98304edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|98304}}
{{ED intro}}


== Theory ==
== Theory ==
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=== As a tuning standard ===
=== As a tuning standard ===
A step of 98304edo is known as a '''tridecamu''' (thirteenth MIDI-resolution unit, 13mu, 2<sup>13</sup> = 8192 equal divisions of the [[12edo]] semitone). The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or &minus;) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
A step of 98304edo is known as a '''tridecamu''' (thirteenth MIDI-resolution unit, 13mu, 2<sup>13</sup> = 8192 equal divisions of the [[12edo]] semitone). The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.


=== Prime harmonics ===
=== Prime harmonics ===
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{| class="wikitable right-1 right-2 center-3 right-4 right-5"
{| class="wikitable right-1 right-2 center-3 right-4 right-5"
|-
|-
! rowspan="2"| #
! rowspan="2" | #
! rowspan="2"| Cents
! rowspan="2" | Cents
! colspan="2"| JI Interval
! colspan="2" | JI interval
! rowspan="2"| Error<br>(Cents)
! rowspan="2" | Error<br>(cents)
|-
|-
! Ratio
! Ratio
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| [[1/1]]
| [[1/1]]
| 0.0000
| 0.0000
| &plusmn;0.00000
| ±0.00000
|-
|-
| 3995
| 3995
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| [[36/35]]
| [[36/35]]
| 48.7704
| 48.7704
| &minus;0.00329
| −0.00329
|-
|-
| 4111
| 4111
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| 35/34
| 35/34
| 50.1842
| 50.1842
| &minus;0.00111
| −0.00111
|-
|-
| 4234
| 4234
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| [[33/32]]
| [[33/32]]
| 53.2729
| 53.2729
| &minus;0.00146
| −0.00146
|-
|-
| 5158
| 5158
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| [[27/26]]
| [[27/26]]
| 65.3373
| 65.3373
| &minus;0.00531
| −0.00531
|-
|-
| 5562
| 5562
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| [[26/25]]
| [[26/25]]
| 67.9002
| 67.9002
| &minus;0.00473
| −0.00473
|-
|-
| 6598
| 6598
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| [[18/17]]
| [[18/17]]
| 98.9546
| 98.9546
| &minus;0.00440
| −0.00440
|-
|-
| 8598
| 8598
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| [[16/15]]
| [[16/15]]
| 111.7313
| 111.7313
| &minus;0.00033
| −0.00033
|-
|-
| 9785
| 9785
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| [[14/13]]
| [[14/13]]
| 128.2982
| 128.2982
| &minus;0.00235
| −0.00235
|-
|-
| 11352
| 11352
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| [[12/11]]
| [[12/11]]
| 150.6371
| 150.6371
| &minus;0.00229
| −0.00229
|-
|-
| 13517
| 13517
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| [[11/10]]
| [[11/10]]
| 165.0042
| 165.0042
| &minus;0.00179
| −0.00179
|-
|-
| 14943
| 14943
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| [[19/17]]
| [[19/17]]
| 192.5576
| 192.5576
| &minus;0.00390
| −0.00390
|-
|-
| 16704
| 16704
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| [[9/8]]
| [[9/8]]
| 203.9100
| 203.9100
| &minus;0.00375
| −0.00375
|-
|-
| 17751
| 17751
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| [[7/6]]
| [[7/6]]
| 266.8709
| 266.8709
| &minus;0.00079
| −0.00079
|-
|-
| 23049
| 23049
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| [[13/11]]
| [[13/11]]
| 289.2097
| 289.2097
| &minus;0.00074
| −0.00074
|-
|-
| 24372
| 24372
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| [[19/16]]
| [[19/16]]
| 297.5130
| 297.5130
| &minus;0.00325
| −0.00325
|-
|-
| 25857
| 25857
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| [[6/5]]
| [[6/5]]
| 315.6413
| 315.6413
| &minus;0.00408
| −0.00408
|-
|-
| 27536
| 27536
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| [[19/15]]
| [[19/15]]
| 409.2443
| 409.2443
| &minus;0.00358
| −0.00358
|-
|-
| 34202
| 34202
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| [[14/11]]
| [[14/11]]
| 417.5080
| 417.5080
| &minus;0.00308
| −0.00308
|-
|-
| 35642
| 35642
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| [[9/7]]
| [[9/7]]
| 435.0841
| 435.0841
| &minus;0.00109
| −0.00109
|-
|-
| 36566
| 36566
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| [[22/17]]
| [[22/17]]
| 446.3625
| 446.3625
| &minus;0.00023
| −0.00023
|-
|-
| 37209
| 37209
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| [[13/10]]
| [[13/10]]
| 454.2139
| 454.2139
| &minus;0.00252
| −0.00252
|-
|-
| 38046
| 38046
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| [[21/16]]
| [[21/16]]
| 470.7809
| 470.7809
| &minus;0.00454
| −0.00454
|-
|-
| 40800
| 40800
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| [[19/14]]
| [[19/14]]
| 528.6871
| 528.6871
| &minus;0.00059
| −0.00059
|-
|-
| 43987
| 43987
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| [[15/11]]
| [[15/11]]
| 536.9508
| 536.9508
| &minus;0.00009
| −0.00009
|-
|-
| 44484
| 44484
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| [[18/13]]
| [[18/13]]
| 563.3823
| 563.3823
| &minus;0.00343
| −0.00343
|-
|-
| 47719
| 47719
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| [[7/5]]
| [[7/5]]
| 582.5122
| 582.5122
| &minus;0.00487
| −0.00487
|-
|-
| 48906
| 48906
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| [[24/17]]
| [[24/17]]
| 596.9996
| 596.9996
| &minus;0.00252
| −0.00252
|-
|-
| 49398
| 49398
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| [[3/2]]
| [[3/2]]
| 701.9550
| 701.9550
| &minus;0.00188
| −0.00188
|-
|-
| 62662
| 62662
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| [[8/5]]
| [[8/5]]
| 813.6863
| 813.6863
| &minus;0.00220
| −0.00220
|-
|-
| 72447
| 72447
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| [[7/4]]
| [[7/4]]
| 968.8259
| 968.8259
| &minus;0.00266
| −0.00266
|-
|-
| 98304
| 98304
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| [[Octave|2/1]]
| [[Octave|2/1]]
| 1200.0000
| 1200.0000
| &plusmn;0.00000
| ±0.00000
|}
|}


Latest revision as of 22:37, 20 February 2025

← 98303edo 98304edo 98305edo →
Prime factorization 215 × 3
Step size 0.012207 ¢ 
Fifth 57504\98304 (701.953 ¢) (→ 599\1024)
Semitones (A1:m2) 9312:7392 (113.7 ¢ : 90.23 ¢)
Consistency limit 19
Distinct consistency limit 19

98304 equal divisions of the octave (abbreviated 98304edo or 98304ed2), also called 98304-tone equal temperament (98304tet) or 98304 equal temperament (98304et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 98304 equal parts of about 0.0122 ¢ each. Each step represents a frequency ratio of 21/98304, or the 98304th root of 2.

Theory

98304edo is consistent to the 19-odd-limit. The equal temperament tempers out [45 2 -28 6⟩, [-54 15 -16 24⟩, and [-29 135 -18 -51⟩ in the 7-limit; [-4 15 -14 7 -2⟩, [-3 28 2 -9 -6⟩, [-50 0 -2 17 2⟩, and [30 10 8 -2 -17⟩ in the 11-limit; 123201/123200, 32427005625/32426652544, 278924131584/278916015625, 37744795080531/37744172597248, and 156905298045000/156904157228819 in the 13-limit; 2000033/2000000, 154002541/154001250, 303464448/303460625, 338676338/338671875, 791249550/791243563, and 176846618624/176846076825 in the 17-limit; 89376/89375, 104976/104975, 709632/709631, 5836831/5836800, 494190983/494190000, 1206902781/1206878450, and 21867094832/21867015625 in the 19-limit.

As a tuning standard

A step of 98304edo is known as a tridecamu (thirteenth MIDI-resolution unit, 13mu, 213 = 8192 equal divisions of the 12edo semitone). The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

Prime harmonics

Approximation of prime harmonics in 98304edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00000 -0.00188 +0.00220 -0.00266 +0.00042 -0.00032 +0.00065 -0.00325 -0.00286 +0.00044 -0.00383
Relative (%) +0.0 -15.4 +18.1 -21.8 +3.4 -2.6 +5.3 -26.6 -23.5 +3.6 -31.4
Steps
(reduced) 		98304
(0) 		155808
(57504) 		228255
(31647) 		275974
(79366) 		340076
(45164) 		363768
(68856) 		401814
(8598) 		417588
(24372) 		444684
(51468) 		477559
(84343) 		487017
(93801)

Selected intervals

# Cents JI interval Error
(cents)
Ratio Cents
0 0.0000 1/1 0.0000 ±0.00000
3995 48.7671 36/35 48.7704 −0.00329
4111 50.1831 35/34 50.1842 −0.00111
4234 51.6846 34/33 51.6825 +0.00210
4364 53.2715 33/32 53.2729 −0.00146
5158 62.9639 28/27 62.9609 +0.00296
5352 65.3320 27/26 65.3373 −0.00531
5562 67.8955 26/25 67.9002 −0.00473
6598 80.5420 22/21 80.5370 +0.00496
7275 88.8062 20/19 88.8007 +0.00545
7668 93.6035 19/18 93.6030 +0.00050
8106 98.9502 18/17 98.9546 −0.00440
8598 104.9561 17/16 104.9554 +0.00065
9153 111.7310 16/15 111.7313 −0.00033
9785 119.4458 15/14 119.4428 +0.00299
10510 128.2959 14/13 128.2982 −0.00235
11352 138.5742 13/12 138.5727 +0.00156
12340 150.6348 12/11 150.6371 −0.00229
13517 165.0024 11/10 165.0042 −0.00179
14943 182.4097 10/9 182.4037 +0.00596
15774 192.5537 19/17 192.5576 −0.00390
16704 203.9063 9/8 203.9100 −0.00375
17751 216.6870 17/15 216.6867 +0.00032
18938 231.1768 8/7 231.1741 +0.00266
20295 247.7417 15/13 247.7411 +0.00065
20792 253.8086 22/19 253.8049 +0.00367
21862 266.8701 7/6 266.8709 −0.00079
23049 281.3599 20/17 281.3583 +0.00156
23692 289.2090 13/11 289.2097 −0.00074
24372 297.5098 19/16 297.5130 −0.00325
25857 315.6372 6/5 315.6413 −0.00408
27536 336.1328 17/14 336.1295 +0.00331
28460 347.4121 11/9 347.4079 +0.00417
29448 359.4727 16/13 359.4723 +0.00032
31647 386.3159 5/4 386.3137 +0.00220
33132 404.4434 24/19 404.4420 +0.00137
33525 409.2407 19/15 409.2443 −0.00358
34202 417.5049 14/11 417.5080 −0.00308
35642 435.0830 9/7 435.0841 −0.00109
36566 446.3623 22/17 446.3625 −0.00023
37209 454.2114 13/10 454.2139 −0.00252
38046 464.4287 17/13 464.4277 +0.00096
38566 470.7764 21/16 470.7809 −0.00454
40800 498.0469 4/3 498.0450 +0.00188
43310 528.6865 19/14 528.6871 −0.00059
43987 536.9507 15/11 536.9508 −0.00009
44484 543.0176 26/19 543.0146 +0.00293
45164 551.3184 11/8 551.3179 +0.00042
46152 563.3789 18/13 563.3823 −0.00343
47719 582.5073 7/5 582.5122 −0.00487
48906 596.9971 24/17 596.9996 −0.00252
49398 603.0029 17/12 603.0004 +0.00252
50585 617.4927 10/7 617.4878 +0.00487
57504 701.9531 3/2 701.9550 −0.00188
62662 764.9170 14/9 764.9159 +0.00109
66657 813.6841 8/5 813.6863 −0.00220
72447 884.3628 5/3 884.3587 +0.00408
76442 933.1299 12/7 933.1291 +0.00079
79366 968.8232 7/4 968.8259 −0.00266
98304 1200.0000 2/1 1200.0000 ±0.00000

See also

External links

Retrieved from "https://en.xen.wiki/index.php?title=98304edo&oldid=182095"