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Line 37: Line 37:
! Generators
! Generators
! Notation (1/1 = J)
! Notation (1/1 = J)
! Interval category name
! [[TAMNAMS]] name
! In L's and s's
! Generators
! Generators
! Notation of 2/1 inverse  
! Notation of 2/1 inverse  
! Interval category name
! [[TAMNAMS]] name
! In L's and s's
|-
|-
| colspan="6" style="text-align:left" | The 7-note MOS has the following intervals (from some root):
| colspan="8" style="text-align:left" | The 7-note MOS has the following intervals (from some root):
|-
|-
| 0
| 0
| J
| J
| perfect unison
| perfect unison
| 0L + 0s
| 0
| 0
| J
| J
| octave
| octave
| 4L + 3s
|-
|-
| 1
| 1
| L
| L
| perfect smithird
| perfect smithird
| 1L + 1s
| -1
| -1
| O
| O
| perfect smisixth
| perfect smisixth
| 3L + 2s
|-
|-
| 2
| 2
| N
| N
| minor smififth (aka minor fifth)
| minor mosfifth (aka minor fifth)
| 2L + 2s
| -2
| -2
| M
| M
| major smifourth (aka major fourth)
| major mosfourth (aka major fourth)
| 2L + 1s
|-
|-
| 3
| 3
| P
| P
| minor smiseventh
| minor mosseventh
| 3L + 3s
| -3
| -3
| K
| K
| major smisecond
| major mossecond
| 1L + 0s
|-
|-
| 4
| 4
| K@
| K@
| minor smisecond
| minor mossecond
| 0L + 1s
| -4
| -4
| Q&
| Q&
| major smiseventh
| major mosseventh
| 4L + 2s
|-
|-
| 5
| 5
| M@
| M@
| minor smifourth (aka minor fourth)
| minor mosfourth (aka minor fourth)
| 1L + 2s
| -5
| -5
| N&
| N&
| major smififth (aka major fifth)
| major mosfifth (aka major fifth)
| 3L + 1s
|-
|-
| 6
| 6
| O@
| O@
| diminished smisixth
| diminished mossixth
| 2L + 3s
| -6
| -6
| L&
| L&
| augmented smithird
| augmented mosthird
| 2L + 0s
|-
|-
| colspan="6" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]], [[4L 7s]], or [[11edo]]) also has the following intervals (from some root):
| colspan="8" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]], [[4L 7s]], or [[11edo]]) also has the following intervals (from some root):
|-
|-
| 7
| 7
| J@
| J@
| diminished smioctave
| diminished mosoctave
| 5L + 2s
| -7
| -7
| J&
| J&
| augmented smiunison; smichroma; smicomma (in parasoft smitonic contexts)
| augmented mosunison; smichroma; smicomma (in parasoft smitonic contexts)
| 1L - 1s
|-
|-
| 8
| 8
| L@
| L@
| diminished smithird
| diminished mosthird
| 0L + 2s
| -8
| -8
| O&
| O&
| augmented smisixth
| augmented mossixth
| 4L + 1s
|-
|-
| 9
| 9
| N@
| N@
| diminished smififth
| diminished mosfifth
| 1L + 3s
| -9
| -9
| M&
| M&
| augmented smifourth
| augmented mosfourth
| 3L + 0s
|-
|-
| 10
| 10
| P@
| P@
| diminished smiseventh
| diminished mosseventh
| 2L + 4s
| -10
| -10
| K&
| K&
| augmented smisecond
| augmented mossecond
| 2L - 1s
|}
|}


Line 130: Line 154:
Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:  
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 large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]).
* The augmented smithird (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.
* The augmented 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 tunings have both minor fifths and major fifths about equally off a just fifth, and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like smichromas. For this reason, parasoft might be the most accessible smitonic tuning range.
Parasoft smitonic tunings have both minor fifths and major fifths about equally off a just fifth, and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like smichromas. For this reason, parasoft might be the most accessible smitonic tuning range.


Parasoft smitonic EDOs include [[18edo]], [[25edo]], and [[43edo]].
Parasoft smitonic EDOs include [[18edo]], [[25edo]], 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 smisixth) and falling fifths (666.7c, a major smififth) almost equally off from a just fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 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 smisixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* [[25edo]] can be used to make the augmented smithird a good [[5/4]] (384¢).
* [[25edo]] can be used to make the augmented 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.  
The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.  
Line 201: Line 225:
| +11
| +11
|-
|-
| minor smi2nd
| minor mos2nd
| 2\18, 133.3
| 2\18, 133.3
| 3\25, 144.0
| 3\25, 144.0
Line 209: Line 233:
| +4
| +4
|-
|-
| major smi2nd
| major mos2nd
| 3\18, 200.0
| 3\18, 200.0
| 4\25, 192.0
| 4\25, 192.0
Line 217: Line 241:
| -3
| -3
|-
|-
| aug. smi2nd
| aug. mos2nd
| 4\18, 266.7
| 4\18, 266.7
| 5\25, 240.0
| 5\25, 240.0
Line 241: Line 265:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. smi3rd
| aug. mos3rd
| 6\18, 400.0
| 6\18, 400.0
| 8\25, 384.4
| 8\25, 384.4
Line 249: Line 273:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| doubly aug. smi3rd
| doubly aug. mos3rd
| 7\18, 466.7  
| 7\18, 466.7  
| 9\25, 432.0
| 9\25, 432.0
Line 265: Line 289:
| +12
| +12
|-
|-
| minor smi4th
| minor mos4th
| 7\18, 466.7
| 7\18, 466.7
| 10\25, 480.0
| 10\25, 480.0
Line 273: Line 297:
| +5
| +5
|-
|-
| major smi4th
| major mos4th
| 8\18, 533.3  
| 8\18, 533.3  
| 11\25, 528.0
| 11\25, 528.0
Line 281: Line 305:
| -2
| -2
|-
|-
| aug. smi4th
| aug. mos4th
| 9\18, 600.0
| 9\18, 600.0
| 12\25, 576.0
| 12\25, 576.0
Line 297: Line 321:
| +9
| +9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi5th
| minor mos5th
| 10\18, 666.7
| 10\18, 666.7
| 14\25, 672.0
| 14\25, 672.0
Line 305: Line 329:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi5th
| major mos5th
| 11\18, 733.3
| 11\18, 733.3
| 15\25, 720.0
| 15\25, 720.0
Line 313: Line 337:
| -5
| -5
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. smi5th
| aug. mos5th
| 12\18, 800.0
| 12\18, 800.0
| 16\25, 768.0
| 16\25, 768.0
Line 345: Line 369:
| -1
| -1
|-
|-
| aug. smi6th
| aug. mos6th
| 14\18, 933.3
| 14\18, 933.3
| 19\25, 912.0
| 19\25, 912.0
Line 361: Line 385:
| +10
| +10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi7th
| minor mos7th
| 15\18, 1000.0
| 15\18, 1000.0
| 21\25, 1008.0
| 21\25, 1008.0
Line 369: Line 393:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi7th
| major mos7th
| 16\18, 1066.7
| 16\18, 1066.7
| 22\25, 1056.0
| 22\25, 1056.0
Line 377: Line 401:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. smi7th
| aug. mos7th
| 17\18, 1133.3
| 17\18, 1133.3
| 23\25, 1104.0
| 23\25, 1104.0
Line 437: Line 461:
| 0
| 0
|-
|-
| minor smi2nd
| minor mos2nd
| 3\29, 124.1
| 3\29, 124.1
| K@
| K@
Line 443: Line 467:
| +4
| +4
|-
|-
| major smi2nd
| major mos2nd
| 5\29, 206.9
| 5\29, 206.9
| K
| K
Line 455: Line 479:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. smi3rd
| aug. mos3rd
| 10\29, 413.8
| 10\29, 413.8
| L&
| L&
Line 461: Line 485:
| -6
| -6
|-
|-
| minor smi4th
| minor mos4th
| 11\29, 455.2
| 11\29, 455.2
| M@
| M@
Line 467: Line 491:
| +5
| +5
|-
|-
| major smi4th
| major mos4th
| 13\29, 537.9
| 13\29, 537.9
| M
| M
Line 473: Line 497:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi5th
| minor mos5th
| 16\29, 662.1
| 16\29, 662.1
| N
| N
Line 479: Line 503:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi5th
| major mos5th
| 18\26, 744.8
| 18\26, 744.8
| N&
| N&
Line 497: Line 521:
| -1
| -1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi7th
| minor mos7th
| 24\29, 993.1
| 24\29, 993.1
| P
| P
Line 503: Line 527:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi7th
| major mos7th
| 26\28, 1075.9
| 26\28, 1075.9
| P&
| P&
Line 511: Line 535:


=== Hypohard ===
=== 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. This set of JI approximations is called [[orgone]] temperament.
[[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 mosfourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations. This set of JI approximations is called [[orgone]] temperament.


Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Line 561: Line 585:
| 0
| 0
|-
|-
| minor smi2nd
| minor mos2nd
| 1\11, 109.1
| 1\11, 109.1
| 1\15, 80.0
| 1\15, 80.0
Line 569: Line 593:
| +4
| +4
|-
|-
| major smi2nd
| major mos2nd
| 2\11, 218.2
| 2\11, 218.2
| 3\15, 240.0
| 3\15, 240.0
Line 585: Line 609:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. smi3rd
| aug. mos3rd
| 4\11, 436.4
| 4\11, 436.4
| 6\15, 480.0
| 6\15, 480.0
Line 593: Line 617:
| -6
| -6
|-
|-
| minor smi4th
| minor mos4th
| 4\11, 436.4
| 4\11, 436.4
| 5\15, 400.0
| 5\15, 400.0
Line 601: Line 625:
| +5
| +5
|-
|-
| major smi4th
| major mos4th
| 5\11, 545.5
| 5\11, 545.5
| 7\15, 560.0
| 7\15, 560.0
Line 609: Line 633:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi5th
| minor mos5th
| 6\11, 656.6
| 6\11, 656.6
| 8\15, 640.0
| 8\15, 640.0
Line 617: Line 641:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi5th
| major mos5th
| 7\11, 763.6
| 7\11, 763.6
| 10\15, 800.0
| 10\15, 800.0
Line 641: Line 665:
| -1
| -1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi7th
| minor mos7th
| 9\11, 981.8
| 9\11, 981.8
| 12\15, 960.0
| 12\15, 960.0
Line 649: Line 673:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi7th
| major mos7th
| 10\11, 1090.9
| 10\11, 1090.9
| 14\15, 1120.0
| 14\15, 1120.0
Line 711: Line 735:
| 0
| 0
|-
|-
| minor smi2nd
| minor mos2nd
| 1\19, 63.2
| 1\19, 63.2
| 2\34, 70.6
| 2\34, 70.6
Line 719: Line 743:
| +4
| +4
|-
|-
| major smi2nd
| major mos2nd
| 4\19, 252.6
| 4\19, 252.6
| 7\34, 247.1
| 7\34, 247.1
Line 735: Line 759:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. smi3rd
| aug. mos3rd
| 8\19, 505.3
| 8\19, 505.3
| 14\34, 494.1
| 14\34, 494.1
Line 743: Line 767:
| -6
| -6
|-
|-
| minor smi4th
| minor mos4th
| 6\19, 378.9
| 6\19, 378.9
| 11\34, 388.2
| 11\34, 388.2
Line 751: Line 775:
| +5
| +5
|-
|-
| major smi4th
| major mos4th
| 9\19, 568.4
| 9\19, 568.4
| 16\34, 564.7
| 16\34, 564.7
Line 759: Line 783:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi5th
| minor mos5th
| 10\19, 631.6
| 10\19, 631.6
| 18\34, 635.3
| 18\34, 635.3
Line 767: Line 791:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi5th
| major mos5th
| 16\19, 821.1
| 16\19, 821.1
| 23\34, 811.8
| 23\34, 811.8
Line 791: Line 815:
| -1
| -1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor smi7th
| minor mos7th
| 15\19, 947.4
| 15\19, 947.4
| 27\34, 952.9
| 27\34, 952.9
Line 799: Line 823:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major smi7th
| major mos7th
| 18\19, 1136.8
| 18\19, 1136.8
| 32\34, 1129.4
| 32\34, 1129.4
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