Subharmonics 18–36: Difference between revisions

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{{Infobox IFDO|steps=18}}
{{Infobox subharmonics|18}}
'''18ifdo''' ([[IFDO|inverse-arithmetic frequency division of the octave]]), or '''18udo''' ([[utonal division]] of the octave) would divide the [[octave]] into eighteen inverse-arithmetically equal parts. It is a superset of [[17ifdo]] and a subset of [[19ifdo]], its inverse is [[18afdo]]. As a [[scale]] it may also be known as mode 18 of the subharmonic series or the Under-18 scale.
{{Subharmonics intro|18}}


Directly above its root it contains copies of [[6ifdo]] and [[9ifdo]], making it a comprehensive scale for under-3 harmony, the [[utonal]] version of over-3 [[primodality]].  
Directly above its root it contains copies of [[Subharmonics 6–12|subharmonics 6–12]] and [[Subharmonics 9–18|subharmonics 9–18]], making it a comprehensive scale for under-3 harmony, the [[utonal]] version of over-3 [[primodality]].  


Above its root it contains a fairly even mix of [[5-limit]] intervals familiar to [[12edo]] listeners, and higher limit [[xenharmonic]] sonorities.
Above its root it contains a fairly even mix of [[5-limit]] intervals familiar to [[12edo]] listeners, and higher limit [[xenharmonic]] sonorities.


Coincidentally, 18ifdo has the property that its first step is similar in size to a [[24edo]]-step, and its 18th step is similar in size to a [[12edo]]-step.
Coincidentally, this scale has the property that its first step is similar in size to a [[24edo]]-step, and its 18th step is similar in size to a [[12edo]]-step.


The first serious attempt made at documenting and using 18ifdo was made by the [[Monthly Tunings]] Facebook group  (a smaller group within the larger [[Xenharmonic Alliance]]). They voted it as the monthly tuning for September 2026, and made the first attempts to explore this tuning.
The first serious attempt made at documenting and using this scale was made by the [[Monthly Tunings]] Facebook group  (a smaller group within the larger [[Xenharmonic Alliance]]). They voted it as the monthly tuning for September 2026, and made the first attempts to explore this tuning.


== Intervals ==
== Intervals ==
Line 132: Line 132:
| [[File:Jid_2_1_pluck_adu_dr220.mp3]]
| [[File:Jid_2_1_pluck_adu_dr220.mp3]]
|}
|}
{{todo|complete table|comment=add audio examples|inline=1}}


== Modes ==
== Modes ==
Interval values given in [[cents]].
Interval values given in [[cents]].
'''Orange''' indicates intervals in the subminor and minor third [[interval region]]s.
'''Yellow''' indicates intervals in the major third region.
'''Teal''' indicates intervals in the perfect fourth region.
'''Pink''' indicates intervals in the semiaugmented fourth (near 11/8) region.
'''Blue''' indicates intervals in the perfect fifth interval region.
'''Green''' indicates intervals in the major sixth interval region.
'''Red''' indicates intervals in the subminor seventh ('barbershop seventh') interval region.
''(Interval region boundaries are subjective and approximate. See [[interval region]] for discussion.)''


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{| class="wikitable" style="text-align:center;"
Line 141: Line 158:
|-
|-
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| 0 || 48.77 || 98.955 || 150.637 || 203.91 || style="background:#FFD6A5; font-weight:bold;" | 258.874 || style="background:#FFD6A5; font-weight:bold;" | 315.641 || style="background:#FDFFB6; font-weight:bold;" | 374.333 || 435.084 || style="background:#9BF6FF; font-weight:bold;" | 498.045 || style="background:#FFC6FF; font-weight:bold;" | 563.382 || 631.283 || style="background:#A0C4FF; font-weight:bold;" | 701.955 || 775.636 || 852.592 || 933.129 || 1017.596 || 1106.397 || 0
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|-
! 48.77
! 48.77
| 1151.23 || 0 || 50.185 || 101.867 || 155.14 || 210.104 || style="background:#FFFF00; font-weight:bold;" | 266.871 || 325.563 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 386.314 || 449.275 || 514.612 || 582.513 || 653.185 || 726.866 || 803.822 || 884.359 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 968.826 || 1057.627 || 1151.23
| 1151.23 || 0 || 50.185 || 101.867 || 155.14 || 210.104 || style="background:#FFD6A5; font-weight:bold;" | 266.871 || style="background:#FFD6A5; font-weight:bold;" | 325.563 || style="background:#FDFFB6; font-weight:bold;" | 386.314 || 449.275 || style="background:#9BF6FF; font-weight:bold;" | 514.612 || 582.513 || 653.185 || 726.866 || 803.822 || style="background:#CAFFBF; font-weight:bold;" | 884.359 || style="background:#FFADAD; font-weight:bold;" | 968.826 || 1057.627 || 1151.23
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|-
! 98.955
! 98.955
| 1101.045 || 1149.815 || 0 || 51.682 || 104.955 || 159.919 || 216.686 || style="background:#FFFF00; font-weight:bold;" | 275.378 || 336.129 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 399.09 || 464.427 || 532.328 || 603 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 676.681 || 753.637 || 834.174 || 918.641 || 1007.442 || 1101.045
| 1101.045 || 1149.815 || 0 || 51.682 || 104.955 || 159.919 || 216.686 || style="background:#FFD6A5; font-weight:bold;" | 275.378 || style="background:#FFD6A5; font-weight:bold;" | 336.129 || style="background:#FDFFB6; font-weight:bold;" | 399.09 || 464.427 || style="background:#FFC6FF; font-weight:bold;" | 532.328 || 603 || 676.681 || 753.637 || 834.174 || 918.641 || 1007.442 || 1101.045
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|-
! 150.637
! 150.637
| 1049.363 || 1098.133 || 1148.318 || 0 || 53.273 || 108.237 || 165.004 || 223.696 || style="background:#FFFF00; font-weight:bold;" | 284.447 || 347.408 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 412.745 || 480.646 || 551.318 || 624.999 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955 || 782.492 || 866.959 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 955.76 || 1049.363
| 1049.363 || 1098.133 || 1148.318 || 0 || 53.273 || 108.237 || 165.004 || 223.696 || style="background:#FFD6A5; font-weight:bold;" | 284.447 || 347.408 || 412.745 || style="background:#9BF6FF; font-weight:bold;" | 480.646 || style="background:#FFC6FF; font-weight:bold;" | 551.318 || 624.999 || style="background:#A0C4FF; font-weight:bold;" | 701.955 || 782.492 || style="background:#CAFFBF; font-weight:bold;" | 866.959 || style="background:#FFADAD; font-weight:bold;" | 955.76 || 1049.363
|-
|-
! 203.91
! 203.91
| style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 996.09 || 1044.86 || 1095.045 || 1146.727 || 0 || 54.964 || 111.731 || 170.423 || 231.174 || style="background:#FFFF00; font-weight:bold;" | 294.135 || 359.472 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 427.373 || 498.045 || 571.726 || 648.682 || 729.219 || 813.686 || 902.487 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 996.09
| style="background:#FFADAD; font-weight:bold;" | 996.09 || 1044.86 || 1095.045 || 1146.727 || 0 || 54.964 || 111.731 || 170.423 || 231.174 || style="background:#FFD6A5; font-weight:bold;" | 294.135 || 359.472 || 427.373 || style="background:#9BF6FF; font-weight:bold;" | 498.045 || 571.726 || 648.682 || 729.219 || 813.686 || style="background:#CAFFBF; font-weight:bold;" | 902.487 || style="background:#FFADAD; font-weight:bold;" | 996.09
|-
|-
! 258.874
! 258.874
| 941.126 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 989.896 || 1040.081 || 1091.763 || 1145.036 || 0 || 56.767 || 115.459 || 176.21 || 239.171 || style="background:#FFFF00; font-weight:bold;" | 304.508 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 372.409 || 443.081 || 516.762 || 593.718 || 674.255 || 758.722 || 847.523 || 941.126
| 941.126 || style="background:#FFADAD; font-weight:bold;" | 989.896 || 1040.081 || 1091.763 || 1145.036 || 0 || 56.767 || 115.459 || 176.21 || 239.171 || style="background:#FFD6A5; font-weight:bold;" | 304.508 || style="background:#FDFFB6; font-weight:bold;" | 372.409 || 443.081 || style="background:#9BF6FF; font-weight:bold;" | 516.762 || 593.718 || 674.255 || 758.722 || 847.523 || 941.126
|-
|-
! 315.641
! 315.641
| 884.359 || 933.129 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 983.314 || 1034.996 || 1088.269 || 1143.233 || 0 || 58.692 || 119.443 || 182.404 || 247.741 || style="background:#FFFF00; font-weight:bold;" | 315.642 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 386.314 || 459.995 || 536.951 || 617.488 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955 || 790.756 || 884.359
| style="background:#CAFFBF; font-weight:bold;" | 884.359 || 933.129 || style="background:#FFADAD; font-weight:bold;" | 983.314 || 1034.996 || 1088.269 || 1143.233 || 0 || 58.692 || 119.443 || 182.404 || 247.741 || style="background:#FFD6A5; font-weight:bold;" | 315.642 || style="background:#FDFFB6; font-weight:bold;" | 386.314 || 459.995 || style="background:#FFC6FF; font-weight:bold;" | 536.951 || 617.488 || style="background:#A0C4FF; font-weight:bold;" | 701.955 || 790.756 || style="background:#CAFFBF; font-weight:bold;" | 884.359
|-
|-
! 374.333
! 374.333
| 825.667 || 874.437 || 924.622 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 976.304 || 1029.577 || 1084.541 || 1141.308 || 0 || 60.751 || 123.712 || 189.049 || style="background:#FFFF00; font-weight:bold;" | 256.95 || 327.622 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 401.303 || 478.259 || 558.796 || 643.263 || 732.064 || 825.667
| 825.667 || style="background:#CAFFBF; font-weight:bold;" | 874.437 || 924.622 || style="background:#FFADAD; font-weight:bold;" | 976.304 || 1029.577 || 1084.541 || 1141.308 || 0 || 60.751 || 123.712 || 189.049 || style="background:#FFD6A5; font-weight:bold;" | 256.95 || style="background:#FFD6A5; font-weight:bold;" | 327.622 || style="background:#FDFFB6; font-weight:bold;" | 401.303 || style="background:#9BF6FF; font-weight:bold;" | 478.259 || style="background:#FFC6FF; font-weight:bold;" | 558.796 || 643.263 || 732.064 || 825.667
|-
|-
! 435.084
! 435.084
| 764.916 || 813.686 || 863.871 || 915.553 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 968.826 || 1023.79 || 1080.557 || 1139.249 || 0 || 62.961 || 128.298 || 196.199 || style="background:#FFFF00; font-weight:bold;" | 266.871 || 340.552 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 417.508 || 498.045 || 582.512 || 671.313 || 764.916
| 764.916 || 813.686 || style="background:#CAFFBF; font-weight:bold;" | 863.871 || 915.553 || style="background:#FFADAD; font-weight:bold;" | 968.826 || 1023.79 || 1080.557 || 1139.249 || 0 || 62.961 || 128.298 || 196.199 || style="background:#FFD6A5; font-weight:bold;" | 266.871 || 340.552 || 417.508 || style="background:#9BF6FF; font-weight:bold;" | 498.045 || 582.512 || 671.313 || 764.916
|-
|-
! 498.045
! 498.045
| style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955 || 750.725 || 800.91 || 852.592 || 905.865 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 960.829 || 1017.596 || 1076.288 || 1137.039 || 0 || 65.337 || 133.238 || 203.91 || style="background:#FFFF00; font-weight:bold;" | 277.591 || 354.547 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 435.084 || 519.551 || 608.352 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955
| style="background:#A0C4FF; font-weight:bold;" | 701.955 || 750.725 || 800.91 || 852.592 || style="background:#CAFFBF; font-weight:bold;" | 905.865 || style="background:#FFADAD; font-weight:bold;" | 960.829 || 1017.596 || 1076.288 || 1137.039 || 0 || 65.337 || 133.238 || 203.91 || style="background:#FFD6A5; font-weight:bold;" | 277.591 || 354.547 || 435.084 || style="background:#9BF6FF; font-weight:bold;" | 519.551 || 608.352 || style="background:#A0C4FF; font-weight:bold;" | 701.955
|-
|-
! 563.382
! 563.382
| 636.618 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 685.388 || 735.573 || 787.255 || 840.528 || 895.492 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 952.259 || 1010.951 || 1071.702 || 1134.663 || 0 || 67.901 || 138.573 || 212.254 || style="background:#FFFF00; font-weight:bold;" | 289.21 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 369.747 || 454.214 || 543.015 || 636.618
| 636.618 || style="background:#A0C4FF; font-weight:bold;" | 685.388 || 735.573 || 787.255 || 840.528 || style="background:#CAFFBF; font-weight:bold;" | 895.492 || style="background:#FFADAD; font-weight:bold;" | 952.259 || 1010.951 || 1071.702 || 1134.663 || 0 || 67.901 || 138.573 || 212.254 || style="background:#FFD6A5; font-weight:bold;" | 289.21 || style="background:#FDFFB6; font-weight:bold;" | 369.747 || 454.214 || style="background:#FFC6FF; font-weight:bold;" | 543.015 || 636.618
|-
|-
! 631.283
! 631.283
| 568.717 || 617.487 || 667.672 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 719.354 || 772.627 || 827.591 || 884.358 || 943.05 || 1003.801 || 1066.762 || 1132.099 || 0 || 70.672 || 144.353 || 221.309 || style="background:#FFFF00; font-weight:bold;" | 301.846 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 386.313 || 475.114 || 568.717
| style="background:#FFC6FF; font-weight:bold;" | 568.717 || 617.487 || 667.672 || style="background:#A0C4FF; font-weight:bold;" | 719.354 || 772.627 || 827.591 || style="background:#CAFFBF; font-weight:bold;" | 884.358 || 943.05 || 1003.801 || 1066.762 || 1132.099 || 0 || 70.672 || 144.353 || 221.309 || style="background:#FFD6A5; font-weight:bold;" | 301.846 || style="background:#FDFFB6; font-weight:bold;" | 386.313 || 475.114 || style="background:#FFC6FF; font-weight:bold;" | 568.717
|-
|-
! 701.955
! 701.955
| 498.045 || 546.815 || 597 || 648.682 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955 || 756.919 || 813.686 || 872.378 || 933.129 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 996.09 || 1061.427 || 1129.328 || 0 || 73.681 || 150.637 || 231.174 || style="background:#FFFF00; font-weight:bold;" | 315.641 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 404.442 || 498.045
| style="background:#9BF6FF; font-weight:bold;" | 498.045 || style="background:#FFC6FF; font-weight:bold;" | 546.815 || 597 || 648.682 || style="background:#A0C4FF; font-weight:bold;" | 701.955 || 756.919 || 813.686 || style="background:#CAFFBF; font-weight:bold;" | 872.378 || 933.129 || style="background:#FFADAD; font-weight:bold;" | 996.09 || 1061.427 || 1129.328 || 0 || 73.681 || 150.637 || 231.174 || style="background:#FFD6A5; font-weight:bold;" | 315.641 || style="background:#FDFFB6; font-weight:bold;" | 404.442 || style="background:#9BF6FF; font-weight:bold;" | 498.045
|-
|-
! 775.636
! 775.636
| style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 424.364 || 473.134 || 523.319 || 575.001 || 628.274 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 683.238 || 740.005 || 798.697 || 859.448 || 922.409 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 987.746 || 1055.647 || 1126.319 || 0 || 76.956 || 157.493 || 241.96 || 330.761 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 424.364
| 424.364 || 473.134 || 523.319 || 575.001 || 628.274 || style="background:#A0C4FF; font-weight:bold;" | 683.238 || 740.005 || 798.697 || style="background:#CAFFBF; font-weight:bold;" | 859.448 || 922.409 || style="background:#FFADAD; font-weight:bold;" | 987.746 || 1055.647 || 1126.319 || 0 || 76.956 || 157.493 || 241.96 || style="background:#FFD6A5; font-weight:bold;" | 330.761 || 424.364
|-
|-
! 852.592
! 852.592
| 347.408 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 396.178 || 446.363 || 498.045 || 551.318 || 606.282 || 663.049 || 721.741 || 782.492 || 845.453 || 910.79 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 978.691 || 1049.363 || 1123.044 || 0 || 80.537 || 165.004 || style="background:#FFFF00; font-weight:bold;" | 253.805 || 347.408
| 347.408 || style="background:#FDFFB6; font-weight:bold;" | 396.178 || 446.363 || style="background:#9BF6FF; font-weight:bold;" | 498.045 || style="background:#FFC6FF; font-weight:bold;" | 551.318 || 606.282 || 663.049 || style="background:#A0C4FF; font-weight:bold;" | 721.741 || 782.492 || 845.453 || 910.79 || style="background:#FFADAD; font-weight:bold;" | 978.691 || 1049.363 || 1123.044 || 0 || 80.537 || 165.004 || style="background:#FFD6A5; font-weight:bold;" | 253.805 || 347.408
|-
|-
! 933.129
! 933.129
| style="background:#FFFF00; font-weight:bold;" | 266.871 || style="background:#FFFF00; font-weight:bold;" | 315.641 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 365.826 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 417.508 || 470.781 || 525.745 || 582.512 || 641.204 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955 || 764.916 || 830.253 || 898.154 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 968.826 || 1042.507 || 1119.463 || 0 || 84.467 || 173.268 || style="background:#FFFF00; font-weight:bold;" | 266.871
| style="background:#FFD6A5; font-weight:bold;" | 266.871 || style="background:#FFD6A5; font-weight:bold;" | 315.641 || style="background:#FDFFB6; font-weight:bold;" | 365.826 || 417.508 || 470.781 || 525.745 || 582.512 || 641.204 || style="background:#A0C4FF; font-weight:bold;" | 701.955 || 764.916 || 830.253 || style="background:#CAFFBF; font-weight:bold;" | 898.154 || style="background:#FFADAD; font-weight:bold;" | 968.826 || 1042.507 || 1119.463 || 0 || 84.467 || 173.268 || style="background:#FFD6A5; font-weight:bold;" | 266.871
|-
|-
! 1017.596
! 1017.596
| 182.404 || 231.174 || style="background:#FFFF00; font-weight:bold;" | 281.359 || 333.041 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 386.314 || 441.278 || 498.045 || 556.737 || 617.488 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 680.449 || 745.786 || 813.687 || 884.359 || style="background:#00B050; font-weight:bold; color:#FFFFFF;" | 958.04 || 1034.996 || 1115.533 || 0 || 88.801 || 182.404
| 182.404 || 231.174 || style="background:#FFD6A5; font-weight:bold;" | 281.359 || style="background:#FFD6A5; font-weight:bold;" | 333.041 || style="background:#FDFFB6; font-weight:bold;" | 386.314 || 441.278 || style="background:#9BF6FF; font-weight:bold;" | 498.045 || style="background:#FFC6FF; font-weight:bold;" | 556.737 || 617.488 || style="background:#A0C4FF; font-weight:bold;" | 680.449 || 745.786 || 813.687 || style="background:#CAFFBF; font-weight:bold;" | 884.359 || style="background:#FFADAD; font-weight:bold;" | 958.04 || 1034.996 || 1115.533 || 0 || 88.801 || 182.404
|-
|-
! 1106.397
! 1106.397
| 93.603 || 142.373 || 192.558 || 244.24 || style="background:#FFFF00; font-weight:bold;" | 297.513 || 352.477 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 409.244 || 467.936 || 528.687 || 591.648 || 656.985 || 724.886 || 795.558 || 869.239 || 946.195 || 1026.732 || 1111.199 || 0 || 93.603
| 93.603 || 142.373 || 192.558 || 244.24 || style="background:#FFD6A5; font-weight:bold;" | 297.513 || 352.477 || style="background:#FDFFB6; font-weight:bold;" | 409.244 || 467.936 || 528.687 || 591.648 || 656.985 || 724.886 || 795.558 || style="background:#CAFFBF; font-weight:bold;" | 869.239 || 946.195 || 1026.732 || 1111.199 || 0 || 93.603
|-
|-
! 1200
! 1200
| 0 || 48.77 || 98.955 || 150.637 || 203.91 || style="background:#FFFF00; font-weight:bold;" | 258.874 || style="background:#FFFF00; font-weight:bold;" | 315.641 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 374.333 || style="background:#7030A0; font-weight:bold; color:#FFFFFF;" | 435.084 || 498.045 || 563.382 || 631.283 || style="background:#FF0000; font-weight:bold; color:#FFFFFF;" | 701.955 || 775.636 || 852.592 || 933.129 || 1017.596 || 1106.397 || 0
| 0 || 48.77 || 98.955 || 150.637 || 203.91 || style="background:#FFD6A5; font-weight:bold;" | 258.874 || style="background:#FFD6A5; font-weight:bold;" | 315.641 || style="background:#FDFFB6; font-weight:bold;" | 374.333 || 435.084 || style="background:#9BF6FF; font-weight:bold;" | 498.045 || style="background:#FFC6FF; font-weight:bold;" | 563.382 || 631.283 || style="background:#A0C4FF; font-weight:bold;" | 701.955 || 775.636 || 852.592 || 933.129 || 1017.596 || 1106.397 || 0
|}
|}


Line 204: Line 221:
* Fireflies{{idio}}: 36/30-36/24-36/21-36/20-36/18
* Fireflies{{idio}}: 36/30-36/24-36/21-36/20-36/18
* Han-Kumoi: 36/32-36/27-36/24-36/23-36/18
* Han-Kumoi: 36/32-36/27-36/24-36/23-36/18
* Minor Blues: 36/30-36/27-36/26-36/24-36/20-36/19-36/18
* Minor Blues: 36/30-36/27-36/26-36/24-36/20-36/18
* Minor hexatonic: 36/32-36/30-36/27-36/24-36/20-36/18
* Minor hexatonic: 36/32-36/30-36/27-36/24-36/20-36/18
* Minor melodic pentatonic: 36/32-36/30-36/24-36/19-36/18
* Minor melodic pentatonic: 36/32-36/30-36/24-36/19-36/18
Line 216: Line 233:
* LTN file: [[File:18ifdo.ltn]]
* LTN file: [[File:18ifdo.ltn]]
* Image file: [[File:Miller 18ifdo mapping.png]]
* Image file: [[File:Miller 18ifdo mapping.png]]
== Music ==
; [[Noah Jordan]]
* [https://m.youtube.com/watch?v=voic4mft41M ''FILMBOARD: Microtonal Study #1 - 18ifdo — Inverse-Arithmetic Frequency Division of the Octave''] (2026)
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=zwMpZU5wo1A ''Four microtonal improvisations in 18ifdo''] (2026)
; [[Claudi Meneghin]]
* [https://www.youtube.com/shorts/e0da1CkFNZI ''A 2-in-1 canon in F for organ (short version), in 18-IFDO, for baroque oboe and bassoon''] (2026)
{{Navbox subharmonics}}

Latest revision as of 00:52, 11 October 2026

Subharmonics 18–36
Prime factorization 2 × 32
Fifth 36/24 (701.955c)

The subharmonics 18–36 are the subharmonics 18 through 36 of the subharmonic series. Above the root they form the ratios 36/36, 36/35, …, 36/18 and span one octave. Used as a scale, this set is also called mode 18 of the subharmonic series. The inverse consists of harmonics 18–36.

Directly above its root it contains copies of subharmonics 6–12 and subharmonics 9–18, making it a comprehensive scale for under-3 harmony, the utonal version of over-3 primodality.

Above its root it contains a fairly even mix of 5-limit intervals familiar to 12edo listeners, and higher limit xenharmonic sonorities.

Coincidentally, this scale has the property that its first step is similar in size to a 24edo-step, and its 18th step is similar in size to a 12edo-step.

The first serious attempt made at documenting and using this scale was made by the Monthly Tunings Facebook group (a smaller group within the larger Xenharmonic Alliance). They voted it as the monthly tuning for September 2026, and made the first attempts to explore this tuning.

Intervals

# Cents Ratio Interval name Audio
0 0.00 1/1 perfect unison
1 48.770 36/35 septimal quarter tone
2 98.955 18/17 small septendecimal semitone, Arabic lute index finger
3 150.637 12/11 small undecimal neutral second
4 203.910 9/8 Pythagorean (whole) tone, major (whole) tone, octave-reduced 9th harmonic
5 258.874 36/31 tricesimoprimal semifourth, tricesimoprimal subminor third
6 315.641 6/5 classic minor third, just minor third
7 374.333 36/29 vicesimononal major third
8 435.084 9/7 septimal supermajor third
9 498.045 4/3 just perfect fourth, octave-reduced 3rd subharmonic
10 563.382 18/13 tridecimal augmented fourth
11 631.283 36/25 pental diminished fifth, classic diminshed fifth
12 701.955 3/2 just perfect fifth, octave-reduced 3rd harmonic
13 775.636 36/23 vicesimotertial augmented fifth
14 852.592 18/11 undecimal neutral sixth
15 933.129 12/7 septimal supermajor sixth
16 1017.596 9/5 classic minor seventh, pental minor seventh
17 1106.397 36/19 undevicesimal major seventh, Boethius' major seventh
18 1200.000 2/1 octave
Todo: complete table

add audio examples

Modes

Interval values given in cents.

Orange indicates intervals in the subminor and minor third interval regions.

Yellow indicates intervals in the major third region.

Teal indicates intervals in the perfect fourth region.

Pink indicates intervals in the semiaugmented fourth (near 11/8) region.

Blue indicates intervals in the perfect fifth interval region.

Green indicates intervals in the major sixth interval region.

Red indicates intervals in the subminor seventh ('barbershop seventh') interval region.

(Interval region boundaries are subjective and approximate. See interval region for discussion.)

0 48.77 98.955 150.637 203.91 258.874 315.641 374.333 435.084 498.045 563.382 631.283 701.955 775.636 852.592 933.129 1017.596 1106.397 1200
0 0 48.77 98.955 150.637 203.91 258.874 315.641 374.333 435.084 498.045 563.382 631.283 701.955 775.636 852.592 933.129 1017.596 1106.397 0
48.77 1151.23 0 50.185 101.867 155.14 210.104 266.871 325.563 386.314 449.275 514.612 582.513 653.185 726.866 803.822 884.359 968.826 1057.627 1151.23
98.955 1101.045 1149.815 0 51.682 104.955 159.919 216.686 275.378 336.129 399.09 464.427 532.328 603 676.681 753.637 834.174 918.641 1007.442 1101.045
150.637 1049.363 1098.133 1148.318 0 53.273 108.237 165.004 223.696 284.447 347.408 412.745 480.646 551.318 624.999 701.955 782.492 866.959 955.76 1049.363
203.91 996.09 1044.86 1095.045 1146.727 0 54.964 111.731 170.423 231.174 294.135 359.472 427.373 498.045 571.726 648.682 729.219 813.686 902.487 996.09
258.874 941.126 989.896 1040.081 1091.763 1145.036 0 56.767 115.459 176.21 239.171 304.508 372.409 443.081 516.762 593.718 674.255 758.722 847.523 941.126
315.641 884.359 933.129 983.314 1034.996 1088.269 1143.233 0 58.692 119.443 182.404 247.741 315.642 386.314 459.995 536.951 617.488 701.955 790.756 884.359
374.333 825.667 874.437 924.622 976.304 1029.577 1084.541 1141.308 0 60.751 123.712 189.049 256.95 327.622 401.303 478.259 558.796 643.263 732.064 825.667
435.084 764.916 813.686 863.871 915.553 968.826 1023.79 1080.557 1139.249 0 62.961 128.298 196.199 266.871 340.552 417.508 498.045 582.512 671.313 764.916
498.045 701.955 750.725 800.91 852.592 905.865 960.829 1017.596 1076.288 1137.039 0 65.337 133.238 203.91 277.591 354.547 435.084 519.551 608.352 701.955
563.382 636.618 685.388 735.573 787.255 840.528 895.492 952.259 1010.951 1071.702 1134.663 0 67.901 138.573 212.254 289.21 369.747 454.214 543.015 636.618
631.283 568.717 617.487 667.672 719.354 772.627 827.591 884.358 943.05 1003.801 1066.762 1132.099 0 70.672 144.353 221.309 301.846 386.313 475.114 568.717
701.955 498.045 546.815 597 648.682 701.955 756.919 813.686 872.378 933.129 996.09 1061.427 1129.328 0 73.681 150.637 231.174 315.641 404.442 498.045
775.636 424.364 473.134 523.319 575.001 628.274 683.238 740.005 798.697 859.448 922.409 987.746 1055.647 1126.319 0 76.956 157.493 241.96 330.761 424.364
852.592 347.408 396.178 446.363 498.045 551.318 606.282 663.049 721.741 782.492 845.453 910.79 978.691 1049.363 1123.044 0 80.537 165.004 253.805 347.408
933.129 266.871 315.641 365.826 417.508 470.781 525.745 582.512 641.204 701.955 764.916 830.253 898.154 968.826 1042.507 1119.463 0 84.467 173.268 266.871
1017.596 182.404 231.174 281.359 333.041 386.314 441.278 498.045 556.737 617.488 680.449 745.786 813.687 884.359 958.04 1034.996 1115.533 0 88.801 182.404
1106.397 93.603 142.373 192.558 244.24 297.513 352.477 409.244 467.936 528.687 591.648 656.985 724.886 795.558 869.239 946.195 1026.732 1111.199 0 93.603
1200 0 48.77 98.955 150.637 203.91 258.874 315.641 374.333 435.084 498.045 563.382 631.283 701.955 775.636 852.592 933.129 1017.596 1106.397 0

Scales

  • Akebono II: 36/34-36/27-36/24-36/23-36/18
  • Equiheptatonic: 36/33-36/30-36/27-36/24-36/22-36/20-36/18
  • Equipentatonic: 36/31-36/27-36/24-36/21-36/18
  • Fireflies[idiosyncratic term]: 36/30-36/24-36/21-36/20-36/18
  • Han-Kumoi: 36/32-36/27-36/24-36/23-36/18
  • Minor Blues: 36/30-36/27-36/26-36/24-36/20-36/18
  • Minor hexatonic: 36/32-36/30-36/27-36/24-36/20-36/18
  • Minor melodic pentatonic: 36/32-36/30-36/24-36/19-36/18
  • Mixolydian pentatonic: 36/32-36/27-36/24-36/19-36/18
  • Neutral 2 neutral 6 pentatonic (sounds cold/ancient): 36/33-36/27-36/24-36/22-36/18
  • Sharpened Dorian[idiosyncratic term] (sounds metallic/futuristic): 36/32-36/30-36/27-36/24-36/21-36/20-36/18

Instruments

Lumatone mappings

Herman Miller, 7 Sep 2026

Music

Noah Jordan
Budjarn Lambeth
Claudi Meneghin
View • Talk • EditUndertone scales 
Small modes 3 • 11 • 16 • 18
Families 3/: 3 • 18
Related Subharmonic series • Harmonic series • Overtone scale • Ringer scale
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