Nicetone: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
→Tunings: Step label casing
ArrowHead294 (talk | contribs)
14,512 edits
m Table formatting
Line 7: Line 7:
Nicetone can be tuned as a [[5-limit]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[10/9]], and s represents [[16/15]].
Nicetone can be tuned as a [[5-limit]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[10/9]], and s represents [[16/15]].


{| class="wikitable" style="margin-left: auto; margin-right: auto;"
{| class="wikitable" style="margin: auto auto auto auto;"
|+ Comparison of Pythagorean and Ptolemaic diatonic scales in 53edo
|+ style="font-size: 105%;" | Comparison of Pythagorean and Ptolemaic diatonic scales in 53edo
|-
|-
! Name !! Structure !! Step Sizes !! Graphical Representation
! Name !! Structure !! Step Sizes !! Graphical Representation
Line 16: Line 16:
| Pythagorean || rowspan="2" | 5L 2s || 9\53, 4\53 || ├────────┼────────┼───┼────────┼────────┼────────┼───┤
| Pythagorean || rowspan="2" | 5L 2s || 9\53, 4\53 || ├────────┼────────┼───┼────────┼────────┼────────┼───┤
|-
|-
| Meantone,<br>{{frac|5}}&nbsp;comma || 7\43, 4\43 || style="border: none; display: inline-block; transform: scale(calc(53 / 43), 1) translate(-2px, 0); transform-origin: left center;" | ├──────┼──────┼───┼──────┼──────┼──────┼───┤
| Meantone, <br />{{frac|1|5}}&nbsp;comma || 7\43, 4\43 || style="border: none; display: inline-block; transform: scale(calc(53 / 43), 1) translate(-2px, 0); transform-origin: left center;" | ├──────┼──────┼───┼──────┼──────┼──────┼───┤
|}
|}


Line 35: Line 35:


{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
|+ Interval sizes in nicetone
|+ style="font-size: 105%;" | Interval sizes in nicetone
|-
|-
!colspan=2|Interval class
! colspan="2" | Interval class
! Sizes
! Sizes
! 5-limit JI
! 5-limit JI
! [[15edo]] <br>(L:M:s = 3:2:1)
! [[15edo]] <br />({{nowrap|L:M:s {{=}} 3:2:1}})
! [[41edo]] <br>(L:M:s = 7:6:4)
! [[41edo]] <br />({{nowrap|L:M:s ={{=}} 7:6:4}})
|- bgcolor="#eaeaff"
|- style="background-color: #eaeaff;"
!rowspan=3|Second <br>([[TAMNAMS|1-step]])
! rowspan="3" | Second <br />([[TAMNAMS|1-step]])
!| <small>small</small>
! style="font-size: 0.75em;" | Small
| s
| s
| 16/15, 111.73¢
| 16/15, 111.73¢
| 1\15, 80.00¢
| 1\15, 80.00¢
| 4\41, 117.07¢
| 4\41, 117.07¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!|<small>medium</small>
! style="font-size: 0.75em;" | Medium
| M
| M
| 10/9, 182.40¢
| 10/9, 182.40¢
| 2\15, 160.00¢
| 2\15, 160.00¢
| 6\41, 175.61¢
| 6\41, 175.61¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!|<small>large</small>
! style="font-size: 0.75em;" | Large
| L
| L
| 9/8, 203.91¢
| 9/8, 203.91¢
Line 62: Line 62:
| 7\41, 204.88¢
| 7\41, 204.88¢
|-
|-
!rowspan=3|Third <br>([[TAMNAMS|2-step]])
! rowspan="3" | Third <br />([[TAMNAMS|2-step]])
!|<small>small</small>
! style="font-size: 0.75em;" | Small
| M + s
| M + s
| 32/27, 294.13¢
| 32/27, 294.13¢
Line 69: Line 69:
| 10\41, 292.68¢
| 10\41, 292.68¢
|-
|-
!|<small>medium</small>
! style="font-size: 0.75em;" | Medium
| L + s
| L + s
| 6/5, 315.64¢
| 6/5, 315.64¢
Line 75: Line 75:
| 11\41, 321.95¢
| 11\41, 321.95¢
|-
|-
!|<small>large</small>
! style="font-size: 0.75em;" | Large
| L + M
| L + M
| 5/4, 386.31¢
| 5/4, 386.31¢
| 5\15, 400.00¢
| 5\15, 400.00¢
| 13\41, 380.49¢
| 13\41, 380.49¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!rowspan=3|Fourth <br>([[TAMNAMS|3-step]])
! rowspan="3" | Fourth <br />([[TAMNAMS|3-step]])
!|<small>small</small>
! style="font-size: 0.75em;" | Small
| L + M + s
| L + M + s
| 4/3, 498.04¢
| 4/3, 498.04¢
| 6\15, 480.00¢
| 6\15, 480.00¢
| 17\41, 497.56¢
| 17\41, 497.56¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!|<small>medium</small>
! style="font-size: 0.75em;" | Medium
| 2L + s
| 2L + s
| 27/20, 519.55¢
| 27/20, 519.55¢
| 7\15, 560.00¢
| 7\15, 560.00¢
| 18\41, 526.83¢
| 18\41, 526.83¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!|<small>large</small>
! style="font-size: 0.75em;" | Large
| 2L + M
| 2L + M
| 45/32, 590.22¢
| 45/32, 590.22¢
Line 100: Line 100:
| 20\41, 585.37¢
| 20\41, 585.37¢
|-
|-
!rowspan=3|Fifth <br>([[TAMNAMS|4-step]])
! rowspan="3" | Fifth <br />([[TAMNAMS|4-step]])
!|<small>small</small>
! style="font-size: 0.75em;" | Small
| L + M + 2s
| L + M + 2s
| 64/45, 609.78¢
| 64/45, 609.78¢
Line 107: Line 107:
| 21\41, 614.63¢
| 21\41, 614.63¢
|-
|-
!|<small>medium</small>
! style="font-size: 0.75em;" | Medium
| L + 2M + s
| L + 2M + s
| 40/27, 680.45¢
| 40/27, 680.45¢
Line 113: Line 113:
| 23\41, 673.17¢
| 23\41, 673.17¢
|-
|-
!|<small>large</small>
! style="font-size: 0.75em;" | Large
| 2L + M + s
| 2L + M + s
| 3/2, 701.96¢
| 3/2, 701.96¢
| 9\15, 720.00¢
| 9\15, 720.00¢
| 24\41, 702.44¢
| 24\41, 702.44¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!rowspan=3|Sixth <br>([[TAMNAMS|5-step]])
! rowspan="3" | Sixth <br />([[TAMNAMS|5-step]])
!|<small>small</small>
! style="font-size: 0.75em;" | Small
| 2L + M + 2s
| 2L + M + 2s
| 8/5, 813.69¢
| 8/5, 813.69¢
| 10\15, 800.00¢
| 10\15, 800.00¢
| 28\41, 819.51¢
| 28\41, 819.51¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!|<small>medium</small>
! style="font-size: 0.75em;" | Medium
| 2L + 2M + s
| 2L + 2M + s
| 5/3, 884.36¢
| 5/3, 884.36¢
| 11\15, 880.00¢
| 11\15, 880.00¢
| 30\41, 878.05¢
| 30\41, 878.05¢
|- bgcolor="#eaeaff"
|- style="background-color: "#eaeaff;"
!|<small>large</small>
! style="font-size: 0.75em;" | Large
| 3L + M + s
| 3L + M + s
| 27/16, 905.87¢
| 27/16, 905.87¢
Line 138: Line 138:
| 31\41, 907.32¢
| 31\41, 907.32¢
|-
|-
!rowspan=3|Seventh <br>([[TAMNAMS|6-step]])
! rowspan="3" | Seventh <br />([[TAMNAMS|6-step]])
!| <small>small</small>
! style="font-size: 0.75em;" | Small
| 2L + 2M + 2s
| 2L + 2M + 2s
| 16/9, 996.09¢
| 16/9, 996.09¢
Line 145: Line 145:
| 34\41, 995.12¢
| 34\41, 995.12¢
|-
|-
!|<small>medium</small>
! style="font-size: 0.75em;" | Medium
| 3L + M + 2s
| 3L + M + 2s
| 9/5, 1017.60¢
| 9/5, 1017.60¢
Line 151: Line 151:
| 35\41, 1024.39¢
| 35\41, 1024.39¢
|-
|-
!|<small>large</small>
! style="font-size: 0.75em;" | Large
| 3L + 2M + s
| 3L + 2M + s
| 15/8, 1088.27¢
| 15/8, 1088.27¢
Line 163: Line 163:
The modes are arranged by brightest to darkest.
The modes are arranged by brightest to darkest.
{| class="wikitable"
{| class="wikitable"
|+Nicetone modes
|+ style="font-size: 105%;" | Nicetone modes
!Left handed
|-
!Right handed
! Left handed !! Right handed
|-
|-
|LMLsMLs <br>LH Nice-Lydian
| LMLsMLs <br />LH Nice-Lydian
|LMLsLMs <br>RH Nice-Lydian
| LMLsLMs <br />RH Nice-Lydian
|-
|-
|MLsLMLs <br>LH Nice-Ionian
| MLsLMLs <br />LH Nice-Ionian
|LMsLMLs <br>RH Nice-Ionian
| LMsLMLs <br />RH Nice-Ionian
|-
|-
|MLsMLsL <br>LH Nice-Mixolydian
| MLsMLsL <br />LH Nice-Mixolydian
|MLsLMsL <br>RH Nice-Mixolydian
| MLsLMsL <br />RH Nice-Mixolydian
|-
|-
|LsLMLsM <br>LH Nice-Dorian
| LsLMLsM <br />LH Nice-Dorian
|MsLMLsL <br>RH Nice-Dorian
| MsLMLsL <br />RH Nice-Dorian
|-
|-
|LsMLsLM <br>LH Nice-Aeolian
| LsMLsLM <br />LH Nice-Aeolian
|LsLMsLM <br>RH Nice-Aeolian
| LsLMsLM <br />RH Nice-Aeolian
|-
|-
|sLMLsML <br>LH Nice-Phrygian
| sLMLsML <br />LH Nice-Phrygian
|sLMLsLM <br>RH Nice-Phrygian
| sLMLsLM <br />RH Nice-Phrygian
|-
|-
|sMLsLML <br>LH Nice-Locrian
| sMLsLML <br />LH Nice-Locrian
|sLMsLML <br>RH Nice-Locrian
| sLMsLML <br />RH Nice-Locrian
|}
|}


Line 193: Line 193:


{| class="wikitable"
{| class="wikitable"
|+Tuning range of nicetone
|+ style="font-size: 105%;" | Tuning range of nicetone
|-
!  
!  
! Tuning range (in [[octave]]s)
! Tuning range (in [[octave]]s)
|-
|-
! Outer generator <br>(''G''<sub>1</sub> = 2L + M + s)
! Outer generator <br />(''G''<sub>1</sub> = 2L + M + s)
| <math>\displaystyle \frac{4}{7} &lt; G_\text{1} &lt; \frac{2}{3}</math>
| <math>\displaystyle \frac{4}{7} &lt; G_\text{1} &lt; \frac{2}{3}</math>
|-
|-
! RH inner generator <br>(''G''<sub>2R</sub> = L + M)
! RH inner generator <br />(''G''<sub>2R</sub> = L + M)
| <math>\displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 4 G_\text{1} - 2 \text{ for } \frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5}</math> <br><math>\displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 1 - G_\text{1} \text{ for }\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3}</math>
| <math>\displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 4 G_\text{1} - 2 \text{ for } \frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5}</math> <br><math>\displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 1 - G_\text{1} \text{ for }\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3}</math>
|-
|-
! LH inner generator <br>(''G''<sub>2L</sub> = L + s)
! LH inner generator <br />(''G''<sub>2L</sub> = L + s)
| <math>\displaystyle 2 - 3 G_\text{1} &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \text{ for }\frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5}</math> <br><math>\displaystyle 2 G_\text{1} - 1 &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \text{ for }\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3}</math>
| <math>\displaystyle 2 - 3 G_\text{1} &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \text{ for }\frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5}</math> <br><math>\displaystyle 2 G_\text{1} - 1 &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \text{ for }\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3}</math>
|-
|-
! Large step <br>(L = 2''G''<sub>1</sub> - 1)
! Large step <br />(L = 2''G''<sub>1</sub> - 1)
| <math>\displaystyle \frac{1}{7} &lt; L &lt; \frac{1}{3}</math>
| <math>\displaystyle \frac{1}{7} &lt; L &lt; \frac{1}{3}</math>
|-
|-
! Middle step <br>(M = 1 - ''G''<sub>1</sub> - ''G''<sub>2L</sub>)
! Middle step <br />(M = 1 - ''G''<sub>1</sub> - ''G''<sub>2L</sub>)
| <math>\displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; L \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; \frac{1}{2} (1 - 3 L) \text{ for } \frac{1}{5} &le; L &lt; \frac{1}{3}</math>
| <math>\displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; L \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; \frac{1}{2} (1 - 3 L) \text{for} \frac{1}{5} &le; L &lt; \frac{1}{3}</math>
|-
|-
! Small step <br>(s = 1 - ''G''<sub>1</sub> - ''G''<sub>2R</sub>)
! Small step <br />(s = 1 - ''G''<sub>1</sub> - ''G''<sub>2R</sub>)
| <math>\displaystyle \frac{1}{2} (1 - 5 L) &lt; s &lt; \frac{1}{4} (1 - 3 L) \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle 0 &lt; s &lt; \frac{1}{4} (1 - 3 L) \text{ for } \frac{1}{5} &le; L &lt; \frac{1}{3}</math>
| <math>\displaystyle \frac{1}{2} (1 - 5 L) &lt; s &lt; \frac{1}{4} (1 - 3 L) \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5}</math> <br><math>\displaystyle 0 &lt; s &lt; \frac{1}{4} (1 - 3 L) \text{for} \frac{1}{5} &le; L &lt; \frac{1}{3}</math>
|}
|}


{| class="wikitable"
{| class="wikitable"
|+Common Nicetone tunings
|+ style="font-size: 105%;" | Common Nicetone tunings
! rowspan="2" | Tuning
|-
! rowspan="2" | L:M:s
! rowspan="2" | Tuning !! rowspan="2" | L:M:s !! colspan="3" | Size of step (¢) !! colspan="2" | Inner generator !! rowspan="2" | Outer generator <br />({{nowrap|2L + M + s}}) !! rowspan="2" | Comments
! colspan="3" | Size of step (¢)
! colspan="2" | Inner generator
! rowspan="2" | Outer generator <br>(2L+M+s)
! rowspan="2" | Comments
|-
|-
! L
! L !! M !! s !! LH ({{nowrap|L + s}}) !! RH ({{nowrap|L + M}})
! M
! s
! LH (L+s)
! RH (L+M)
|-
|-
| 5-limit JI || ||203.910||182.404||111.731||315.641||386.314||701.955||L=9/8, M=10/9, s=16/15
| 5-limit JI || || 203.910 || 182.404 || 111.731 || 315.641 || 386.314 || 701.955 || {{nowrap|L {{=}} 9/8}}, {{nowrap|M {{=}} 10/9}}, {{nowrap|s {{=}} 16/15}}
|-
|-
|[[15edo]]||3:2:1||240.000||160.000||80.000||320.000||400.000||720.000||5-limit patent val
| [[15edo]] || 3:2:1 || 240.000 || 160.000 || 80.000 || 320.000 || 400.000 || 720.000 || 5-limit patent val
|-
|-
|[[18edo]]||4:2:1||266.667||133.333||66.667||333.333||400.000||733.333||5-limit patent val
| [[18edo]] || 4:2:1 || 266.667 || 133.333 || 66.667 || 333.333 || 400.000 || 733.333 || 5-limit patent val
|-
|-
|[[20edo]]||4:3:1||240.000||180.000||60.000||300.000||420.000||720.000||
| [[20edo]] || 4:3:1 || 240.000 || 180.000 || 60.000 || 300.000 || 420.000 || 720.000 ||
|-
|-
|[[21edo]]||5:2:1||285.714||114.286||57.143||342.857||400.000||742.857||
| [[21edo]] || 5:2:1 || 285.714 || 114.286 || 57.143 || 342.857 || 400.000 || 742.857 ||
|-
|-
|[[22edo]]||4:3:2||218.182||163.636||109.091||327.273||381.818||709.091||5-limit patent val
| [[22edo]] || 4:3:2 || 218.182 || 163.636 || 109.091 || 327.273 || 381.818 || 709.091 || 5-limit patent val
|-
|-
|[[23edo]]||5:3:1||260.870||156.522||52.174||313.043||417.391||730.435||
| [[23edo]] || 5:3:1 || 260.870 || 156.522 || 52.174 || 313.043 || 417.391 || 730.435 ||
|-
|-
|[[24edo]]||6:2:1||300.000||100.000||50.000||350.000||400.000||750.000||
| [[24edo]] || 6:2:1 || 300.000 || 100.000 || 50.000 || 350.000 || 400.000 || 750.000 ||
|-
|-
| rowspan="2" |[[25edo]]||5:3:2||240.000||144.000||96.000||336.000||384.000||720.000||5-limit patent val
| rowspan="2" | [[25edo]] || 5:3:2 || 240.000 || 144.000 || 96.000 || 336.000 || 384.000 || 720.000 || 5-limit patent val
|-
|-
|5:4:1||240.000||192.000||48.000||288.000||432.000||720.000||
| 5:4:1 || 240.000 || 192.000 || 48.000 || 288.000 || 432.000 || 720.000 ||
|-
|-
|[[26edo]]||6:3:1||276.923||138.462||46.154||323.077||415.385||738.462||
| [[26edo]] || 6:3:1 || 276.923 || 138.462 || 46.154 || 323.077 || 415.385 || 738.462 ||
|-
|-
| rowspan="2" |[[27edo]]||5:4:2||222.222||177.778||88.889||311.111||400.000||711.111||5-limit patent val
| rowspan="2" | [[27edo]] || 5:4:2 || 222.222 || 177.778 || 88.889 || 311.111 || 400.000 || 711.111 || 5-limit patent val
|-
|-
|7:2:1||311.111||88.889||44.444||355.556||400.000||755.556||
| 7:2:1 || 311.111 || 88.889 || 44.444 || 355.556 || 400.000 || 755.556 ||
|-
|-
| rowspan="2" |[[28edo]]||6:3:2||257.143||128.571||85.714||342.857||385.714||728.571||
| rowspan="2" | [[28edo]] || 6:3:2 || 257.143 || 128.571 || 85.714 || 342.857 || 385.714 || 728.571 ||
|-
|-
|6:4:1||257.143||171.429||42.857||300.000||428.571||728.571||
| 6:4:1 || 257.143 || 171.429 || 42.857 || 300.000 || 428.571 || 728.571 ||
|-
|-
| rowspan="2" |[[29edo]]||5:4:3||206.897||165.517||124.138||331.034||372.414||703.448||5-limit patent val
| rowspan="2" | [[29edo]] || 5:4:3 || 206.897 || 165.517 || 124.138 || 331.034 || 372.414 || 703.448 || 5-limit patent val
|-
|-
|7:3:1||289.655||124.138||41.379||331.034||413.793||744.828||
| 7:3:1 || 289.655 || 124.138 || 41.379 || 331.034 || 413.793 || 744.828 ||
|-
|-
| rowspan="2" |[[30edo]]||6:5:1||240.000||200.000||40.000||280.000||440.000||720.000||
| rowspan="2" | [[30edo]] || 6:5:1 || 240.000 || 200.000 || 40.000 || 280.000 || 440.000 || 720.000 ||
|-
|-
|8:2:1||320.000||80.000||40.000||360.000||400.000||760.000||
| 8:2:1 || 320.000 || 80.000 || 40.000 || 360.000 || 400.000 || 760.000 ||
|-
|-
| rowspan="2" |[[31edo]]||7:3:2||270.968||116.129||77.419||348.387||387.097||735.484||
| rowspan="2" | [[31edo]] || 7:3:2 || 270.968 || 116.129 || 77.419 || 348.387 || 387.097 || 735.484 ||
|-
|-
|7:4:1||270.968||154.839||38.710||309.677||425.806||735.484||
| 7:4:1 || 270.968 || 154.839 || 38.710 || 309.677 || 425.806 || 735.484 ||
|-
|-
| rowspan="3" |[[32edo]]||6:4:3||225.000||150.000||112.500||337.500||375.000||712.500||5-limit patent val
| rowspan="3" | [[32edo]] || 6:4:3 || 225.000 || 150.000 || 112.500 || 337.500 || 375.000 || 712.500 || 5-limit patent val
|-
|-
|6:5:2||225.000||187.500||75.000||300.000||412.500||712.500||
| 6:5:2 || 225.000 || 187.500 || 75.000 || 300.000 || 412.500 || 712.500 ||
|-
|-
|8:3:1||300.000||112.500||37.500||337.500||412.500||750.000||
| 8:3:1 || 300.000 || 112.500 || 37.500 || 337.500 || 412.500 || 750.000 ||
|-
|-
| rowspan="3" |[[33edo]]||7:4:2||254.545||145.455||72.727||327.273||400.000||727.273||
| rowspan="3" | [[33edo]] || 7:4:2 || 254.545 || 145.455 || 72.727 || 327.273 || 400.000 || 727.273 ||
|-
|-
|7:5:1||254.545||181.818||36.364||290.909||436.364||727.273||
| 7:5:1 || 254.545 || 181.818 || 36.364 || 290.909 || 436.364 || 727.273 ||
|-
|-
|9:2:1||327.273||72.727||36.364||363.636||400.000||763.636||
| 9:2:1 || 327.273 || 72.727 || 36.364 || 363.636 || 400.000 || 763.636 ||
|-
|-
| rowspan="3" |[[34edo]]||6:5:3||211.765||176.471||105.882||317.647||388.235||705.882||5-limit patent val
| rowspan="3" | [[34edo]] || 6:5:3 || 211.765 || 176.471 || 105.882 || 317.647 || 388.235 || 705.882 || 5-limit patent val
|-
|-
|8:3:2||282.353||105.882||70.588||352.941||388.235||741.176||
| 8:3:2 || 282.353 || 105.882 || 70.588 || 352.941 || 388.235 || 741.176 ||
|-
|-
|8:4:1||282.353||141.176||35.294||317.647||423.529||741.176||
| 8:4:1 || 282.353 || 141.176 || 35.294 || 317.647 || 423.529 || 741.176 ||
|-
|-
| rowspan="4" |[[35edo]]||7:4:3||240.000||137.143||102.857||342.857||377.143||720.000||
| rowspan="4" | [[35edo]] || 7:4:3 || 240.000 || 137.143 || 102.857 || 342.857 || 377.143 || 720.000 ||
|-
|-
|7:5:2||240.000||171.429||68.571||308.571||411.429||720.000||
| 7:5:2 || 240.000 || 171.429 || 68.571 || 308.571 || 411.429 || 720.000 ||
|-
|-
|7:6:1||240.000||205.714||34.286||274.286||445.714||720.000||
| 7:6:1 || 240.000 || 205.714 || 34.286 || 274.286 || 445.714 || 720.000 ||
|-
|-
|9:3:1||308.571||102.857||34.286||342.857||411.429||754.286||
| 9:3:1 || 308.571 || 102.857 || 34.286 || 342.857 || 411.429 || 754.286 ||
|-
|-
| rowspan="3" |[[36edo]]||6:5:4||200.000||166.667||133.333||333.333||366.667||700.000||
| rowspan="3" | [[36edo]] || 6:5:4 || 200.000 || 166.667 || 133.333 || 333.333 || 366.667 || 700.000 ||
|-
|-
|8:5:1||266.667||166.667||33.333||300.000||433.333||733.333||
| 8:5:1 || 266.667 || 166.667 || 33.333 || 300.000 || 433.333 || 733.333 ||
|-
|-
|10:2:1||333.333||66.667||33.333||366.667||400.000||766.667||
| 10:2:1 || 333.333 || 66.667 || 33.333 || 366.667 || 400.000 || 766.667 ||
|-
|-
| rowspan="4" |[[37edo]]||7:5:3||227.027||162.162||97.297||324.324||389.189||713.514||5-limit patent val
| rowspan="4" | [[37edo]] || 7:5:3 || 227.027 || 162.162 || 97.297 || 324.324 || 389.189 || 713.514 || 5-limit patent val
|-
|-
|7:6:2||227.027||194.595||64.865||291.892||421.622||713.514||
| 7:6:2 || 227.027 || 194.595 || 64.865 || 291.892 || 421.622 || 713.514 ||
|-
|-
|9:3:2||291.892||97.297||64.865||356.757||389.189||745.946||
| 9:3:2 || 291.892 || 97.297 || 64.865 || 356.757 || 389.189 || 745.946 ||
|-
|-
|9:4:1||291.892||129.730||32.432||324.324||421.622||745.946||
| 9:4:1 || 291.892 || 129.730 || 32.432 || 324.324 || 421.622 || 745.946 ||
|-
|-
| rowspan="4" |[[38edo]]||8:4:3||252.632||126.316||94.737||347.368||378.947||726.316||
| rowspan="4" | [[38edo]] || 8:4:3 || 252.632 || 126.316 || 94.737 || 347.368 || 378.947 || 726.316 ||
|-
|-
|8:5:2||252.632||157.895||63.158||315.789||410.526||726.316||
| 8:5:2 || 252.632 || 157.895 || 63.158 || 315.789 || 410.526 || 726.316 ||
|-
|-
|8:6:1||252.632||189.474||31.579||284.211||442.105||726.316||
| 8:6:1 || 252.632 || 189.474 || 31.579 || 284.211 || 442.105 || 726.316 ||
|-
|-
|10:3:1||315.789||94.737||31.579||347.368||410.526||757.895||
| 10:3:1 || 315.789 || 94.737 || 31.579 || 347.368 || 410.526 || 757.895 ||
|-
|-
| rowspan="5" |[[39edo]]||7:5:4||215.385||153.846||123.077||338.462||369.231||707.692||
| rowspan="5" | [[39edo]] || 7:5:4 || 215.385 || 153.846 || 123.077 || 338.462 || 369.231 || 707.692 ||
|-
|-
|7:6:3||215.385||184.615||92.308||307.692||400.000||707.692||5-limit patent val
| 7:6:3 || 215.385 || 184.615 || 92.308 || 307.692 || 400.000 || 707.692 || 5-limit patent val
|-
|-
|9:4:2||276.923||123.077||61.538||338.462||400.000||738.462||
| 9:4:2 || 276.923 || 123.077 || 61.538 || 338.462 || 400.000 || 738.462 ||
|-
|-
|9:5:1||276.923||153.846||30.769||307.692||430.769||738.462||
| 9:5:1 || 276.923 || 153.846 || 30.769 || 307.692 || 430.769 || 738.462 ||
|-
|-
|11:2:1||338.462||61.538||30.769||369.231||400.000||769.231||
| 11:2:1 || 338.462 || 61.538 || 30.769 || 369.231 || 400.000 || 769.231 ||
|-
|-
| rowspan="4" |[[40edo]]||8:5:3||240.000||150.000||90.000||330.000||390.000||720.000||
| rowspan="4" | [[40edo]] || 8:5:3 || 240.000 || 150.000 || 90.000 || 330.000 || 390.000 || 720.000 ||
|-
|-
|8:7:1||240.000||210.000||30.000||270.000||450.000||720.000||
| 8:7:1 || 240.000 || 210.000 || 30.000 || 270.000 || 450.000 || 720.000 ||
|-
|-
|10:3:2||300.000||90.000||60.000||360.000||390.000||750.000||
| 10:3:2 || 300.000 || 90.000 || 60.000 || 360.000 || 390.000 || 750.000 ||
|-
|-
|10:4:1||300.000||120.000||30.000||330.000||420.000||750.000||
| 10:4:1 || 300.000 || 120.000 || 30.000 || 330.000 || 420.000 || 750.000 ||
|-
|-
| rowspan="5" |[[41edo]]||7:6:4||204.878||175.610||117.073||321.951||380.488||702.439||5-limit patent val
| rowspan="5" | [[41edo]] || 7:6:4 || 204.878 || 175.610 || 117.073 || 321.951 || 380.488 || 702.439 || 5-limit patent val
|-
|-
|9:4:3||263.415||117.073||87.805||351.220||380.488||731.707||
| 9:4:3 || 263.415 || 117.073 || 87.805 || 351.220 || 380.488 || 731.707 ||
|-
|-
|9:5:2||263.415||146.341||58.537||321.951||409.756||731.707||
| 9:5:2 || 263.415 || 146.341 || 58.537 || 321.951 || 409.756 || 731.707 ||
|-
|-
|9:6:1||263.415||175.610||29.268||292.683||439.024||731.707||
| 9:6:1 || 263.415 || 175.610 || 29.268 || 292.683 || 439.024 || 731.707 ||
|-
|-
|11:3:1||321.951||87.805||29.268||351.220||409.756||760.976||
| 11:3:1 || 321.951 || 87.805 || 29.268 || 351.220 || 409.756 || 760.976 ||
|-
|-
| rowspan="5" |[[42edo]]||8:5:4||228.571||142.857||114.286||342.857||371.429||714.286||
| rowspan="5" | [[42edo]] || 8:5:4 || 228.571 || 142.857 || 114.286 || 342.857 || 371.429 || 714.286 ||
|-
|-
|8:6:3||228.571||171.429||85.714||314.286||400.000||714.286||5-limit patent val
| 8:6:3 || 228.571 || 171.429 || 85.714 || 314.286 || 400.000 || 714.286 || 5-limit patent val
|-
|-
|8:7:2||228.571||200.000||57.143||285.714||428.571||714.286||
| 8:7:2 || 228.571 || 200.000 || 57.143 || 285.714 || 428.571 || 714.286 ||
|-
|-
|10:5:1||285.714||142.857||28.571||314.286||428.571||742.857||
| 10:5:1 || 285.714 || 142.857 || 28.571 || 314.286 || 428.571 || 742.857 ||
|-
|-
|12:2:1||342.857||57.143||28.571||371.429||400.000||771.429||
| 12:2:1 || 342.857 || 57.143 || 28.571 || 371.429 || 400.000 || 771.429 ||
|-
|-
| rowspan="6" |[[43edo]]||7:6:5||195.349||167.442||139.535||334.884||362.791||697.674||
| rowspan="6" | [[43edo]] || 7:6:5 || 195.349 || 167.442 || 139.535 || 334.884 || 362.791 || 697.674 ||
|-
|-
|9:5:3||251.163||139.535||83.721||334.884||390.698||725.581||
| 9:5:3 || 251.163 || 139.535 || 83.721 || 334.884 || 390.698 || 725.581 ||
|-
|-
|9:6:2||251.163||167.442||55.814||306.977||418.605||725.581||
| 9:6:2 || 251.163 || 167.442 || 55.814 || 306.977 || 418.605 || 725.581 ||
|-
|-
|9:7:1||251.163||195.349||27.907||279.070||446.512||725.581||
| 9:7:1 || 251.163 || 195.349 || 27.907 || 279.070 || 446.512 || 725.581 ||
|-
|-
|11:3:2||306.977||83.721||55.814||362.791||390.698||753.488||
| 11:3:2 || 306.977 || 83.721 || 55.814 || 362.791 || 390.698 || 753.488 ||
|-
|-
|11:4:1||306.977||111.628||27.907||334.884||418.605||753.488||
| 11:4:1 || 306.977 || 111.628 || 27.907 || 334.884 || 418.605 || 753.488 ||
|-
|-
| rowspan="5" |[[44edo]]||8:7:3||218.182||190.909||81.818||300.000||409.091||709.091||
| rowspan="5" | [[44edo]] || 8:7:3 || 218.182 || 190.909 || 81.818 || 300.000 || 409.091 || 709.091 ||
|-
|-
|10:4:3||272.727||109.091||81.818||354.545||381.818||736.364||
| 10:4:3 || 272.727 || 109.091 || 81.818 || 354.545 || 381.818 || 736.364 ||
|-
|-
|10:5:2||272.727||136.364||54.545||327.273||409.091||736.364||
| 10:5:2 || 272.727 || 136.364 || 54.545 || 327.273 || 409.091 || 736.364 ||
|-
|-
|10:6:1||272.727||163.636||27.273||300.000||436.364||736.364||
| 10:6:1 || 272.727 || 163.636 || 27.273 || 300.000 || 436.364 || 736.364 ||
|-
|-
|12:3:1||327.273||81.818||27.273||354.545||409.091||763.636||
| 12:3:1 || 327.273 || 81.818 || 27.273 || 354.545 || 409.091 || 763.636 ||
|-
|-
| rowspan="6" |[[45edo]]||9:5:4||240.000||133.333||106.667||346.667||373.333||720.000||
| rowspan="6" | [[45edo]] || 9:5:4 || 240.000 || 133.333 || 106.667 || 346.667 || 373.333 || 720.000 ||
|-
|-
|9:7:2||240.000||186.667||53.333||293.333||426.667||720.000||
| 9:7:2 || 240.000 || 186.667 || 53.333 || 293.333 || 426.667 || 720.000 ||
|-
|-
|9:8:1||240.000||213.333||26.667||266.667||453.333||720.000||
| 9:8:1 || 240.000 || 213.333 || 26.667 || 266.667 || 453.333 || 720.000 ||
|-
|-
|11:4:2||293.333||106.667||53.333||346.667||400.000||746.667||
| 11:4:2 || 293.333 || 106.667 || 53.333 || 346.667 || 400.000 || 746.667 ||
|-
|-
|11:5:1||293.333||133.333||26.667||320.000||426.667||746.667||
| 11:5:1 || 293.333 || 133.333 || 26.667 || 320.000 || 426.667 || 746.667 ||
|-
|-
|13:2:1||346.667||53.333||26.667||373.333||400.000||773.333||
| 13:2:1 || 346.667 || 53.333 || 26.667 || 373.333 || 400.000 || 773.333 ||
|-
|-
| rowspan="6" |[[46edo]]||8:6:5||208.696||156.522||130.435||339.130||365.217||704.348||
| rowspan="6" | [[46edo]] || 8:6:5 || 208.696 || 156.522 || 130.435 || 339.130 || 365.217 || 704.348 ||
|-
|-
|8:7:4||208.696||182.609||104.348||313.043||391.304||704.348||5-limit patent val
| 8:7:4 || 208.696 || 182.609 || 104.348 || 313.043 || 391.304 || 704.348 || 5-limit patent val
|-
|-
|10:5:3||260.870||130.435||78.261||339.130||391.304||730.435||
| 10:5:3 || 260.870 || 130.435 || 78.261 || 339.130 || 391.304 || 730.435 ||
|-
|-
|10:7:1||260.870||182.609||26.087||286.957||443.478||730.435||
| 10:7:1 || 260.870 || 182.609 || 26.087 || 286.957 || 443.478 || 730.435 ||
|-
|-
|12:3:2||313.043||78.261||52.174||365.217||391.304||756.522||
| 12:3:2 || 313.043 || 78.261 || 52.174 || 365.217 || 391.304 || 756.522 ||
|-
|-
|12:4:1||313.043||104.348||26.087||339.130||417.391||756.522||
| 12:4:1 || 313.043 || 104.348 || 26.087 || 339.130 || 417.391 || 756.522 ||
|-
|-
| rowspan="7" |[[47edo]]||9:6:4||229.787||153.191||102.128||331.915||382.979||714.894||
| rowspan="7" | [[47edo]] || 9:6:4 || 229.787 || 153.191 || 102.128 || 331.915 || 382.979 || 714.894 ||
|-
|-
|9:7:3||229.787||178.723||76.596||306.383||408.511||714.894||
| 9:7:3 || 229.787 || 178.723 || 76.596 || 306.383 || 408.511 || 714.894 ||
|-
|-
|9:8:2||229.787||204.255||51.064||280.851||434.043||714.894||
| 9:8:2 || 229.787 || 204.255 || 51.064 || 280.851 || 434.043 || 714.894 ||
|-
|-
|11:4:3||280.851||102.128||76.596||357.447||382.979||740.426||
| 11:4:3 || 280.851 || 102.128 || 76.596 || 357.447 || 382.979 || 740.426 ||
|-
|-
|11:5:2||280.851||127.660||51.064||331.915||408.511||740.426||
| 11:5:2 || 280.851 || 127.660 || 51.064 || 331.915 || 408.511 || 740.426 ||
|-
|-
|11:6:1||280.851||153.191||25.532||306.383||434.043||740.426||
| 11:6:1 || 280.851 || 153.191 || 25.532 || 306.383 || 434.043 || 740.426 ||
|-
|-
|13:3:1||331.915||76.596||25.532||357.447||408.511||765.957||
| 13:3:1 || 331.915 || 76.596 || 25.532 || 357.447 || 408.511 || 765.957 ||
|-
|-
| rowspan="7" |[[48edo]]||8:7:5||200.000||175.000||125.000||325.000||375.000||700.000||5-limit patent val
| rowspan="7" | [[48edo]] || 8:7:5 || 200.000 || 175.000 || 125.000 || 325.000 || 375.000 || 700.000 || 5-limit patent val
|-
|-
|10:5:4||250.000||125.000||100.000||350.000||375.000||725.000||
| 10:5:4 || 250.000 || 125.000 || 100.000 || 350.000 || 375.000 || 725.000 ||
|-
|-
|10:6:3||250.000||150.000||75.000||325.000||400.000||725.000||
| 10:6:3 || 250.000 || 150.000 || 75.000 || 325.000 || 400.000 || 725.000 ||
|-
|-
|10:7:2||250.000||175.000||50.000||300.000||425.000||725.000||
| 10:7:2 || 250.000 || 175.000 || 50.000 || 300.000 || 425.000 || 725.000 ||
|-
|-
|10:8:1||250.000||200.000||25.000||275.000||450.000||725.000||
| 10:8:1 || 250.000 || 200.000 || 25.000 || 275.000 || 450.000 || 725.000 ||
|-
|-
|12:5:1||300.000||125.000||25.000||325.000||425.000||750.000||
| 12:5:1 || 300.000 || 125.000 || 25.000 || 325.000 || 425.000 || 750.000 ||
|-
|-
|14:2:1||350.000||50.000||25.000||375.000||400.000||775.000||
| 14:2:1 || 350.000 || 50.000 || 25.000 || 375.000 || 400.000 || 775.000 ||
|-
|-
| rowspan="8" |[[49edo]]||9:6:5||220.408||146.939||122.449||342.857||367.347||710.204||
| rowspan="8" | [[49edo]] || 9:6:5 || 220.408 || 146.939 || 122.449 || 342.857 || 367.347 || 710.204 ||
|-
|-
|9:7:4||220.408||171.429||97.959||318.367||391.837||710.204||5-limit patent val
| 9:7:4 || 220.408 || 171.429 || 97.959 || 318.367 || 391.837 || 710.204 || 5-limit patent val
|-
|-
|9:8:3||220.408||195.918||73.469||293.878||416.327||710.204||
| 9:8:3 || 220.408 || 195.918 || 73.469 || 293.878 || 416.327 || 710.204 ||
|-
|-
|11:5:3||269.388||122.449||73.469||342.857||391.837||734.694||
| 11:5:3 || 269.388 || 122.449 || 73.469 || 342.857 || 391.837 || 734.694 ||
|-
|-
|11:6:2||269.388||146.939||48.980||318.367||416.327||734.694||
| 11:6:2 || 269.388 || 146.939 || 48.980 || 318.367 || 416.327 || 734.694 ||
|-
|-
|11:7:1||269.388||171.429||24.490||293.878||440.816||734.694||
| 11:7:1 || 269.388 || 171.429 || 24.490 || 293.878 || 440.816 || 734.694 ||
|-
|-
|13:3:2||318.367||73.469||48.980||367.347||391.837||759.184||
| 13:3:2 || 318.367 || 73.469 || 48.980 || 367.347 || 391.837 || 759.184 ||
|-
|-
|13:4:1||318.367||97.959||24.490||342.857||416.327||759.184||
| 13:4:1 || 318.367 || 97.959 || 24.490 || 342.857 || 416.327 || 759.184 ||
|-
|-
| rowspan="7" |[[50edo]]||8:7:6||192.000||168.000||144.000||336.000||360.000||696.000||
| rowspan="7" | [[50edo]] || 8:7:6 || 192.000 || 168.000 || 144.000 || 336.000 || 360.000 || 696.000 ||
|-
|-
|10:7:3||240.000||168.000||72.000||312.000||408.000||720.000||
| 10:7:3 || 240.000 || 168.000 || 72.000 || 312.000 || 408.000 || 720.000 ||
|-
|-
|10:9:1||240.000||216.000||24.000||264.000||456.000||720.000||
| 10:9:1 || 240.000 || 216.000 || 24.000 || 264.000 || 456.000 || 720.000 ||
|-
|-
|12:4:3||288.000||96.000||72.000||360.000||384.000||744.000||
| 12:4:3 || 288.000 || 96.000 || 72.000 || 360.000 || 384.000 || 744.000 ||
|-
|-
|12:5:2||288.000||120.000||48.000||336.000||408.000||744.000||
| 12:5:2 || 288.000 || 120.000 || 48.000 || 336.000 || 408.000 || 744.000 ||
|-
|-
|12:6:1||288.000||144.000||24.000||312.000||432.000||744.000||
| 12:6:1 || 288.000 || 144.000 || 24.000 || 312.000 || 432.000 || 744.000 ||
|-
|-
|14:3:1||336.000||72.000||24.000||360.000||408.000||768.000||
| 14:3:1 || 336.000 || 72.000 || 24.000 || 360.000 || 408.000 || 768.000 ||
|-
|-
| rowspan="9" |[[51edo]]||9:7:5||211.765||164.706||117.647||329.412||376.471||705.882||5-limit patent val
| rowspan="9" | [[51edo]] || 9:7:5 || 211.765 || 164.706 || 117.647 || 329.412 || 376.471 || 705.882 || 5-limit patent val
|-
|-
|9:8:4||211.765||188.235||94.118||305.882||400.000||705.882||
| 9:8:4 || 211.765 || 188.235 || 94.118 || 305.882 || 400.000 || 705.882 ||
|-
|-
|11:5:4||258.824||117.647||94.118||352.941||376.471||729.412||
| 11:5:4 || 258.824 || 117.647 || 94.118 || 352.941 || 376.471 || 729.412 ||
|-
|-
|11:6:3||258.824||141.176||70.588||329.412||400.000||729.412||
| 11:6:3 || 258.824 || 141.176 || 70.588 || 329.412 || 400.000 || 729.412 ||
|-
|-
|11:7:2||258.824||164.706||47.059||305.882||423.529||729.412||
| 11:7:2 || 258.824 || 164.706 || 47.059 || 305.882 || 423.529 || 729.412 ||
|-
|-
|11:8:1||258.824||188.235||23.529||282.353||447.059||729.412||
| 11:8:1 || 258.824 || 188.235 || 23.529 || 282.353 || 447.059 || 729.412 ||
|-
|-
|13:4:2||305.882||94.118||47.059||352.941||400.000||752.941||
| 13:4:2 || 305.882 || 94.118 || 47.059 || 352.941 || 400.000 || 752.941 ||
|-
|-
|13:5:1||305.882||117.647||23.529||329.412||423.529||752.941||
| 13:5:1 || 305.882 || 117.647 || 23.529 || 329.412 || 423.529 || 752.941 ||
|-
|-
|15:2:1||352.941||47.059||23.529||376.471||400.000||776.471||
| 15:2:1 || 352.941 || 47.059 || 23.529 || 376.471 || 400.000 || 776.471 ||
|-
|-
| rowspan="8" |[[52edo]]||10:6:5||230.769||138.462||115.385||346.154||369.231||715.385||
| rowspan="8" | [[52edo]] || 10:6:5 || 230.769 || 138.462 || 115.385 || 346.154 || 369.231 || 715.385 ||
|-
|-
|10:7:4||230.769||161.538||92.308||323.077||392.308||715.385||
| 10:7:4 || 230.769 || 161.538 || 92.308 || 323.077 || 392.308 || 715.385 ||
|-
|-
|10:8:3||230.769||184.615||69.231||300.000||415.385||715.385||
| 10:8:3 || 230.769 || 184.615 || 69.231 || 300.000 || 415.385 || 715.385 ||
|-
|-
|10:9:2||230.769||207.692||46.154||276.923||438.462||715.385||
| 10:9:2 || 230.769 || 207.692 || 46.154 || 276.923 || 438.462 || 715.385 ||
|-
|-
|12:5:3||276.923||115.385||69.231||346.154||392.308||738.462||
| 12:5:3 || 276.923 || 115.385 || 69.231 || 346.154 || 392.308 || 738.462 ||
|-
|-
|12:7:1||276.923||161.538||23.077||300.000||438.462||738.462||
| 12:7:1 || 276.923 || 161.538 || 23.077 || 300.000 || 438.462 || 738.462 ||
|-
|-
|14:3:2||323.077||69.231||46.154||369.231||392.308||761.538||
| 14:3:2 || 323.077 || 69.231 || 46.154 || 369.231 || 392.308 || 761.538 ||
|-
|-
|14:4:1||323.077||92.308||23.077||346.154||415.385||761.538||
| 14:4:1 || 323.077 || 92.308 || 23.077 || 346.154 || 415.385 || 761.538 ||
|-
|-
| rowspan="10" |[[53edo]]||9:7:6||203.774||158.491||135.849||339.623||362.264||701.887||
| rowspan="10" | [[53edo]] || 9:7:6 || 203.774 || 158.491 || 135.849 || 339.623 || 362.264 || 701.887 ||
|-
|-
|9:8:5||203.774||181.132||113.208||316.981||384.906||701.887||5-limit patent val
| 9:8:5 || 203.774 || 181.132 || 113.208 || 316.981 || 384.906 || 701.887 || 5-limit patent val
|-
|-
|11:6:4||249.057||135.849||90.566||339.623||384.906||724.528||
| 11:6:4 || 249.057 || 135.849 || 90.566 || 339.623 || 384.906 || 724.528 ||
|-
|-
|11:7:3||249.057||158.491||67.925||316.981||407.547||724.528||
| 11:7:3 || 249.057 || 158.491 || 67.925 || 316.981 || 407.547 || 724.528 ||
|-
|-
|11:8:2||249.057||181.132||45.283||294.340||430.189||724.528||
| 11:8:2 || 249.057 || 181.132 || 45.283 || 294.340 || 430.189 || 724.528 ||
|-
|-
|11:9:1||249.057||203.774||22.642||271.698||452.830||724.528||
| 11:9:1 || 249.057 || 203.774 || 22.642 || 271.698 || 452.830 || 724.528 ||
|-
|-
|13:4:3||294.340||90.566||67.925||362.264||384.906||747.170||
| 13:4:3 || 294.340 || 90.566 || 67.925 || 362.264 || 384.906 || 747.170 ||
|-
|-
|13:5:2||294.340||113.208||45.283||339.623||407.547||747.170||
| 13:5:2 || 294.340 || 113.208 || 45.283 || 339.623 || 407.547 || 747.170 ||
|-
|-
|13:6:1||294.340||135.849||22.642||316.981||430.189||747.170||
| 13:6:1 || 294.340 || 135.849 || 22.642 || 316.981 || 430.189 || 747.170 ||
|-
|-
|15:3:1||339.623||67.925||22.642||362.264||407.547||769.811||
| 15:3:1 || 339.623 || 67.925 || 22.642 || 362.264 || 407.547 || 769.811 ||
|-
|-
| rowspan="7" |[[54edo]]||10:7:5||222.222||155.556||111.111||333.333||377.778||711.111||5-limit patent val
| rowspan="7" | [[54edo]] || 10:7:5 || 222.222 || 155.556 || 111.111 || 333.333 || 377.778 || 711.111 || 5-limit patent val
|-
|-
|10:9:3||222.222||200.000||66.667||288.889||422.222||711.111||
| 10:9:3 || 222.222 || 200.000 || 66.667 || 288.889 || 422.222 || 711.111 ||
|-
|-
|12:5:4||266.667||111.111||88.889||355.556||377.778||733.333||
| 12:5:4 || 266.667 || 111.111 || 88.889 || 355.556 || 377.778 || 733.333 ||
|-
|-
|12:7:2||266.667||155.556||44.444||311.111||422.222||733.333||
| 12:7:2 || 266.667 || 155.556 || 44.444 || 311.111 || 422.222 || 733.333 ||
|-
|-
|12:8:1||266.667||177.778||22.222||288.889||444.444||733.333||
| 12:8:1 || 266.667 || 177.778 || 22.222 || 288.889 || 444.444 || 733.333 ||
|-
|-
|14:5:1||311.111||111.111||22.222||333.333||422.222||755.556||
| 14:5:1 || 311.111 || 111.111 || 22.222 || 333.333 || 422.222 || 755.556 ||
|-
|-
|16:2:1||355.556||44.444||22.222||377.778||400.000||777.778||
| 16:2:1 || 355.556 || 44.444 || 22.222 || 377.778 || 400.000 || 777.778 ||  
|}
|}


Line 546: Line 539:


== See also ==
== See also ==
* [[Blackdye]] – a 10-note scale that is an extension to nicetone.
* [[Blackdye]] &ndash; A 10-note scale that is an extension to nicetone.
* [[Zarlino]] – a 5-limit JI scale with the same pattern.
* [[Zarlino]] &ndash; A 5-limit JI scale with the same pattern.
* [[Omnidiatonic]]­­ ­– sister 2L 3M 2s scale
* [[Omnidiatonic]]­­ ­&ndash; Sister 2L 3M 2s scale
* [[Antinicetone]] ­– sister 2L 2M 3s scale
* [[Antinicetone]] &ndash; Sister 2L 2M 3s scale
* [[5L 2s]] – LM-equalized version of nicetone
* [[5L 2s]] &ndash; LM-equalized version of nicetone
** [[5L 2s Muddles]] – other diatonic muddles
** [[5L 2s Muddles]] &ndash; Other diatonic muddles
* [[3L 4s]] – MS-equalized version of nicetone
* [[3L 4s]] &ndash; MS-equalized version of nicetone
* [[3L 2s]] – collapsed version of nicetone
* [[3L 2s]] &ndash; Collapsed version of nicetone


[[Category:Rank-3 scales]]
[[Category:Rank-3 scales]]
[[Category:7-tone scales]]
[[Category:7-tone scales]]
[[Category:GO scales]]
[[Category:GO scales]]

Revision as of 14:32, 9 September 2024

Nicetone (also known as the Zarlino pattern or Ptolemaic diatonic) is a 7-note maximum-variety-3 scale with the step signature 3L 2M 2s. Nicetone is a chiral scale with left-handed (LH, step pattern LMLsMLs) and right-handed (RH, step pattern LMLsLMs) variants that are rotationally non-equivalent. 15edo is the first equal division that supports nicetone.

Nicetone has the same pattern of the 5-limit Zarlino scale, though it encompasses the whole range of 3L 2M 2s. It's also a subset of the 5L 2m 3s blackdye scale.

Nicetone is intermediate between the 5L 2s diatonic scale and the 3L 4s neutral scale.

Nicetone can be tuned as a 5-limit JI scale or a tempered version thereof, where L represents 9/8, M represents 10/9, and s represents 16/15.

Comparison of Pythagorean and Ptolemaic diatonic scales in 53edo
Name Structure Step Sizes Graphical Representation
Ptolemaic 3L 2M 2s 9\53, 8\53, 5\53 ├────────┼───────┼────┼────────┼───────┼────────┼────┤
Pythagorean 5L 2s 9\53, 4\53 ├────────┼────────┼───┼────────┼────────┼────────┼───┤
Meantone,
1⁄5 comma		 7\43, 4\43 ├──────┼──────┼───┼──────┼──────┼──────┼───┤
Comparison with mosh and antipentic in 33edo
Name Structure Step Sizes Graphical Representation
Mosh 3L 4s 7\33, 3\33 ├──┼──────┼──┼──────┼──┼──┼──────┤
Nicetone 3L 2M 2s 7\33, 4\33, 2\33 ├───┼──────┼─┼──────┼───┼─┼──────┤
Antipentic 3L 2s 7\33, 6\33 ├─────┼──────┼──────┼─────┼──────┤

Intervals

The following is a table of nicetone 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 nicetone using the following expressions.

Interval sizes in nicetone
Interval class Sizes 5-limit JI 15edo
(L:M:s = 3:2:1) 		41edo
(= 7:6:4)
Second
(1-step) 		Small s 16/15, 111.73¢ 1\15, 80.00¢ 4\41, 117.07¢
Medium M 10/9, 182.40¢ 2\15, 160.00¢ 6\41, 175.61¢
Large L 9/8, 203.91¢ 3\15, 240.00¢ 7\41, 204.88¢
Third
(2-step) 		Small M + s 32/27, 294.13¢ 3\15, 240.00¢ 10\41, 292.68¢
Medium L + s 6/5, 315.64¢ 4\15, 320.00¢ 11\41, 321.95¢
Large L + M 5/4, 386.31¢ 5\15, 400.00¢ 13\41, 380.49¢
Fourth
(3-step) 		Small L + M + s 4/3, 498.04¢ 6\15, 480.00¢ 17\41, 497.56¢
Medium 2L + s 27/20, 519.55¢ 7\15, 560.00¢ 18\41, 526.83¢
Large 2L + M 45/32, 590.22¢ 8\15, 640.00¢ 20\41, 585.37¢
Fifth
(4-step) 		Small L + M + 2s 64/45, 609.78¢ 7\15, 560.00¢ 21\41, 614.63¢
Medium L + 2M + s 40/27, 680.45¢ 8\15, 640.00¢ 23\41, 673.17¢
Large 2L + M + s 3/2, 701.96¢ 9\15, 720.00¢ 24\41, 702.44¢
Sixth
(5-step) 		Small 2L + M + 2s 8/5, 813.69¢ 10\15, 800.00¢ 28\41, 819.51¢
Medium 2L + 2M + s 5/3, 884.36¢ 11\15, 880.00¢ 30\41, 878.05¢
Large 3L + M + s 27/16, 905.87¢ 12\15, 960.00¢ 31\41, 907.32¢
Seventh
(6-step) 		Small 2L + 2M + 2s 16/9, 996.09¢ 12\15, 960.00¢ 34\41, 995.12¢
Medium 3L + M + 2s 9/5, 1017.60¢ 13\15, 1040.00¢ 35\41, 1024.39¢
Large 3L + 2M + s 15/8, 1088.27¢ 14\15, 1120.00¢ 37\41, 1082.93¢

Modes

Nicetone has 14 modes total, with 7 LH and 7 RH modes. The names are based on their diatonic (5L 2s) counterparts.

The modes are arranged by brightest to darkest.

Nicetone modes
Left handed Right handed
LMLsMLs
LH Nice-Lydian 		LMLsLMs
RH Nice-Lydian
MLsLMLs
LH Nice-Ionian 		LMsLMLs
RH Nice-Ionian
MLsMLsL
LH Nice-Mixolydian 		MLsLMsL
RH Nice-Mixolydian
LsLMLsM
LH Nice-Dorian 		MsLMLsL
RH Nice-Dorian
LsMLsLM
LH Nice-Aeolian 		LsLMsLM
RH Nice-Aeolian
sLMLsML
LH Nice-Phrygian 		sLMLsLM
RH Nice-Phrygian
sMLsLML
LH Nice-Locrian 		sLMsLML
RH Nice-Locrian

Tunings

Tuning range of nicetone
Tuning range (in octaves)
Outer generator
(G1 = 2L + M + s) 		[math]\displaystyle{ \displaystyle \frac{4}{7} &lt; G_\text{1} &lt; \frac{2}{3} }[/math]
RH inner generator
(G2R = L + M) 		[math]\displaystyle{ \displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 4 G_\text{1} - 2 \text{ for } \frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5} }[/math]
[math]\displaystyle{ \displaystyle \frac{1}{2} G_\text{1} &lt; G_\text{2R} &lt; 1 - G_\text{1} \text{ for }\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3} }[/math]
LH inner generator
(G2L = L + s) 		[math]\displaystyle{ \displaystyle 2 - 3 G_\text{1} &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \text{ for }\frac{4}{7} &lt; G_\text{1} &le; \frac{3}{5} }[/math]
[math]\displaystyle{ \displaystyle 2 G_\text{1} - 1 &lt; G_\text{2L} &lt; \frac{1}{2} G_\text{1} \text{ for }\frac{3}{5} &le; G_\text{1} &lt; \frac{2}{3} }[/math]
Large step
(L = 2G1 - 1) 		[math]\displaystyle{ \displaystyle \frac{1}{7} &lt; L &lt; \frac{1}{3} }[/math]
Middle step
(M = 1 - G1 - G2L) 		[math]\displaystyle{ \displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; L \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5} }[/math]
[math]\displaystyle{ \displaystyle \frac{1}{4} (1 - 3 L) &lt; M &lt; \frac{1}{2} (1 - 3 L) \text{for} \frac{1}{5} &le; L &lt; \frac{1}{3} }[/math]
Small step
(s = 1 - G1 - G2R) 		[math]\displaystyle{ \displaystyle \frac{1}{2} (1 - 5 L) &lt; s &lt; \frac{1}{4} (1 - 3 L) \text{ for } \frac{1}{7} &lt; L &le; \frac{1}{5} }[/math]
[math]\displaystyle{ \displaystyle 0 &lt; s &lt; \frac{1}{4} (1 - 3 L) \text{for} \frac{1}{5} &le; L &lt; \frac{1}{3} }[/math]
Common Nicetone tunings
Tuning L:M:s Size of step (¢) Inner generator Outer generator
(2L + M + s)		 Comments
L M s LH (L + s) RH (L + M)
5-limit JI 203.910 182.404 111.731 315.641 386.314 701.955 L = 9/8, M = 10/9, s = 16/15
15edo 3:2:1 240.000 160.000 80.000 320.000 400.000 720.000 5-limit patent val
18edo 4:2:1 266.667 133.333 66.667 333.333 400.000 733.333 5-limit patent val
20edo 4:3:1 240.000 180.000 60.000 300.000 420.000 720.000
21edo 5:2:1 285.714 114.286 57.143 342.857 400.000 742.857
22edo 4:3:2 218.182 163.636 109.091 327.273 381.818 709.091 5-limit patent val
23edo 5:3:1 260.870 156.522 52.174 313.043 417.391 730.435
24edo 6:2:1 300.000 100.000 50.000 350.000 400.000 750.000
25edo 5:3:2 240.000 144.000 96.000 336.000 384.000 720.000 5-limit patent val
5:4:1 240.000 192.000 48.000 288.000 432.000 720.000
26edo 6:3:1 276.923 138.462 46.154 323.077 415.385 738.462
27edo 5:4:2 222.222 177.778 88.889 311.111 400.000 711.111 5-limit patent val
7:2:1 311.111 88.889 44.444 355.556 400.000 755.556
28edo 6:3:2 257.143 128.571 85.714 342.857 385.714 728.571
6:4:1 257.143 171.429 42.857 300.000 428.571 728.571
29edo 5:4:3 206.897 165.517 124.138 331.034 372.414 703.448 5-limit patent val
7:3:1 289.655 124.138 41.379 331.034 413.793 744.828
30edo 6:5:1 240.000 200.000 40.000 280.000 440.000 720.000
8:2:1 320.000 80.000 40.000 360.000 400.000 760.000
31edo 7:3:2 270.968 116.129 77.419 348.387 387.097 735.484
7:4:1 270.968 154.839 38.710 309.677 425.806 735.484
32edo 6:4:3 225.000 150.000 112.500 337.500 375.000 712.500 5-limit patent val
6:5:2 225.000 187.500 75.000 300.000 412.500 712.500
8:3:1 300.000 112.500 37.500 337.500 412.500 750.000
33edo 7:4:2 254.545 145.455 72.727 327.273 400.000 727.273
7:5:1 254.545 181.818 36.364 290.909 436.364 727.273
9:2:1 327.273 72.727 36.364 363.636 400.000 763.636
34edo 6:5:3 211.765 176.471 105.882 317.647 388.235 705.882 5-limit patent val
8:3:2 282.353 105.882 70.588 352.941 388.235 741.176
8:4:1 282.353 141.176 35.294 317.647 423.529 741.176
35edo 7:4:3 240.000 137.143 102.857 342.857 377.143 720.000
7:5:2 240.000 171.429 68.571 308.571 411.429 720.000
7:6:1 240.000 205.714 34.286 274.286 445.714 720.000
9:3:1 308.571 102.857 34.286 342.857 411.429 754.286
36edo 6:5:4 200.000 166.667 133.333 333.333 366.667 700.000
8:5:1 266.667 166.667 33.333 300.000 433.333 733.333
10:2:1 333.333 66.667 33.333 366.667 400.000 766.667
37edo 7:5:3 227.027 162.162 97.297 324.324 389.189 713.514 5-limit patent val
7:6:2 227.027 194.595 64.865 291.892 421.622 713.514
9:3:2 291.892 97.297 64.865 356.757 389.189 745.946
9:4:1 291.892 129.730 32.432 324.324 421.622 745.946
38edo 8:4:3 252.632 126.316 94.737 347.368 378.947 726.316
8:5:2 252.632 157.895 63.158 315.789 410.526 726.316
8:6:1 252.632 189.474 31.579 284.211 442.105 726.316
10:3:1 315.789 94.737 31.579 347.368 410.526 757.895
39edo 7:5:4 215.385 153.846 123.077 338.462 369.231 707.692
7:6:3 215.385 184.615 92.308 307.692 400.000 707.692 5-limit patent val
9:4:2 276.923 123.077 61.538 338.462 400.000 738.462
9:5:1 276.923 153.846 30.769 307.692 430.769 738.462
11:2:1 338.462 61.538 30.769 369.231 400.000 769.231
40edo 8:5:3 240.000 150.000 90.000 330.000 390.000 720.000
8:7:1 240.000 210.000 30.000 270.000 450.000 720.000
10:3:2 300.000 90.000 60.000 360.000 390.000 750.000
10:4:1 300.000 120.000 30.000 330.000 420.000 750.000
41edo 7:6:4 204.878 175.610 117.073 321.951 380.488 702.439 5-limit patent val
9:4:3 263.415 117.073 87.805 351.220 380.488 731.707
9:5:2 263.415 146.341 58.537 321.951 409.756 731.707
9:6:1 263.415 175.610 29.268 292.683 439.024 731.707
11:3:1 321.951 87.805 29.268 351.220 409.756 760.976
42edo 8:5:4 228.571 142.857 114.286 342.857 371.429 714.286
8:6:3 228.571 171.429 85.714 314.286 400.000 714.286 5-limit patent val
8:7:2 228.571 200.000 57.143 285.714 428.571 714.286
10:5:1 285.714 142.857 28.571 314.286 428.571 742.857
12:2:1 342.857 57.143 28.571 371.429 400.000 771.429
43edo 7:6:5 195.349 167.442 139.535 334.884 362.791 697.674
9:5:3 251.163 139.535 83.721 334.884 390.698 725.581
9:6:2 251.163 167.442 55.814 306.977 418.605 725.581
9:7:1 251.163 195.349 27.907 279.070 446.512 725.581
11:3:2 306.977 83.721 55.814 362.791 390.698 753.488
11:4:1 306.977 111.628 27.907 334.884 418.605 753.488
44edo 8:7:3 218.182 190.909 81.818 300.000 409.091 709.091
10:4:3 272.727 109.091 81.818 354.545 381.818 736.364
10:5:2 272.727 136.364 54.545 327.273 409.091 736.364
10:6:1 272.727 163.636 27.273 300.000 436.364 736.364
12:3:1 327.273 81.818 27.273 354.545 409.091 763.636
45edo 9:5:4 240.000 133.333 106.667 346.667 373.333 720.000
9:7:2 240.000 186.667 53.333 293.333 426.667 720.000
9:8:1 240.000 213.333 26.667 266.667 453.333 720.000
11:4:2 293.333 106.667 53.333 346.667 400.000 746.667
11:5:1 293.333 133.333 26.667 320.000 426.667 746.667
13:2:1 346.667 53.333 26.667 373.333 400.000 773.333
46edo 8:6:5 208.696 156.522 130.435 339.130 365.217 704.348
8:7:4 208.696 182.609 104.348 313.043 391.304 704.348 5-limit patent val
10:5:3 260.870 130.435 78.261 339.130 391.304 730.435
10:7:1 260.870 182.609 26.087 286.957 443.478 730.435
12:3:2 313.043 78.261 52.174 365.217 391.304 756.522
12:4:1 313.043 104.348 26.087 339.130 417.391 756.522
47edo 9:6:4 229.787 153.191 102.128 331.915 382.979 714.894
9:7:3 229.787 178.723 76.596 306.383 408.511 714.894
9:8:2 229.787 204.255 51.064 280.851 434.043 714.894
11:4:3 280.851 102.128 76.596 357.447 382.979 740.426
11:5:2 280.851 127.660 51.064 331.915 408.511 740.426
11:6:1 280.851 153.191 25.532 306.383 434.043 740.426
13:3:1 331.915 76.596 25.532 357.447 408.511 765.957
48edo 8:7:5 200.000 175.000 125.000 325.000 375.000 700.000 5-limit patent val
10:5:4 250.000 125.000 100.000 350.000 375.000 725.000
10:6:3 250.000 150.000 75.000 325.000 400.000 725.000
10:7:2 250.000 175.000 50.000 300.000 425.000 725.000
10:8:1 250.000 200.000 25.000 275.000 450.000 725.000
12:5:1 300.000 125.000 25.000 325.000 425.000 750.000
14:2:1 350.000 50.000 25.000 375.000 400.000 775.000
49edo 9:6:5 220.408 146.939 122.449 342.857 367.347 710.204
9:7:4 220.408 171.429 97.959 318.367 391.837 710.204 5-limit patent val
9:8:3 220.408 195.918 73.469 293.878 416.327 710.204
11:5:3 269.388 122.449 73.469 342.857 391.837 734.694
11:6:2 269.388 146.939 48.980 318.367 416.327 734.694
11:7:1 269.388 171.429 24.490 293.878 440.816 734.694
13:3:2 318.367 73.469 48.980 367.347 391.837 759.184
13:4:1 318.367 97.959 24.490 342.857 416.327 759.184
50edo 8:7:6 192.000 168.000 144.000 336.000 360.000 696.000
10:7:3 240.000 168.000 72.000 312.000 408.000 720.000
10:9:1 240.000 216.000 24.000 264.000 456.000 720.000
12:4:3 288.000 96.000 72.000 360.000 384.000 744.000
12:5:2 288.000 120.000 48.000 336.000 408.000 744.000
12:6:1 288.000 144.000 24.000 312.000 432.000 744.000
14:3:1 336.000 72.000 24.000 360.000 408.000 768.000
51edo 9:7:5 211.765 164.706 117.647 329.412 376.471 705.882 5-limit patent val
9:8:4 211.765 188.235 94.118 305.882 400.000 705.882
11:5:4 258.824 117.647 94.118 352.941 376.471 729.412
11:6:3 258.824 141.176 70.588 329.412 400.000 729.412
11:7:2 258.824 164.706 47.059 305.882 423.529 729.412
11:8:1 258.824 188.235 23.529 282.353 447.059 729.412
13:4:2 305.882 94.118 47.059 352.941 400.000 752.941
13:5:1 305.882 117.647 23.529 329.412 423.529 752.941
15:2:1 352.941 47.059 23.529 376.471 400.000 776.471
52edo 10:6:5 230.769 138.462 115.385 346.154 369.231 715.385
10:7:4 230.769 161.538 92.308 323.077 392.308 715.385
10:8:3 230.769 184.615 69.231 300.000 415.385 715.385
10:9:2 230.769 207.692 46.154 276.923 438.462 715.385
12:5:3 276.923 115.385 69.231 346.154 392.308 738.462
12:7:1 276.923 161.538 23.077 300.000 438.462 738.462
14:3:2 323.077 69.231 46.154 369.231 392.308 761.538
14:4:1 323.077 92.308 23.077 346.154 415.385 761.538
53edo 9:7:6 203.774 158.491 135.849 339.623 362.264 701.887
9:8:5 203.774 181.132 113.208 316.981 384.906 701.887 5-limit patent val
11:6:4 249.057 135.849 90.566 339.623 384.906 724.528
11:7:3 249.057 158.491 67.925 316.981 407.547 724.528
11:8:2 249.057 181.132 45.283 294.340 430.189 724.528
11:9:1 249.057 203.774 22.642 271.698 452.830 724.528
13:4:3 294.340 90.566 67.925 362.264 384.906 747.170
13:5:2 294.340 113.208 45.283 339.623 407.547 747.170
13:6:1 294.340 135.849 22.642 316.981 430.189 747.170
15:3:1 339.623 67.925 22.642 362.264 407.547 769.811
54edo 10:7:5 222.222 155.556 111.111 333.333 377.778 711.111 5-limit patent val
10:9:3 222.222 200.000 66.667 288.889 422.222 711.111
12:5:4 266.667 111.111 88.889 355.556 377.778 733.333
12:7:2 266.667 155.556 44.444 311.111 422.222 733.333
12:8:1 266.667 177.778 22.222 288.889 444.444 733.333
14:5:1 311.111 111.111 22.222 333.333 422.222 755.556
16:2:1 355.556 44.444 22.222 377.778 400.000 777.778

Supersets

Nicetone has following generator-offset MV3 supersets:

  • Sephipechroid: 13-note 3L 5M 5s scale (LMsMLsMsLMsMs and LMsMLsMsMLsMs)
  • Interoneichro: 13-note 5L 3M 5s scale (LMsLsLMsLsMLs and LMsLsMLsLsMLs)
  • Sephimechroid: 13-note 5L 5M 3s scale (LMLMsLMLsMLMs and LMLsMLMLsMLMs)
  • Beatloid: 17-note 5L 5M 7s scale (LMsLsMLsMsLMsLsMs and LMsLsMLsMsLsMLsMs)
  • Enharoid: 17-note 5L 7M 5s scale (LMsLMsMLMsLMsMLsM and LMsLMsMLsMLMsMLsM)
  • Moharoid: 17-note 7L 5M 5s scale (LMLsLMsLMLsMLsLMs and LMLsMLsLMLsMLsLMs)

Remarkable non-MV3 generator-offset supersets include blackdye (10-note, LmLsLmLsLs).

See also

  • Blackdye – A 10-note scale that is an extension to nicetone.
  • Zarlino – A 5-limit JI scale with the same pattern.
  • Omnidiatonic­­ ­– Sister 2L 3M 2s scale
  • Antinicetone – Sister 2L 2M 3s scale
  • 5L 2s – LM-equalized version of nicetone
  • 3L 4s – MS-equalized version of nicetone
  • 3L 2s – Collapsed version of nicetone
Retrieved from "https://en.xen.wiki/index.php?title=Nicetone&oldid=154442"