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'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M/N@''' '''N/M&''' N&/O@ '''O/P@''' '''P/O@''' P&/J@ '''J'''
'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M/N@''' '''N/M&''' N&/O@ '''O/P@''' '''P/O@''' P&/J@ '''J'''
== Tuning ranges ==
=== Parasoft ===
[[Parasoft]] smitonic tunings  have step ratios between 5/4 and 3/2, which implies a generator sharper than 5\18 = 333.33¢ and flatter than 9\32 = 337.5¢.
Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]).
* The major mosthird (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.
Parasoft smitonic EDOs include [[18edo]], [[25edo]], [[32edo]], and [[43edo]].
* 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted [[19edo]] diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* [[25edo]] can be used to make the major mosthird a good [[5/4]] (384¢).
The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[18edo]]
! [[25edo]]
! [[32edo]]
! Optimal (2.9.5 [[POTE]]) tuning
|-
| generator (g)
| 5\18, 333.33
| 7\25, 336.00
| 9\32, 337.50
| 335.84
|-
| L (octave - 3g)
| 3\18, 200.00
| 4\25, 192.00
| 5\32, 187.50
| 193.16
|-
| s (4g - octave)
| 2\18, 133.33
| 3\25, 144.00
| 4\32, 150.00
| 143.36
|}
=== Hyposoft ===
[[Hyposoft]] tunings of smitonic  have [[step ratio]]s between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.
The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "[[parapyth]] smitonic" or "[[archy]] smitonic", in analogy to parasoft smitonic being meantone smitonic.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[11edo]]
! [[18edo]]
! [[29edo]]
|-
| generator (g)
| 3\11, 327.27
| 5\18, 333.33
| 8\29, 331.03
|-
| L (octave - 3g)
| 2\11, 218.18
| 3\18, 200.00
| 5\29, 206.90
|-
| s (4g - octave)
| 1\11, 109.09
| 2\18, 133.33
| 3\29, 124.14
|}
=== Hypohard ===
[[Hypohard]] tunings have [[step ratio]]s between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate [[8/7]], and the major smifourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations.
Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various hypohard smitonic tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[11edo]]
! [[15edo]]
! [[26edo]]
! JI intervals represented
|-
| generator (g)
| 3\11, 327.27
| 4\15, 320.00
| 7\26, 323.08
| 77/64
|-
| L (octave - 3g)
| 2\11, 218.18
| 3\15, 240.00
| 5\26, 230.77
| 8/7
|-
| s (4g - octave)
| 1\11, 109.09
| 1\15, 80.00
| 2\26, 92.31
| 128/121, (16/15)
|}
=== Parahard ===
In [[parahard]] smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach a perfect fifth. The 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the [[4L 7s]] 11-note MOS, is suggested for getting 5-limit harmony.
EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[15edo]]
! [[19edo]]
! [[34edo]]
! [[53edo]]
! JI intervals represented
|-
| generator (g)
| 4\15, 320.00
| 5\19, 315.79
| 9\34, 317.65
| 316.98
| 6/5
|-
| L (octave - 3g)
| 3\15, 240.00
| 4\19, 252.63
| 7\34, 247.06
| 249.06
| 15/13, 23/20
|-
| s (4g - octave)
| 1\15, 80.00
| 1\19, 63.16
| 2\34, 70.59
| 67.92
| 25/24
|}
== Intervals ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = J)
! Heptatonic interval category name
! Generators
! Notation of 2/1 inverse
! Heptatonic interval category name
|-
| colspan="6" style="text-align:left" | The 7-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| 0
| J
| octave
|-
| 1
| L
| perfect smithird
| -1
| O
| perfect smisixth
|-
| 2
| N
| minor smififth (aka minor fifth)
| -2
| M
| major smifourth (aka major fourth)
|-
| 3
| P
| minor smiseventh
| -3
| K
| major smisecond
|-
| 4
| K@
| minor smisecond
| -4
| Q&
| major smiseventh
|-
| 5
| M@
| minor smifourth (aka minor fourth)
| -5
| N&
| major smififth (aka major fifth)
|-
| 6
| O@
| diminished smisixth
| -6
| L&
| augmented smithird
|-
| colspan="6" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]] or [[4L 7s]]) also has the following intervals (from some root):
|-
| 7
| J@
| diminished octave
| -7
| J&
| augmented unison
|-
| 8
| L@
| diminished smithird
| -8
| O&
| augmented smisixth
|-
| 9
| N@
| diminished smififth
| -9
| M&
| augmented smifourth
|-
| 10
| P@
| diminished smiseventh
| -10
| K&
| augmented smisecond
|}
== Modes ==
== Pseudo-diatonic theory ==
=== Hypohard ===
=== Parasoft ===
== Primodal theory ==
=== Primodal chords ===
=== Nejis ===
<!--
== Rank-2 temperaments ==
=== Myna (27&31) ===
=== Kleismic (19&15, 2.3.5.7) ===
=== Orgone (15&11, 2.7.11) ===
=== Sixix (18&25) ===
[[Sixix]] can be viewed as a [[dual-fifth]] temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):
* It has both a sharp fifth and a flat fifth but no near-just 3/2.
* Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.
* The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in [[porcupine]] harmony. Tempering out this comma leads to [[7edo]].
-->
== Samples ==
[[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic (WIP)
== Scale tree ==
== Scale tree ==
The spectrum looks like this:
The spectrum looks like this:
Line 475: Line 727:
| style="text-align:center;" |  
| style="text-align:center;" |  
|}
|}
== Tuning ranges ==
=== Parasoft ===
[[Parasoft]] smitonic tunings  have step ratios between 5/4 and 3/2, which implies a generator sharper than 5\18 = 333.33¢ and flatter than 9\32 = 337.5¢.
Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]).
* The major mosthird (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.
Parasoft smitonic EDOs include [[18edo]], [[25edo]], [[32edo]], and [[43edo]].
* 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted [[19edo]] diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* [[25edo]] can be used to make the major mosthird a good [[5/4]] (384¢).
The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[18edo]]
! [[25edo]]
! [[32edo]]
! Optimal (2.9.5 [[POTE]]) tuning
|-
| generator (g)
| 5\18, 333.33
| 7\25, 336.00
| 9\32, 337.50
| 335.84
|-
| L (octave - 3g)
| 3\18, 200.00
| 4\25, 192.00
| 5\32, 187.50
| 193.16
|-
| s (4g - octave)
| 2\18, 133.33
| 3\25, 144.00
| 4\32, 150.00
| 143.36
|}
=== Hyposoft ===
[[Hyposoft]] tunings of smitonic  have [[step ratio]]s between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.
The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "[[parapyth]] smitonic" or "[[archy]] smitonic", in analogy to parasoft smitonic being meantone smitonic.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[11edo]]
! [[18edo]]
! [[29edo]]
|-
| generator (g)
| 3\11, 327.27
| 5\18, 333.33
| 8\29, 331.03
|-
| L (octave - 3g)
| 2\11, 218.18
| 3\18, 200.00
| 5\29, 206.90
|-
| s (4g - octave)
| 1\11, 109.09
| 2\18, 133.33
| 3\29, 124.14
|}
=== Hypohard ===
[[Hypohard]] tunings have [[step ratio]]s between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate [[8/7]], and the major smifourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations.
Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various hypohard smitonic tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[11edo]]
! [[15edo]]
! [[26edo]]
! JI intervals represented
|-
| generator (g)
| 3\11, 327.27
| 4\15, 320.00
| 7\26, 323.08
| 77/64
|-
| L (octave - 3g)
| 2\11, 218.18
| 3\15, 240.00
| 5\26, 230.77
| 8/7
|-
| s (4g - octave)
| 1\11, 109.09
| 1\15, 80.00
| 2\26, 92.31
| 128/121, (16/15)
|}
=== Parahard ===
In [[parahard]] smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach a perfect fifth. The 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the [[4L 7s]] 11-note MOS, is suggested for getting 5-limit harmony.
EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[15edo]]
! [[19edo]]
! [[34edo]]
! [[53edo]]
! JI intervals represented
|-
| generator (g)
| 4\15, 320.00
| 5\19, 315.79
| 9\34, 317.65
| 316.98
| 6/5
|-
| L (octave - 3g)
| 3\15, 240.00
| 4\19, 252.63
| 7\34, 247.06
| 249.06
| 15/13, 23/20
|-
| s (4g - octave)
| 1\15, 80.00
| 1\19, 63.16
| 2\34, 70.59
| 67.92
| 25/24
|}
== Intervals ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = J)
! Heptatonic interval category name
! Generators
! Notation of 2/1 inverse
! Heptatonic interval category name
|-
| colspan="6" style="text-align:left" | The 7-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| 0
| J
| octave
|-
| 1
| L
| perfect smithird
| -1
| O
| perfect smisixth
|-
| 2
| N
| minor smififth (aka minor fifth)
| -2
| M
| major smifourth (aka major fourth)
|-
| 3
| P
| minor smiseventh
| -3
| K
| major smisecond
|-
| 4
| K@
| minor smisecond
| -4
| Q&
| major smiseventh
|-
| 5
| M@
| minor smifourth (aka minor fourth)
| -5
| N&
| major smififth (aka major fifth)
|-
| 6
| O@
| diminished smisixth
| -6
| L&
| augmented smithird
|-
| colspan="6" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]] or [[4L 7s]]) also has the following intervals (from some root):
|-
| 7
| J@
| diminished octave
| -7
| J&
| augmented unison
|-
| 8
| L@
| diminished smithird
| -8
| O&
| augmented smisixth
|-
| 9
| N@
| diminished smififth
| -9
| M&
| augmented smifourth
|-
| 10
| P@
| diminished smiseventh
| -10
| K&
| augmented smisecond
|}
== Modes ==
== Pseudo-diatonic theory ==
=== Hypohard ===
=== Parasoft ===
== Primodal theory ==
=== Primodal chords ===
=== Nejis ===
<!--
== Rank-2 temperaments ==
=== Myna (27&31) ===
=== Kleismic (19&15, 2.3.5.7) ===
=== Orgone (15&11, 2.7.11) ===
=== Sixix (18&25) ===
[[Sixix]] can be viewed as a [[dual-fifth]] temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):
* It has both a sharp fifth and a flat fifth but no near-just 3/2.
* Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.
* The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in [[porcupine]] harmony. Tempering out this comma leads to [[7edo]].
-->
== Samples ==
[[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic (WIP)
[[Category:Scales]]
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
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