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Line 48: Line 48:
| 1
| 1
| L
| L
| perfect mosthird
| perfect 2-mosstep
| 1L + 1s
| 1L + 1s
| -1
| -1
| O
| O
| perfect mossixth
| perfect 5-mosstep
| 3L + 2s
| 3L + 2s
|-
|-
| 2
| 2
| N
| N
| minor mosfifth
| minor 4-mosstep
| 2L + 2s
| 2L + 2s
| -2
| -2
| M
| M
| major mosfourth
| major 3-mosstep
| 2L + 1s
| 2L + 1s
|-
|-
| 3
| 3
| P
| P
| minor mosseventh
| minor 6-mosstep
| 3L + 3s
| 3L + 3s
| -3
| -3
| K
| K
| major mossecond
| major 1-mosstep
| 1L + 0s
| 1L + 0s
|-
|-
| 4
| 4
| K@
| K@
| minor mossecond
| minor 1-mosstep
| 0L + 1s
| 0L + 1s
| -4
| -4
| Q&
| Q&
| major mosseventh
| major 6-mosstep
| 4L + 2s
| 4L + 2s
|-
|-
| 5
| 5
| M@
| M@
| minor mosfourth
| minor 3-mosstep
| 1L + 2s
| 1L + 2s
| -5
| -5
| N&
| N&
| major mosfifth
| major 4-mosstep
| 3L + 1s
| 3L + 1s
|-
|-
| 6
| 6
| O@
| O@
| diminished mossixth
| diminished 5-mosstep
| 2L + 3s
| 2L + 3s
| -6
| -6
| L&
| L&
| augmented mosthird
| augmented 2-mosstep
| 2L + 0s
| 2L + 0s
|-
|-
Line 113: Line 113:
| 8
| 8
| L@
| L@
| diminished mosthird
| diminished 2-mosstep
| 0L + 2s
| 0L + 2s
| -8
| -8
| O&
| O&
| augmented mossixth
| augmented 5-mosstep
| 4L + 1s
| 4L + 1s
|-
|-
| 9
| 9
| N@
| N@
| diminished mosfifth
| diminished 4-mosstep
| 1L + 3s
| 1L + 3s
| -9
| -9
| M&
| M&
| augmented mosfourth
| augmented 3-mosstep
| 3L + 0s
| 3L + 0s
|-
|-
| 10
| 10
| P@
| P@
| diminished mosseventh
| diminished 6-mosstep
| 2L + 4s
| 2L + 4s
| -10
| -10
| K&
| K&
| augmented mossecond
| augmented 1-mosstep
| 2L - 1s
| 2L - 1s
|}
|}
Line 141: Line 141:
== Low harmonic entropy scales ==
== Low harmonic entropy scales ==
There are two notable harmonic entropy minima:
There are two notable harmonic entropy minima:
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1 (making the diminished mossixth 3/2)
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1 (making the diminished 5-mosstep 3/2)
* [[Myna]], in which the generator is also 6/5 but now '''10''' of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).
* [[Myna]], in which the generator is also 6/5 but now '''10''' of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).


Line 150: Line 150:
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 mosthird (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 2-mosstep (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 chromas. 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 chromas. 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 mossixth) 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.
* 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 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) 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 mosthird a good [[5/4]] (384¢).
* [[25edo]] can be used to make the augmented 2-mosstep 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 213: Line 213:
| -7
| -7
|-
|-
| dim. mos2nd
| dim. 1-mosstep
| 1\18, 66.7
| 1\18, 66.7
| 2\25, 96.0
| 2\25, 96.0
Line 221: Line 221:
| +11
| +11
|-
|-
| minor mos2nd
| minor 1-mosstep
| 2\18, 133.3
| 2\18, 133.3
| 3\25, 144.0
| 3\25, 144.0
Line 229: Line 229:
| +4
| +4
|-
|-
| major mos2nd
| major 1-mosstep
| 3\18, 200.0
| 3\18, 200.0
| 4\25, 192.0
| 4\25, 192.0
Line 237: Line 237:
| -3
| -3
|-
|-
| aug. mos2nd
| aug. 1-mosstep
| 4\18, 266.7
| 4\18, 266.7
| 5\25, 240.0
| 5\25, 240.0
Line 245: Line 245:
| -10
| -10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos3rd
| dim. 2-mosstep
| 4\18, 266.7
| 4\18, 266.7
| 6\25, 288.0
| 6\25, 288.0
Line 253: Line 253:
| +8
| +8
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| perf. mos3rd
| perf. 2-mosstep
| 5\18, 333.3
| 5\18, 333.3
| 7\25, 336.0
| 7\25, 336.0
Line 261: Line 261:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos3rd
| aug. 2-mosstep
| 6\18, 400.0
| 6\18, 400.0
| 8\25, 384.4
| 8\25, 384.4
Line 269: Line 269:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| doubly aug. mos3rd
| doubly aug. 2-mosstep
| 7\18, 466.7  
| 7\18, 466.7  
| 9\25, 432.0
| 9\25, 432.0
Line 277: Line 277:
| -13
| -13
|-
|-
| dim. mos4th
| dim. 3-mosstep
| 6\18, 400.0
| 6\18, 400.0
| 9\25, 432.0
| 9\25, 432.0
Line 285: Line 285:
| +12
| +12
|-
|-
| minor mos4th
| minor 3-mosstep
| 7\18, 466.7
| 7\18, 466.7
| 10\25, 480.0
| 10\25, 480.0
Line 293: Line 293:
| +5
| +5
|-
|-
| major mos4th
| major 3-mosstep
| 8\18, 533.3  
| 8\18, 533.3  
| 11\25, 528.0
| 11\25, 528.0
Line 301: Line 301:
| -2
| -2
|-
|-
| aug. mos4th
| aug. 3-mosstep
| 9\18, 600.0
| 9\18, 600.0
| 12\25, 576.0
| 12\25, 576.0
Line 309: Line 309:
| -9
| -9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos5th
| dim. 4-mosstep
| 9\18, 600.0
| 9\18, 600.0
| 13\25, 624.0
| 13\25, 624.0
Line 317: Line 317:
| +9
| +9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos5th
| minor 4-mosstep
| 10\18, 666.7
| 10\18, 666.7
| 14\25, 672.0
| 14\25, 672.0
Line 325: Line 325:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos5th
| major 4-mosstep
| 11\18, 733.3
| 11\18, 733.3
| 15\25, 720.0
| 15\25, 720.0
Line 333: Line 333:
| -5
| -5
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos5th
| aug. 4-mosstep
| 12\18, 800.0
| 12\18, 800.0
| 16\25, 768.0
| 16\25, 768.0
Line 341: Line 341:
| -12
| -12
|-
|-
| doubly dim. mos6th
| doubly dim. 5-mosstep
| 11\18, 733.3
| 11\18, 733.3
| 16\25, 768.0
| 16\25, 768.0
Line 349: Line 349:
| +13
| +13
|-
|-
| dim. mos6th
| dim. 5-mosstep
| 12\18, 800.0
| 12\18, 800.0
| 17\25, 816.0
| 17\25, 816.0
Line 357: Line 357:
| +6
| +6
|-
|-
| perf. mos6th
| perf. 5-mosstep
| 13\18, 866.7
| 13\18, 866.7
| 18\25, 864.0
| 18\25, 864.0
Line 365: Line 365:
| -1
| -1
|-
|-
| aug. mos6th
| aug. 5-mosstep
| 14\18, 933.3
| 14\18, 933.3
| 19\25, 912.0
| 19\25, 912.0
Line 373: Line 373:
| -8
| -8
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos7th
| dim. 6-mosstep
| 14\18, 933.3
| 14\18, 933.3
| 20\25, 960.0
| 20\25, 960.0
Line 381: Line 381:
| +10
| +10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos7th
| minor 6-mosstep
| 15\18, 1000.0
| 15\18, 1000.0
| 21\25, 1008.0
| 21\25, 1008.0
Line 389: Line 389:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos7th
| major 6-mosstep
| 16\18, 1066.7
| 16\18, 1066.7
| 22\25, 1056.0
| 22\25, 1056.0
Line 397: Line 397:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos7th
| aug. 6-mosstep
| 17\18, 1133.3
| 17\18, 1133.3
| 23\25, 1104.0
| 23\25, 1104.0
Line 457: Line 457:
| 0
| 0
|-
|-
| minor mos2nd
| minor 1-mosstep
| 3\29, 124.1
| 3\29, 124.1
| K@
| K@
Line 463: Line 463:
| +4
| +4
|-
|-
| major mos2nd
| major 1-mosstep
| 5\29, 206.9
| 5\29, 206.9
| K
| K
Line 469: Line 469:
| -3
| -3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| perf. mos3rd
| perf. 2-mosstep
| 8\29, 331.0
| 8\29, 331.0
| L
| L
Line 475: Line 475:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos3rd
| aug. 2-mosstep
| 10\29, 413.8
| 10\29, 413.8
| L&
| L&
Line 481: Line 481:
| -6
| -6
|-
|-
| minor mos4th
| minor 3-mosstep
| 11\29, 455.2
| 11\29, 455.2
| M@
| M@
Line 487: Line 487:
| +5
| +5
|-
|-
| major mos4th
| major 3-mosstep
| 13\29, 537.9
| 13\29, 537.9
| M
| M
Line 493: Line 493:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos5th
| minor 4-mosstep
| 16\29, 662.1
| 16\29, 662.1
| N
| N
Line 499: Line 499:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos5th
| major 4-mosstep
| 18\26, 744.8
| 18\26, 744.8
| N&
| N&
Line 505: Line 505:
| -5
| -5
|-
|-
| dim. mos6th
| dim. 5-mosstep
| 19\29, 786.2
| 19\29, 786.2
| O@
| O@
Line 511: Line 511:
| +6
| +6
|-
|-
| perf. mos6th
| perf. 5-mosstep
| 21\29, 869.0
| 21\29, 869.0
| O
| O
Line 517: Line 517:
| -1
| -1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos7th
| minor 6-mosstep
| 24\29, 993.1
| 24\29, 993.1
| P
| P
Line 523: Line 523:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos7th
| major 6-mosstep
| 26\28, 1075.9
| 26\28, 1075.9
| P&
| P&
Line 531: Line 531:


=== 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 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 associated with [[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 3-mosstep (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 associated with [[orgone]] temperament.


Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Line 581: Line 581:
| 0
| 0
|-
|-
| minor mos2nd
| minor 1-mosstep
| 1\11, 109.1
| 1\11, 109.1
| 1\15, 80.0
| 1\15, 80.0
Line 589: Line 589:
| +4
| +4
|-
|-
| major mos2nd
| major 1-mosstep
| 2\11, 218.2
| 2\11, 218.2
| 3\15, 240.0
| 3\15, 240.0
Line 597: Line 597:
| -3
| -3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| perf. mos3rd
| perf. 2-mosstep
| 3\11, 327.3
| 3\11, 327.3
| 4\15, 320.0
| 4\15, 320.0
Line 605: Line 605:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos3rd
| aug. 2-mosstep
| 4\11, 436.4
| 4\11, 436.4
| 6\15, 480.0
| 6\15, 480.0
Line 613: Line 613:
| -6
| -6
|-
|-
| minor mos4th
| minor 3-mosstep
| 4\11, 436.4
| 4\11, 436.4
| 5\15, 400.0
| 5\15, 400.0
Line 621: Line 621:
| +5
| +5
|-
|-
| major mos4th
| major 3-mosstep
| 5\11, 545.5
| 5\11, 545.5
| 7\15, 560.0
| 7\15, 560.0
Line 629: Line 629:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos5th
| minor 4-mosstep
| 6\11, 656.6
| 6\11, 656.6
| 8\15, 640.0
| 8\15, 640.0
Line 637: Line 637:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos5th
| major 4-mosstep
| 7\11, 763.6
| 7\11, 763.6
| 10\15, 800.0
| 10\15, 800.0
Line 645: Line 645:
| -5
| -5
|-
|-
| dim. mos6th
| dim. 5-mosstep
| 7\11, 763.6
| 7\11, 763.6
| 9\15, 720.0
| 9\15, 720.0
Line 653: Line 653:
| +6
| +6
|-
|-
| perf. mos6th
| perf. 5-mosstep
| 8\11, 872.7
| 8\11, 872.7
| 11\15, 880.0
| 11\15, 880.0
Line 661: Line 661:
| -1
| -1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos7th
| minor 6-mosstep
| 9\11, 981.8
| 9\11, 981.8
| 12\15, 960.0
| 12\15, 960.0
Line 669: Line 669:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos7th
| major 6-mosstep
| 10\11, 1090.9
| 10\11, 1090.9
| 14\15, 1120.0
| 14\15, 1120.0
Line 733: Line 733:
| 0
| 0
|-
|-
| minor mos2nd
| minor 1-mosstep
| 1\19, 63.2
| 1\19, 63.2
| 2\34, 70.6
| 2\34, 70.6
Line 741: Line 741:
| +4
| +4
|-
|-
| major mos2nd
| major 1-mosstep
| 4\19, 252.6
| 4\19, 252.6
| 7\34, 247.1
| 7\34, 247.1
Line 749: Line 749:
| -3
| -3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| perf. mos3rd
| perf. 2-mosstep
| 5\19, 315.8
| 5\19, 315.8
| 9\34, 317.6
| 9\34, 317.6
Line 757: Line 757:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos3rd
| aug. 2-mosstep
| 8\19, 505.3
| 8\19, 505.3
| 14\34, 494.1
| 14\34, 494.1
Line 765: Line 765:
| -6
| -6
|-
|-
| minor mos4th
| minor 3-mosstep
| 6\19, 378.9
| 6\19, 378.9
| 11\34, 388.2
| 11\34, 388.2
Line 773: Line 773:
| +5
| +5
|-
|-
| major mos4th
| major 3-mosstep
| 9\19, 568.4
| 9\19, 568.4
| 16\34, 564.7
| 16\34, 564.7
Line 781: Line 781:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos5th
| minor 4-mosstep
| 10\19, 631.6
| 10\19, 631.6
| 18\34, 635.3
| 18\34, 635.3
Line 789: Line 789:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos5th
| major 4-mosstep
| 16\19, 821.1
| 16\19, 821.1
| 23\34, 811.8
| 23\34, 811.8
Line 797: Line 797:
| -5
| -5
|-
|-
| dim. mos6th
| dim. 5-mosstep
| 11\19, 694.7
| 11\19, 694.7
| 20\34, 705.9
| 20\34, 705.9
Line 805: Line 805:
| +6
| +6
|-
|-
| perf. mos6th
| perf. 5-mosstep
| 14\19, 884.2
| 14\19, 884.2
| 25\34, 882.4
| 25\34, 882.4
Line 813: Line 813:
| -1
| -1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos7th
| minor 6-mosstep
| 15\19, 947.4
| 15\19, 947.4
| 27\34, 952.9
| 27\34, 952.9
Line 821: Line 821:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos7th
| major 6-mosstep
| 18\19, 1136.8
| 18\19, 1136.8
| 32\34, 1129.4
| 32\34, 1129.4
Retrieved from "https://en.xen.wiki/w/4L_3s"