Diaschismic: Difference between revisions

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'''Diaschismic''', sometimes known as [[srutal ~~vs diaschismic|srutal]]~~ in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]] 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]], is [[tempering out|tempered out]]~~. We also have~~ a ~~whole tone plus a period represent~~ [[8/5]], and ~~in the canonical~~ [[~~extension~~]] to the [[~~7-limit~~]], ~~the fifth is tuned a little sharp such~~ that ~~eight of them octave reduced (an augmented fifth) minus a period approximate~~ [[8/7]], tempering out ~~the starling comma~~, [[~~126~~/~~125~~]].

{{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); [[126/125]], [[2048/2025]] (7-limit); [[126/125]], [[136/135]], [[176/175]], [[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)2, 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''.

~~A stack of twelve perfect fifths octave reduced (a~~ [[~~diesis (scale theory)|diesis~~]]~~) is close in size~~ to ~~a quartertone, and that plus a period can be used to represent~~ [[16/11]]~~. Three more~~ fifths ~~on top of 16/11 give~~ [[~~16/13~~]]~~. Finally~~, ~~since~~ the ~~whole tone has been split in two~~, ~~each can be used to represent~~ [[17/16]]~[[18/17]]. ~~Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125,~~ [[~~136~~/~~135~~]], [[~~176~~/~~175~~]], [[~~196/195~~]], and [[~~256/255~~]].

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]].

See [[Diaschismic family #Diaschismic]] for technical data.

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 ==

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<nowiki/>* In 17-limit CWE tuning, octave-reduced

=== As a ~~detempereament~~ of 12et ===

=== 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. 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 6 ~~generator steps~~, which ~~represent~~ the ~~syntonic~~ comma. 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.

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]]).

== Chords and harmony ==

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.

The ~~13th harmonic~~ is ~~just beyond~~ the ~~specified generator range~~, so the ~~diagram does not show it~~.

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]].

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 syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.

== Scales ==

=== 10-note (proper) ===

== ~~Chords~~ ==

=== 12-note (proper) ===

== Tunings ==

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 5-limit ~~prime~~-~~optimized~~ tunings

|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings

|-

! rowspan="2" |

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{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit ~~prime~~-~~optimized~~ tunings

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings

|-

! rowspan="2" |

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{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 17-limit ~~prime~~-~~optimized~~ tunings

|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings

|-

! rowspan="2" |

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|-

! Edo generator

! [[Eigenmonzo|~~Eigenmonzo~~ (~~unchanged-interval~~)]]*

! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*

! Generator (¢)

! Comments

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|-

|

| 12/7

| 7/6

| 703.681

|

|-

| [[162edo|95\162]]

|

| 703.704

| 162cef val

|-

|

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| 703.965

|

|-

| [[150edo|88\150]]

|

| 704.000

| 150c val

|-

|

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[[Category:Diaschismic family]]

[[Category:Starling temperaments]]

[[Category:~~Hemifamity~~ temperaments]]

[[Category:Aberschismic temperaments]]

@@ Line 1: / Line 1: @@
-'''Diaschismic''', sometimes known as [[srutal vs diaschismic|srutal]] in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]] 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]], is [[tempering out|tempered out]]. We also have a whole tone plus a period represent [[8/5]], and in the canonical [[extension]] to the [[7-limit]], the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]].
+{{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''.
-A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]) is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]]. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, [[136/135]], [[176/175]], [[196/195]], and [[256/255]].
+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]].
-See [[Diaschismic family #Diaschismic]] for technical data.
+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 ==
@@ Line 129: / Line 146: @@
 <nowiki/>* In 17-limit CWE tuning, octave-reduced
-=== As a detempereament of 12et ===
+=== 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. 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 6 generator steps, which represent the syntonic comma. 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.
+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.
-The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.
+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]].
-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 syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.
+== Scales ==
+{{Todo|complete section}}
+=== 10-note (proper) ===
+{{Main| 2L&nbsp;8s }}
-== Chords ==
+=== 12-note (proper) ===
-{{Main| Chords of diaschismic }}
+{{Main| 10L&nbsp;2s }}
 == Tunings ==
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
 |-
 ! rowspan="2" |
@@ Line 169: / Line 197: @@
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
 |-
 ! rowspan="2" |
@@ Line 195: / Line 223: @@
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 17-limit prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings
 |-
 ! rowspan="2" |
@@ Line 224: / Line 252: @@
 |-
 ! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
 ! Comments
@@ Line 299: / Line 327: @@
 |-
 |
-| 12/7
+| 7/6
 | 703.681
 |
+|-
+| [[162edo|95\162]]
+|
+| 703.704
+| 162cef val
 |-
 |
@@ Line 352: / Line 385: @@
 | 703.965
 |
+|-
+| [[150edo|88\150]]
+|
+| 704.000
+| 150c val
 |-
 |
@@ Line 454: / Line 492: @@
 [[Category:Diaschismic family]]
 [[Category:Starling temperaments]]
-[[Category:Hemifamity temperaments]]
+[[Category:Aberschismic temperaments]]