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'''Diasem''' is a [[Maximum variety|max-variety-3]] 7-limit JI scale (or a tempered version of it) that is equivalent to semaphore[9] with two of the small steps made larger and the other two made smaller. This results in better melodic properties than the meantone scales of [[26edo]] and [[31edo]], which both support it. The scale can be generated by an alternating chain of subminor thirds and supermajor seconds. The name "diasem" is a portmanteau of "diatonic" and "semaphore" since it the way it tempers the [[64/63]] is intermediate between [[superpyth]] and [[semaphore]]; it is also a pun based on the [[diesis]], a defining step size in the scale.
{{interwiki
|de = Diasem
|en = Diasem
|es =
|ja =
}}
'''Diasem''' (also denoted 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with [[step signature]] 5L 2m 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed (RH) diasem'' LmLsLmLsL and ''left-handed (LH) diasem'' LsLmLsLmL; these [[step pattern]]s are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales. [[21edo]] is the smallest edo to support a non-degenerate diasem.
 
Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], m represents [[28/27]], and s represents [[64/63]]. This interpretation, or more generally the series of [[generator sequence]] scales generated by GS(7/6, 8/7) or GS(8/7, 7/6), has been named [[Tas]].
 
"Diasem" is a name given by [[groundfault]] (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "[[Semaphore]]") since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[semiquartal]] (5L 4s); it is also a pun based on the [[diesis]], which appears as the small step in the scale in the [[31edo]] and [[36edo]] tunings.


{| class="wikitable"
{| class="wikitable"
|+ Comparison with semaphore and meantone in 62edo
|+ Comparison of diasem with semiquartal and diatonic in 62edo
|-
|-
! Name !! Structure !! Step Sizes !! Graphical Representation
! Name !! Structure !! Step Sizes !! Graphical Representation
|-
|-
| Semaphore || 5L4s || 10\62, 3\62 || ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤
| Semiquartal || 5L 4s || 10\62, 3\62 || ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤
|-
| Diasem || 5L 2m 2s || 10\62, 4\62, 2\62 || ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤
|-
| Diatonic || 5L 2s || 10\62, 6\62 || ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤
|}
 
== Intervals ==
The following is a table of diasem intervals and their abstract sizes in terms of L, m and s. Given concrete sizes of L, m and s in edo steps or cents, you can compute the concrete size of any interval in diasem using the following expressions.
 
{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
|+ Interval sizes in diasem
|-
!colspan=2|Interval class
! Sizes
! 2.3.7 JI
! [[21edo]] (L:m:s = 3:2:1)
! [[31edo]] (L:m:s = 5:2:1)
|- bgcolor="#eaeaff"
!rowspan=3|[[TAMNAMS|1-steps]]
!| <small>small</small>
|| s
| 64/63, 27.26¢
| 1\21, 57.14¢
| 1\31, 38.71¢
|- bgcolor="#eaeaff"
!|<small>medium</small>
| m
| 28/27, 62.96¢
| 2\21, 114.29¢
| 2\31, 77.42¢
|- bgcolor="#eaeaff"
!|<small>large</small>
| L
| 9/8, 203.91¢
| 3\21, 171.43¢
| 5\31, 193.55¢
|-
!rowspan=3|[[TAMNAMS|2-steps]]
!|<small>small</small>
| L + s
| 8/7, 231.17¢
| 4\21, 228.57¢
| 6\31, 232.26¢
|-
!|<small>medium</small>
| L + m
| 7/6, 266.87¢
| 5\21, 285.71¢
| 7\31, 270.97¢
|-
!|<small>large</small>
| 2L
| 81/64, 407.82¢
| 6\21, 342.86¢
| 10\31, 387.10¢
|-  bgcolor="#eaeaff"
!rowspan=3|[[TAMNAMS|3-steps]]
!|<small>small</small>
| L + m + s
| 32/27, 294.14¢
| 6\21, 342.86¢
| 8\31, 309.68¢
|- bgcolor="#eaeaff"
!|<small>medium</small>
| 2L + s
| 9/7, 435.08¢
| 7\21, 400.00¢
| 11\31, 425.81¢
|- bgcolor="#eaeaff"
!|<small>large</small>
| 2L + m
| 21/16, 470.78¢
| 8\21, 457.14¢
| 12\31, 464.52¢
|-
!rowspan=3|[[TAMNAMS|4-steps]]
!|<small>small</small>
| 2L + m + s
| 4/3, 498.04¢
| 9\21, 514.29¢
| 13\31, 503.23¢
|-
!|<small>medium</small>
| 3L + s
| 81/56, 638.99¢
| 10\21, 571.43¢
| 16\31, 619.35¢
|-
!|<small>large</small>
| 3L + m
| 189/128, 674.69¢
| 11\21, 628.57¢
| 17\31, 658.06¢
|- bgcolor="#eaeaff"
!rowspan=3|[[TAMNAMS|5-steps]]
!|<small>small</small>
| 2L + m + 2s
| 256/189, 525.31¢
| 10\21, 571.43¢
| 14\31, 541.94¢
|- bgcolor="#eaeaff"
!|<small>medium</small>
| 2L + 2m + s
| 112/81, 561.01¢
| 11\21, 628.57¢
| 15\31, 580.65¢
|- bgcolor="#eaeaff"
!|<small>large</small>
| 3L + m + s
| 3/2, 701.96¢
| 12\21, 685.71¢
| 18\31, 696.77¢
|-
!rowspan=3|[[TAMNAMS|6-steps]]
!|<small>small</small>
| 3L + m + 2s
| 32/21, 729.22¢
| 13\21, 742.86¢
| 19\31, 735.48¢
|-
!|<small>medium</small>
| 3L + 2m + s
| 14/9, 764.92¢
| 14\21, 800.00¢
| 20\31, 774.19¢
|-
!|<small>large</small>
| 4L + m + s
| 27/16, 905.87¢
| 15\21, 857.14¢
| 23\31, 890.32¢
|-  bgcolor="#eaeaff"
!rowspan=3|[[TAMNAMS|7-steps]]
!|<small>small</small>
| 3L + 2m + 2s
| 128/81, 792.18¢
| 15\21, 857.14¢
| 21\31, 812.90¢
|- bgcolor="#eaeaff"
!|<small>medium</small>
| 4L + m + 2s
| 12/7, 933.13¢
| 16\21, 914.29¢
| 24\31, 929.03¢
|- bgcolor="#eaeaff"
!|<small>large</small>
| 4L + 2m + s
| 7/4, 968.83¢
| 17\21, 971.43¢
| 25\31, 967.74¢
|-
!rowspan=3|[[TAMNAMS|8-steps]]
!|<small>small</small>
| 4L + 2m + 2s
| 16/9, 996.09¢
| 18\21, 1028.57¢
| 26\31, 1006.45¢
|-
!|<small>medium</small>
| 5L + m + 2s
| 54/28, 1137.04¢
| 19\21, 1085.71¢
| 29\31, 1122.58¢
|-
!|<small>large</small>
| 5L + 2m + s
| 63/32, 1172.74¢
| 20\21, 1142.86¢
| 30\31, 1161.29¢
|}
The octave can be called the "perfect 9-step" in [[TAMNAMS]].
 
== Properties ==
Any diasem scale with positive step sizes has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:
* > 4\7 if L > m + s
* = 4\7 if L = m + s
* < 4\7 if L < m + s
(This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.)
 
The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset L + m or L + s (respectively a flat minor third or a sharp major second in tunings of diasem with "reasonable" fifths and small s steps).
 
== Modes ==
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. We have provided names based on the modes of the [[5L 4s]], [[5L 2s]] and [[7L 2s]] temperings of each mode.
 
=== Cyclic order ===
The modes arranged in cyclic order:
{| class="wikitable"
|-
! style="text-align:center;" |Left-handed modes
! style="text-align:center;" |Right-handed modes
|-
| | '''LsLmLsLmL''' <br/>LH Diasem Nucifragan<br/>LH Diasem Mixo<br/>LH Diasem Superaeolian
| | '''LmLsLmLsL''' <br/>RH Diasem Nucifragan<br/>RH Diasem Aeolian<br/>RH Diasem Supermixo
|-
| | '''sLmLsLmLL''' <br/>LH Diasem Pyrrhian<br/>LH Diasem Bright Aeolian<br/>LH Diasem Superlocrian
| | '''mLsLmLsLL''' <br/>RH Diasem Pyrrhian<br/>RH Diasem Locrian<br/>RH Diasem Olympian
|-
| | '''LmLsLmLLs''' <br/>LH Diasem Cornician<br/>LH Diasem Dark Aeolian<br/>LH Diasem Superionian
| | '''LsLmLsLLm''' <br/>RH Diasem Cornician<br/>RH Diasem Ionian<br/>RH Diasem Superaeolian
|-
| | '''mLsLmLLsL''' <br/>LH Diasem Coloean<br/>LH Diasem Locrian<br/>LH Diasem Corinthian
| | '''sLmLsLLmL''' <br/>RH Diasem Coloean<br/>RH Diasem Bright Dorian<br/>RH Diasem Superlocrian
|-
|-
| Diasem || 5L2m2s || 10\62, 4\62, 2\62 || ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤
| | '''LsLmLLsLm''' <br/>LH Diasem Stellerian<br/>LH Diasem Ionian<br/>LH Diasem Superdorian
| | '''LmLsLLmLs''' <br/>RH Diasem Stellerian<br/>RH Diasem Dark Dorian<br/>RH Diasem Superionian
|-
|-
| Meantone || 5L2s || 10\62, 6\62 || ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤
| | '''sLmLLsLmL''' <br/>LH Diasem Frugilegian<br/>LH Diasem Bright Dorian<br/>LH Diasem Superphrygian
| | '''mLsLLmLsL''' <br/>RH Diasem Frugilegian<br/>RH Diasem Phrygian<br/>RH Diasem Corinthian
|-
| | '''LmLLsLmLs''' <br/>LH Diasem Pican<br/>LH Diasem Dark Dorian<br/>LH Diasem Superlydian
| | '''LsLLmLsLm''' <br/>RH Diasem Pican<br/>RH Diasem Lydian<br/>RH Diasem Superdorian
|-
| | '''mLLsLmLsL''' <br/>LH Diasem Coracian<br/>LH Diasem Phrygian<br/>LH Diasem Supermixo
| | '''sLLmLsLmL''' <br/>RH Diasem Coracian<br/>RH Diasem Bright Mixo<br/>RH Diasem Superphrygian
|-
| | '''LLsLmLsLm''' <br/>LH Diasem Cristatan<br/>LH Diasem Lydian<br/>LH Diasem Olympian
| | '''LLmLsLmLs''' <br/>RH Diasem Cristatan<br/>RH Diasem Dark Mixo<br/>RH Diasem Superlydian
|}
|}


Like [[superpyth]], diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths, unlike the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out [[81/80]] are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps.
=== Arranged by generator chain ===
When we arrange the modes in the order given by rotating each mode by the generator (the perfect fifth) we obtain the following families of modes (">" roughly means 'brighter than'):
# RH
## LsLLmLsLm (RH Lydian) > LsLmLsLLm (RH Ionian) > sLLmLsLmL (RH Bright Mixo) > sLmLsLLmL (RH Bright Dorian)
## LLmLsLmLs (RH Dark Mixo) > LmLsLLmLs (RH Dark Dorian) > LmLsLmLsL (RH Aeolian) > mLsLLmLsL (RH Phrygian) > mLsLmLsLL (RH Locrian)
# LH
## LLsLmLsLm (LH Lydian) > LsLmLLsLm (LH Ionian) > LsLmLsLmL (LH Mixo) > sLmLLsLmL (LH Bright Dorian) > sLmLsLmLL (LH Bright Aeolian)
## LmLLsLmLs (LH Dark Dorian) > LmLsLmLLs (LH Dark Aeolian) > mLLsLmLsL (LH Phrygian) > mLsLmLLsL (LH Locrian)
This provides a clear motivation for the diatonic-based mode names.
 
=== Negative-s blackdye ===
Consider right-hand diasem, fixing a choice of positive step sizes. There exists a way of superimposing a left-hand diasem mode on the right hand diasem so that the right-hand diasem and the left-hand diasem overlap in 8 notes, yielding a scale of 10 notes, possibly after changing the mode of right-hand diasem. For example, superimposing LmLsLmLsL (RH Aeolian) and LmLsLmLLs (LH Dark Aeolian) gives LmLsLmLs(L-s)s (in fifth-based notation on C: C D Ebv Fv F G Abv Bbv Bb Cv C, where v denotes lowering by s). Note that the union is achiral. This new scale has two chains of perfect fifths each spanning 5 notes:
 
Chain 1: '''Bb''' (L-s)sLmLs '''F''' LmLs(L-s)S '''C''' LmLsL '''G''' mLs(L-s)sL '''D'''
 
Chain 2: '''Abv''' Ls(L-s)sLm '''Ebv''' LsLmL '''Bbv''' S(L-s)sLmL '''Fv''' sLmLs(L-s) '''Cv'''
 
Notice that we have two generator chains of equal length. To give this scale a generator-offset structure we can treat the large 7-step as the offset of the 10-note scale. We treat L+S as one scale step and consider the scale an interleaving of two pentatonic scales, using the notes of C-D-F-G-Bb for the even numbered notes and Cv-Ebv-Fv-Abv-Bbv for the odd ones. This gives the following ordering: C Cv D Ebv F Fv G Abv Bb Bbv C, or in step sizes, -s L+s M L+s -s L+s M L+s -s L+s. This is formally a [[blackdye]] (sL'mL's'L'mL's'L') pattern, albeit with a negative step size s' = -s! This scale has been called '''negative blackdye''' or '''negative-s blackdye'''.
 
== Alterations ==
* Diasem Melodic Minor LmLsLLsLm
 
== In JI and similar tunings ==
Like [[Superpyth]], JI diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths. On the other hand, the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out [[81/80]] are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what Superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps.
 
=== As a Fokker block ===
[[File:Diasem as fokker block.png|600px|thumbnail|2.3.7 JI diasem as a Fokker block]]
The 2.3.7 JI diasem scale can be viewed as a [[Fokker block]] living in the 2.3.7 octave-equivalent pitch class lattice. The x-axis goes along the 3 direction and the y-axis goes along the 7 direction.
 
The diagram shows the LmLsLmLsL mode. Each dot represents a pitch class of a note in the 2.3.7 lattice. All the notes of the mode are marked as solid purple dots. Notes of the lattice outside the mode are black hollow dots. The red dashed lines are separated by the chroma 49/48, and the blue dotted lines are separated by the chroma 567/512. Note that both 49/48 and 567/512 are tempered out by (the 2.3.7 [[patent val]] of) [[9edo]].
 
The notes of diasem form the {49/48, 567/512} Fokker block, which is a fundamental domain of the 2.3.7 pitch class lattice; it is possible to tile the entire infinite lattice with copies of right-hand diasem transLated by (49/48)<sup>''m''</sup>(567/512)<sup>''n''</sup> for integer ''m'' and ''n''. Including any one of the other three points on the boundary (28/27, 147/128, or 64/63) instead of 9/8 aLso yields Fokker blocks, more specifically, modes of three of the other [[dome]]s of diasem, and transLates of the parallelogram that do not have lattice points on the boundary lead to other domes of this Fokker block. However, only one other choice, 28/27, yields a diasem scale, and it yields the left-handed diasem mode mLLsLmLsL.
 
As a Fokker block, 2.3.7 JI diasem is aLso a product of the tempered 2.3.7 mosses Semaphore[9] (LsLsLsLsL) and septimal Mavila[9] (LLLsLLLsL).
 
== Tunings ==
== Tunings ==
{| class="wikitable"
{{todo|cleanup}} <!-- the table is unreadable in wikitext -->
|+ Common Diasem Tunings
{| class="wikitable sortable"
|+ Diasem tunings
! rowspan="2" |Tuning
! rowspan="2" |Tuning
! rowspan="2" |L:m:s
! rowspan="2" |L:m:s
! rowspan="2" |Good Just Approximations
! rowspan="2" class="unsortable" |Good JI approximations
! rowspan="2" |other comments
! rowspan="2" class="unsortable" |other comments
! colspan="8" |Degrees
! colspan="8" |Degrees of the mode LmLsLmLsL
|-
|-
!1!!2!!3!!4!!5 !!6 !! 7!!8
!1!!2!!3!!4!!5 !!6 !! 7!!8
|-
|-
| || ||
|colspan=4 align=center|2.3.7 subgroup interpretation
| || 9/8||7/6||21/16||4/3||3/2|| 14/9 ||7/4 ||16/9
| 9/8||7/6||21/16||4/3||3/2|| 14/9 ||7/4 ||16/9
|-
|-
|JI||7.479:2.309:1||Just 7/6, 8/7, and 3/2
|JI||7.479:2.309:1||Just 7/6, 8/7, and 3/2
Line 33: Line 291:
| 21edo
| 21edo
|3:2:1
|3:2:1
|
| 16/15, 23/16 and 39/32
|
|
|171.429
|171.429
Line 44: Line 302:
|1028.571
|1028.571
|-
|-
|26edo||4:2:1 ||Neogothic thirds and 8/7
| 26edo || 4:2:1 || 14/11, 8/7 and 11/8
| ||184.615||276.923 ||461.538 ||507.692 ||692.308||784.615|| 969.231||1015.385
| ||184.615||276.923 ||461.538 ||507.692 ||692.308||784.615|| 969.231||1015.385
|-
|-
| 28edo
| 28edo
|4:3:1
|4:3:1
|Pental thirds
|5/4 and 13/7
|
|
| 171.429
| 171.429
Line 56: Line 314:
|514.286
|514.286
|685.714
|685.714
|771.429
|814.286
|985.714
|985.714
|1028.571
|1028.571
Line 62: Line 320:
|30edo
|30edo
| 4:3:2
| 4:3:2
|
|13/8
| cross between Mavila and Semaphore
| [[superdiatonic]] fifth
|160
|160
|280
|280
Line 78: Line 336:
|33edo
|33edo
|5:3:1
|5:3:1
|Septimal and Neogothic thirds and 10/9
|9/7, 13/11 and 10/9
|
|
| 181.818
| 181.818
Line 91: Line 349:
|35edo
|35edo
| 5:3:2
| 5:3:2
5:4:1
|
|
|Uses 21/16 as inconsistent 4/3
|
|171.429
|171.429
|274.286
|274.286
308.571
|445.714
|445.714
480
|514.286
|514.286
|685.714
|685.714
|788.571
|788.571
822.857
|960
|960
994.286
|1028.571
|-
|35edo
| 5:4:1
|
|
|171.429
|308.571
|480
|514.286
|685.714
|822.857
|994.286
|1028.571
|1028.571
|-
|-
Line 113: Line 379:
|5:4:2
|5:4:2
|35/32
|35/32
|cross between Mavila and Semaphore
|[[superdiatonic]] fifth
|162.162
|162.162
|291.892
|291.892
Line 139: Line 405:
|5:4:3
|5:4:3
|
|
| cross between Mavila and Semaphore
| [[superdiatonic]] fifth
|153.846
|153.846
|276.923
|276.923
Line 151: Line 417:
|40edo
|40edo
|6:3:2
|6:3:2
6:4:1
|
|
|Uses 21/16 as inconsistent 4/3
|
| 180
| 180
| 270
| 270
300
|450
|450
480
| 510
| 510
|690
|690
| 780
| 780
810
|960
|960
990
|1020
|-
|40edo
|6:4:1
|
|
| 180
| 300
|480
| 510
|690
|810
|990
|1020
|1020
|-
|-
Line 183: Line 457:
|6:5:1
|6:5:1
|
|
|Uses 21/16 as inconsistent 4/3
|
|171.429
|171.429
| 314.286
| 314.286
Line 208: Line 482:
|44edo
|44edo
|6:4:3
|6:4:3
6:5:2
|11/10 (and 9/7)
|11/10 (and 9/7)
|cross between Mavila and Semaphore
|[[superdiatonic]] fifth
|163.636
|163.636
|272.727
|272.727
300
|436.364
|436.364
463.636
|518.182
|518.182
|681.818
|681.818
|790.909
|790.909
818.182
|954.5455
|954.5455
981.818
|1036.364
|-
|44edo
|6:5:2
|11/10 (and 9/7)
|[[superdiatonic]] fifth
|163.636
|300
|463.636
|518.182
|681.818
|818.182
|981.818
|1036.364
|1036.364
|-
|-
|45edo
|45edo
|7:3:2
|7:3:2
7:4:1
|
|
|
|
|186.667
|186.667
|266.667
|266.667
293.333
|453.333
|453.333
480
|506.667
|506.667
|693.333
|693.333
|773.333
|773.333
800
|960
|960
986.667
|1013.333
|-
|45edo
|7:4:1
|
|
|186.667
|293.333
|480
|506.667
|693.333
|800
|986.667
|1013.333
|1013.333
|-
|-
|46edo
|46edo
|6:5:3
|6:5:3
8:2:1
|Neogothic thirds
|Neogothic thirds
|cross between Mavila and Semaphore
|[[superdiatonic]] fifth
Gentle fifth
|156.522
|156.522
208.696
|286.9565
|286.9565
260.87
|443.478
|443.478
469.565
|521.739
|521.739
495.652
|678.231
|678.231
704.348
|808.696
|808.696
756.522
|965.218
|965.218
|1043.418
|1043.418
991.314
|-
|46edo
|8:2:1
|Neogothic thirds
|[[gentle region|gentle]] fifth
|208.696
|260.87
|469.565
|495.652
|704.348
|756.522
|965.218
|991.314
|-
|-
|47edo
|47edo
|7:4:2
|7:4:2
7:5:1
|
|
|Uses 21/16 as inconsistent 4/3
|
|178.723
|178.723
|280.851
|280.851
306.383
|459.578
|459.578
485.106
|510.638
|510.638
|689.362
|689.362
|791.489
|791.489
817.021
|970.212
|970.212
995.744
|1021.277
|1021.27h
|-
|47edo
|7:5:1
|
|
|178.723
|306.383
|485.106
|510.638
|689.362
|817.021
|995.744
|1021.277
|-
|-
|48edo
|48edo
|6:5:4
|6:5:4
8:3:1
|
|
|cross between Mavila and Semaphore
|[[superdiatonic]] fifth
|150
|150
200
|275
|275
|425
|425
475
|525
|525
500
|675
|675
700
|800
|800
775
|950
|950
975
|1050
|1050
1000
|-
|48edo
|8:3:1
|
|[[superdiatonic]] fifth
|200
|275
|475
|500
|700
|775
|975
|1000
|-
|-
|49edo
|49edo
|7:4:3
|7:4:3
7:5:2
7:6:1
|
|
|Uses 21/16 as inconsistent 4/3
|
|171.429
|171.429
|269.388
|269.388
293.878
318.367
|440.817
|440.817
465.756
489.796
|514.286
|514.286
|685.714
|685.714
|783.6735
|783.6735
808.163
832.653
|955.102
|955.102
979.592
|1028.571
 
|-
1004.082
|49edo
|7:5:2
|
|
|171.429
|293.878
|465.756
|514.286
|685.714
|808.163
|979.592
|1028.571
|-
|49edo
|7:6:1
|
|
|171.429
|318.367
|489.796
|514.286
|685.714
|832.653
|1004.082
|1028.571
|1028.571
|-
|-
|50edo
|50edo
|8:3:2
|8:3:2
8:4:1
|
|
|
|
|192
|192
|264
|264
288
|456
|456
480
|504
|504
|696
|696
|768
|768
792
|960
|960
984
|1008
|-
|50edo
|8:4:1
|
|
|192
|288
|480
|504
|696
|792
|984
|1008
|1008
|-
|-
|51edo
|51edo
|7:5:3
|7:5:3
7:6:2
|
|
|cross between Mavila and Semaphore
|[[superdiatonic]] fifth
|164.706
|164.706
|282.353
|282.353
305.882
|447.059
|447.059
470.588
|517.647
|517.647
|682.353
|682.353
|800
|800
823.529
|964.706
|964.706
988.235
|1035.294
|-
|51edo
|7:6:2
|
|[[superdiatonic]] fifth
|164.706
|305.882
|470.588
|517.647
|682.353
|823.529
|988.235
|1035.294
|1035.294
|-
|-
Line 370: Line 704:
|8:5:1
|8:5:1
|
|
|Uses 21/16 as inconsistent 4/3
|
|184.615
|184.615
|300
|300
Line 382: Line 716:
|53edo
|53edo
|7:5:4
|7:5:4
7:6:3
|27/20
|27/20
|cross between Mavila and Semaphore
|[[superdiatonic]] fifth
|158.491
|158.491
|271.698
|271.698
294.34
|429.189
|429.189
452.831
|520.755
|520.755
|679.245
|679.245
|792.453
|792.453
815.094
|950.944
|950.944
973.585
|1041.509
|1041.509
|-
|-
|54edo
|53edo
|8:4:3
|7:6:3
8:5:2
|27/20
|[[superdiatonic]] fifth
|158.491
|294.34
|452.831
|520.755
|679.245
|815.094
|973.585
|1041.509
|}
 
== Tuning examples ==
=== LsLLmLsLm ===
An example in the RH Diasem Lydian mode LsLLmLsLm. ([[:File:Diasem Lydian Example Score.pdf|score]])
 
[[File:Diasem Lydian Example 14edo.mp3]] [[14edo]], L:M:S = 2:1:1 (degenerate; this is [[basic]] [[semiquartal]])
 
[[File:Diasem Lydian Example 16edo.mp3]] [[16edo]], L:M:S = 2:2:1 (degenerate; this is [[basic]] [[superdiatonic]])
 
[[File:Diasem Lydian Example 19edo.mp3]] [[19edo]], L:M:S = 3:1:1 (degenerate; this is [[hard]] [[semiquartal]])
 
[[File:Diasem Lydian Example 21edo.mp3]] [[21edo]], L:M:S = 3:2:1
 
[[File:Diasem Lydian Example 23edo.mp3]] [[23edo]], L:M:S = 3:2:2 (degenerate; this is [[soft]] [[semiquartal]])
 
[[File:Diasem Lydian Example 24edo.mp3]] [[24edo]], L:M:S = 4:1:1 (degenerate; this is [[superhard]] [[semiquartal]])
 
[[File:Diasem Lydian Example 26edo.mp3]] [[26edo]], L:M:S = 4:2:1
 
[[File:Diasem Lydian Example 28edo.mp3]] [[28edo]], L:M:S = 4:3:1
 
[[File:Diasem Lydian Example 31edo.mp3]] [[31edo]], L:M:S = 5:2:1
 
[[File:Diasem Lydian Example 33edo.mp3]] [[33edo]], L:M:S = 5:3:1
 
[[File:Diasem Lydian Example 33edo semiquartal.mp3]] [[33edo]], L:M:S = 5:2:2 (degenerate; this is [[semihard]] [[semiquartal]])
 
[[File:Diasem Lydian Example 35edo.mp3]] [[35edo]], L:M:S = 5:4:1
 
[[File:Diasem Lydian Example 35edo 5 3 2.mp3]] [[35edo]], L:M:S = 5:3:2
 
[[File:Diasem Lydian Example 36edo.mp3]] [[36edo]], L:M:S = 6:2:1
 
[[File:Diasem Lydian Example 38edo.mp3]] [[38edo]], L:M:S = 6:3:1
 
[[File:Diasem Lydian Example 41edo.mp3]] [[41edo]], L:M:S = 7:2:1
 
===mLsLmLLsL===
An example in LH Diasem Locrian mode mLsLmLLsL. ([[:File:Diasem Locrian Example.pdf|score]])
 
[[File:Diasem Locrian Example 14edo.mp3]] [[14edo]], L:M:S = 2:1:1 (degenerate; this is [[basic]] [[semiquartal]])
 
[[File:Diasem Locrian Example 16edo.mp3]] [[16edo]], L:M:S = 2:2:1 (degenerate; this is [[basic]] [[superdiatonic]])
 
[[File:Diasem Locrian Example 19edo.mp3]] [[19edo]], L:M:S = 3:1:1 (degenerate; this is [[hard]] [[semiquartal]])
 
[[File:Diasem Locrian Example 21edo.mp3]] [[21edo]], L:M:S = 3:2:1
 
[[File:Diasem Locrian Example 24edo.mp3]] [[24edo]], L:M:S = 4:1:1 (degenerate; this is [[superhard]] [[semiquartal]])
 
[[File:Diasem Locrian Example 26edo.mp3]] [[26edo]], L:M:S = 4:2:1
 
[[File:Diasem Locrian Example 28edo.mp3]] [[28edo]], L:M:S = 4:3:1
 
[[File:Diasem Locrian Example 31edo.mp3]] [[31edo]], L:M:S = 5:2:1
 
[[File:Diasem Locrian Example 33edo.mp3]] [[33edo]], L:M:S = 5:3:1
 
[[File:Diasem Locrian Example 35edo.mp3]] [[35edo]], L:M:S = 5:4:1


8:6:1
[[File:Diasem Locrian Example 36edo.mp3]] [[36edo]], L:M:S = 6:2:1
|Septimal thirds
 
Neogothic thirds
[[File:Diasem Locrian Example 38edo.mp3]] [[38edo]], L:M:S = 6:3:1
|Uses 21/16 as inconsistent 4/3
|177.778
|266.667
288.889


311.111
[[File:Diasem Locrian Example 41edo.mp3]] [[41edo]], L:M:S = 7:2:1
|444.444
466.667


488.889
=== mLLsLmLsL ===
|511.111
[[File:Diasem Phrygian Example 21edo.mp3]] [[21edo]], L:M:S = 3:2:1
|688.889
|777.778
800


822.222
[[File:Diasem Phrygian Example 26edo.mp3]] [[26edo]], L:M:S = 4:2:1
|955.556
977.778


1000
[[File:Diasem Phrygian Example 31edo.mp3]] [[31edo]], L:M:S = 5:2:1
|1022.222
|-
|55edo
|7:6:4
|
|cross between Mavila and Semaphore
|152.727
|283.636
|436.364
|523.636
|676.364
|807.273
|960
|1047.273
|-
|56edo
|8:5:3
8:7:1
|
|Golden tuning
Uses 21/16 as inconsistent 4/3
|171.429
|278.571
321.429
|450
492.857
|514.286
|685.714
|792.857
814.286
|964.286
985.714
|1028.571
|-
|57edo
|7:6:5
|
|cross between Mavila and Semaphore
|147.368
|273.684
|421.053
|526.316
|673.684
|800
|947.368
|1052.684
|-
|58edo
|8:5:4
8:6:3


8:7:2
== Supersets ==
|(Septimal and) Neogothic thirds
The diasem scale extends to a 14-note generator-offset scale: LmLsLmLsL and LsLmLsLmL both extend to the scale mcmcmsmcmcmsmc (c = L &minus; m), with two 7-note mosses generated by the diasem's fifths separated by m. This scale is not chiral. This scale extends diasem like how [[blackdye]] is a 10-note non-chiral generator-offset superset of the [[Zarlino]] scale's AG pattern 3L 2m 2s, LmsLmLs. The 14-note superset is one of:
|cross between Mavila and Semaphore
* 5L 7m 2s (if m < c < L)
|165.517
* a 2-step modmos of [[12L 2s]] (if c = m)
|268.9655
* 7L 5m 2s (if s < c < m)
290.394
* [[7L 7s]] (if c = s)
* 7L 2m 5s (if c < s).
5L 7m 2s must have a diatonic fifth, since L > 2m > m + s. The 31edo tuning (c = 3\31, m = 2\31, s = 1\31) of the scale is ideal for the 81/80-tempering 2.3.5.7 interpretation.


311.084
Another superset is scscsmscscsmsc, with c = L - s (5L 2m 7s if c > m). 31edo diasem yields 5L 2m 7s with step ratio 4:2:1.  
|435.222
455.911


476.601
Both these tunings, 5L 7m 2s and 5L 2m 7s, have been named '''crossdye''' ("crossed eyes" referring to the two copies of 5L 2s diatonic + "blackdye", courtesy of [[User:CellularAutomaton|cellularAutomaton]]). 5L 7m 2s has been called ''chromatic crossdye'', and 5L 2m 7s has been called ''dietic crossdye'' and ''whitedye''.
|517.98
|682.02
|786.946
807.6355


828.325
2.3.7 JI diasem also has the following generator-offset, SV3 supersets:
|952.463
* a 19-note superset: mLsmsLmsmLsmsLmsmLs (5L 7m 7s), with L = 2187/2048, m = 28/27, and s = 64/63,
973.153
* a 29-note superset: mLsmLmLsmLmLmsLmLmLsmLmLmsLmL (12L 12m 5s), with L = 28/27, m = 64/63, and s = 531441/524288.


993.842
== See also ==
|1034.483
* [[Blackdye]], a similar diatonic detempering but for 2.3.5
|-
|60edo
|8:7:3
|
|cross between Mavila and Semaphore
|160
|300
|460
|520
|680
|820
|980
|1040
|-
|62edo
|8:7:4
|Neogothic thirds
|cross between Mavila and Semaphore
|154.839
|290.323
|445.161
|522.581
|677.419
|812.903
|967.742
|1045.161
|-
|64edo
|8:7:5
|
|cross between Mavila and Semaphore
|150
|281.25
|431.25
|525
|675
|806.25
|956.25
|1050
|-
|66edo
|8:7:6
|Neogothic thirds
|cross between Mavila and Semaphore
|145.4545
|272.727
|418.182
|527.273
|672.727
|800
|945.4545
|1054.5455
|}


==Links==
==Links==
*[https://sevish.com/scaleworkshop/?name=JI%20Diasem&data=9%2F8%0A7%2F6%0A21%2F16%0A4%2F3%0A3%2F2%0A14%2F9%0A7%2F4%0A16%2F9%0A2%2F1&freq=440&midi=69&vert=5&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=square&ampenv=organ Play JI diasem] - Sevish Scale Workshop
*[https://sevish.com/scaleworkshop/?name=JI%20Diasem&data=9%2F8%0A7%2F6%0A21%2F16%0A4%2F3%0A3%2F2%0A14%2F9%0A7%2F4%0A16%2F9%0A2%2F1&freq=440&midi=69&vert=5&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=square&ampenv=organ Play JI diasem] - Sevish Scale Workshop
*[https://sevish.com/scaleworkshop/?name=Diasem%2026edo&data=46.15384615384615%0A92.3076923076923%0A138.46153846153845%0A184.6153846153846%0A230.76923076923077%0A276.9230769230769%0A323.0769230769231%0A369.2307692307692%0A415.38461538461536%0A461.53846153846155%0A507.6923076923077%0A553.8461538461538%0A600.%0A646.1538461538462%0A692.3076923076923%0A738.4615384615385%0A784.6153846153846%0A830.7692307692307%0A876.9230769230769%0A923.0769230769231%0A969.2307692307692%0A1015.3846153846154%0A1061.5384615384614%0A1107.6923076923076%0A1153.8461538461538%0A1200.&freq=440&midi=69&vert=9&horiz=1&colors=white%20black%20grey%20black%20white%20black%20white%20black%20grey%20black%20white%20white%20black%20grey%20black%20white%20black%20white%20black%20grey%20black%20white%20white%20black%20grey%20black&waveform=square&ampenv=organ Play 26edo diasem] - Sevish Scale Workshop
*[https://sevish.com/scaleworkshop/?name=Diasem%2026edo&data=184.6153846153846%0A276.9230769230769%0A461.53846153846155%0A507.6923076923077%0A692.3076923076923%0A784.6153846153846%0A969.2307692307692%0A1015.3846153846154%0A1200.&freq=440&midi=69&vert=9&horiz=1&colors=white%20black%20grey%20black%20white%20black%20white%20black%20grey%20black%20white%20white%20black%20grey%20black%20white%20black%20white%20black%20grey%20black%20white%20white%20black%20grey%20black&waveform=square&ampenv=organ Play 26edo diasem] - Sevish Scale Workshop


[[Category:Scale]]
[[Category:9-tone scales]]
[[Category:26edo]]
[[Category:26edo]]
[[Category:31edo]]
[[Category:31edo]]
[[Category:36edo]]
[[Category:36edo]]
[[Category:Just intonation]]
[[Category:Just intonation scales]]
[[Category:7-limit]]
[[Category:7-limit]]
[[Category:Maximum variety 3 scales]]
[[Category:Trivalent scales]]
[[Category:GO scales]]
[[Category:Rank-3 scales]]
[[Category:Fokker blocks]]
[[Category:Aberrismic theory]]
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