. While this provides much more harmonic resources, the 2.3.5.17-subgroup version is simpler and has a more flexible tuning range.

An alternative extension to the full 17-limit is srutal, which has a more complex mapping of prime 7 at +15 fifths, tempering out 4375/4374. The mappings for primes 11 and 13 follow through parapyth, and 17 is again mapped to the semitone. This works best for a sharper tuning, between 34edo (with the 34d val) and 46edo. Another option for extension is pajara, which equates the semioctave to 7/5 and 10/7. This inherits the inaccuracy of archy, while providing a much simpler (and arguably more elegant) representation of the 7-limit.

See Diaschismic family #Diaschismic and #Septimal diaschismic for technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

# Period 0 Period 1
Cents* Approximate ratios Cents* Approximate ratios
0 0.0 1/1 600.0 17/12, 24/17
1 703.9 3/2 103.9 16/15, 17/16, 18/17
2 207.7 9/8 807.7 8/5
3 911.6 17/10, 22/13 311.6 6/5
4 415.4 14/11 1015.4 9/5
5 1119.5 21/11, 40/21, 48/25 519.5 27/20
6 623.1 10/7 23.1 56/55, 64/63, 81/80
7 127.0 14/13, 15/14 727.0 32/21
8 830.8 21/13, 34/21 230.8 8/7
9 334.7 17/14, 40/33 934.7 12/7
10 1038.5 20/11 438.5 9/7
11 542.4 15/11 1142.4 27/14, 64/33
12 46.2 36/35, 40/39, 45/44, 50/49 646.2 16/11
13 750.1 17/11, 20/13 150.1 12/11
14 253.9 15/13 853.9 18/11
15 957.8 45/26, 68/39 357.8 16/13
16 461.6 17/13 1061.6 24/13
17 1165.5 51/26, 96/49, 108/55 565.5 18/13

* In 17-limit CWE tuning, octave-reduced

As a detemperament of 12et

Diaschismic as a 58-tone 12et detempering

Diaschismic is naturally considered as a detemperament of the 12 equal temperament. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14 steps. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios 51/50, 55/54, 64/63, 66/65, 78/77, 81/80, and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region. In general, ratios closer to intervals of 12edo, such as 17/16, are simpler in terms of generator steps, while ratios further from 12edo intervals, such as 11/9, are more complex. The 13th harmonic is just beyond the specified generator range, so it does not appear in the diagram.

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise, though it requires using the sharp, second-best approximation of 5/4 (aka the 104c val).

Chords and harmony

See also: Chords of diaschismic

Diaschismic finds the 5-limit triads very simply, with 1–5/4–3/2 (≈0–392–704 ¢) and 1–6/5–3/2 (≈0–312–704 ¢) each occuring six times in the 12-note 10L 2s mos scale. In optimal tunings, the 3/2 perfect fifth and 5/4 major third are both tuned slightly sharp, giving major triads an overall bright sound. Unlike in meantone, the syntonic comma 81/80 is not tempered out, meaning the 9/8 and 10/9 whole tones are distinguished. Due to the sharp fifth, the Pythagorean major and minor triads are neogothic in quality, representing 1–14/11–3/2 (≈0–416–704 ¢) and 1–13/11–3/2 (≈0–288–704 ¢) respectively. It can thus be seen as an expansion on the harmony of 12edo, which adds new flavors and distinctions.

Prime 17 also plays a role: For example, the interval 17/8, which is one octave above 17/16, can be used as a minor ninth in chords such as 1–5/4–3/2–17/8 (≈0–392–704 ¢). Additionally, the semioctave represents 17/12~24/17, and can act as a tritone in diminished triads such as 1–6/5–24/17 (≈0–312–600 ¢) and its melodic inverse 1–20/17–24/17 (≈0–288–600 ¢). The sharp ~9/8 whole tone is equated with a flat ~17/15, and 20/17 is equated with 32/27~13/11. Additionally, the dominant seventh chord can be seen as representing 1–5/4–3/2–30/17 (≈0–392–704–992 ¢), thus becoming dyadically consonant in the 17-odd-limit, meaning any interval between two notes has an interpretation with an odd limit of 17 or less. Its negative harmony version is 1–6/5–3/2–17/10 (≈0–312–704–912 ¢), a minor sixth chord with a relatively simple otonal signature of 10:12:15:17. The 2.3.5.17-subgroup is optimized with a somewhat sharper fifth, around 705 cents, close to 34edo or 80edo.

Scales

10-note (proper)

Main article: 2L 8s

12-note (proper)

Main article: 10L 2s

Tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 705.8655 ¢ CSEE: ~3/2 = 705.5568 ¢ POEE: ~3/2 = 704.9311 ¢
Tenney CTE: ~3/2 = 705.1363 ¢ CWE: ~3/2 = 704.9585 ¢ POTE: ~3/2 = 704.8982 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 704.8398 ¢ CSBE: ~3/2 = 704.7309 ¢ POBE: ~3/2 = 704.7977 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 704.0394 ¢ CSEE: ~3/2 = 703.8161 ¢ POEE: ~3/2 = 703.3785 ¢
Tenney CTE: ~3/2 = 704.0493 ¢ CWE: ~3/2 = 703.7738 ¢ POTE: ~3/2 = 703.6809 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 704.0589 ¢ CSBE: ~3/2 = 703.7520 ¢ POBE: ~3/2 = 703.7438 ¢
17-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 704.0090 ¢ CSEE: ~3/2 = 703.9204 ¢ POEE: ~3/2 = 703.9138 ¢
Tenney CTE: ~3/2 = 704.0164 ¢ CWE: ~3/2 = 703.8520 ¢ POTE: ~3/2 = 703.8121 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 704.0285 ¢ CSBE: ~3/2 = 703.7782 ¢ POBE: ~3/2 = 703.7642 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
17/9 698.955
7\12 700.000 12f val, lower bound of 7-, 9- and 11-odd-limit diamond monotone
3/2 701.955
15/14 702.778
41\70 702.857 70ef val
7/5 702.915
21/20 703.107
15/11 703.359
15/13 703.410
34\58 703.448 Lower bound of 13-, 15-, 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
11/10 703.500
9/7 703.508
13/10 703.522
13/11 703.597
7/6 703.681
95\162 703.704 162cef val
13/9 703.728
11/9 703.757
21/13 703.782
49/48 703.783
13/12 703.839
61\104 703.846 104c val
21/11 703.893
11/6 703.895
7/4 703.897
13/8 703.965
88\150 704.000 150c val
17/14 704.014
17/13 704.027
13/7 704.043
11/8 704.057
36/35 704.064
17/11 704.126
21/16 704.174
21/17 704.272
27\46 704.348 Upper bound of 11-, 13-, 15-, 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
11/7 704.377
9/5 704.399
17/16 704.955
5/3 705.214
25/24 705.866
20\34 705.882 34ef val, upper bound of 7- and 9-odd-limit diamond monotone
17/10 706.214
5/4 706.843
17/15 708.343
15/8 711.731

* Besides the octave

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