Otones12-24: Difference between revisions

Revision as of 19:33, 19 April 2009

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A dedecatonic (12-note) scale borrowed from the overtone series:

12/12, 13/12, 14/12, 15/12, 16/12, 17/12, 18/12, 19/12, 20/12, 21/12, 22/12, 23/12, 24/12

which reduces to:

1/1, 13/12, 7/6, 5/4, 4/3, 17/12, 3/2, 19/12, 5/3, 7/4, 11/6, 23/12, 2/1
in cents:
0, 139, 267, 386, 498, 603, 702, 796, 884, 969, 1049, 1126, 1200

Intervals between adjacent steps gradually get smaller, as is usual for overtone chunks like this:

13:12, 14:13, 15:14, 16:15, 17:16, 18:17, 19:18, 21:20, 22:21, 23:22, 24:23
in cents:
139, 128, 119, 112, 105, 99, 94, 89, 84, 81, 77, 74

==modes== 

For your tuning pleasure, all 12 modes, arranged in a handy-dandy table. The following matrix uses a keyboard mapping that starts the scale on C. Thus, C=1/1, C#=13/12, etc. To find an interval, say the interval from C to F#, first find the lower pitch on the left, C, & follow it across the row to the column of the higher pitch, F# to find 702 cents. To find the name of the interval in just intonation, use the number of the higher pitch as the numerator (18) and the number of the lower pitch (12) as the denominator, then reduce (3/2).

||   || C-12 || C#-13 || D-14 || D#-15 || E-16 || F-17 || F#-18 || G-19 || G#-20 || A-21 || A#-22 || B-23 ||
|| C-12 || 0 || 139 || 267 || 386 || 498 || 603 || 702 || 796 || 884 || 969 || 1049 || 1126 ||
|| C#-13 || 1061 || 0 || 128 || 248 || 359 || 464 || 563 || 657 || 746 || 830 || 911 || 988 ||
|| D-14 || 933 || 1072 || 0 || 119 || 231 || 336 || 435 || 529 || 617 || 702 || 782 || 859 ||
|| D#-15 || 814 || 952 || 1081 || 0 || 112 || 217 || 316 || 409 || 498 || 583 || 663 || 740 ||
|| E-16 || 702 || 841 || 969 || 1088 || 0 || 105 || 204 || 298 || 386 || 471 || 551 || 628 ||
|| F-17 || 597 || 736 || 864 || 983 || 1095 || 0 || 99 || 193 || 281 || 366 || 446 || 523 ||
|| F#-18 || 498 || 637 || 765 || 884 || 996 || 1101 || 0 || 94 || 182 || 267 || 347 || 424 ||
|| G-19 || 404 || 543 || 671 || 791 || 902 || 1007 || 1106 || 0 || 89 || 173 || 254 || 331 ||
|| G#-20 || 316 || 454 || 583 || 702 || 814 || 919 || 1018 || 1111 || 0 || 84 || 165 || 242 ||
|| A-21 || 231 || 370 || 498 || 617 || 729 || 834 || 933 || 1027 || 1116 || 0 || 81 || 157 ||
|| A#-22 || 151 || 289 || 418 || 537 || 649 || 754 || 853 || 946 || 1035 || 1119 || 0 || 77 ||
|| B-23 || 74 || 212 || 341 || 460 || 572 || 677 || 776 || 869 || 958 || 1043 || 1123 || 0 ||

You can see that, due to the varying step sizes, this relatively small scale contains a large number of unique rational intervals up to the 23-limit.

===Inventory of intervals from 0 to 1200 cents:=== 

0 - 1/1 - ...
74 - 24/23 - B-C
77 - 23/22 - A#-B
81 - 22/21 - A-A#
84 - 21/20 - G#-A
89 - 20/19 - G-G#
94 - 19/18 - F#-G
99 - 18/17 - F-F#
105 - 17/16 - E-F
112 - 16/15 - D#-E
119 - 15/14 - D-D#
128 - 14/13 - C#-D
139 - 13/12 - C-C#
151 - 12/11 - A#-C
157 - 23/21 - A-B
165 - 11/10 - A#-B#
173 - 21/19 - G-A
182 - 10/9 - F#-G#
193 - 19/17 - F-G
204 - 9/8 - E-F#
212 - 26/23 - B-C#
217 - 17/15 - D#-F
231 - 8/7 - D-E ; A-C
242 - 23/20 - G#-B
248 - 15/13 - C#-D#
254 - 22/19 - G-A#
267 - 7/6 - C-D ; F#-A
281 - 20/17 - F-G#
289 - 13/11 - A#-C#
316 - 6/5 - D#-F# ; G#-C
331 - 23/19 - G-B
336 - 17/14 - D-F
347 - 11/9 - F#-A#
359 - 16/13 - C#-E
366 - 21/17 - F-A
370 - 26/21 - A-C#
386 - 5/4 - C-D# ; E-G#
404 - 24/19 - G-C
409 - 19/15 - D#-G
418 - 14/11 - A#-D
424 - 23/18 - F#-B
435 - 9/7 - D-F#
446 - 22/17 - F-A#
454 - 13/10 - G#-C#
460 - 30/23 - B-D#
464 - 17/13 - C#-F
471 - 21/16 - E-A
498 - 4/3 - C-E ; D#-G# ; F#-C ; A-D
523 - 23/17 - F-B
529 - 19/14 - D-G
537 - 15/11 - A#-D#
543 - 26/19 - G-C#
551 - 11/8 - E-A
563 - 18/13 - C#-F#
572 - 32/23 - B-E
583 - 7/5 - D#-A ; G#-D
597 - 24/17 - F-C

603 - 17/12 - C-F
617 - 10/7 - D-G# ; A-D#
628 - 23/16 - E-B
637 - 13/9 - F#-C#
649 - 16/11 - A-E
657 - 19/13 - C#-G
663 - 22/15 - D#-A#
671 - 28/19 - G-D
677 - 34/23 - B-F
...

Original HTML content:

<html><head><title>otones12-24</title></head><body>A dedecatonic (12-note) scale borrowed from the overtone series:<br />
<br />
12/12, 13/12, 14/12, 15/12, 16/12, 17/12, 18/12, 19/12, 20/12, 21/12, 22/12, 23/12, 24/12<br />
<br />
which reduces to:<br />
<br />
1/1, 13/12, 7/6, 5/4, 4/3, 17/12, 3/2, 19/12, 5/3, 7/4, 11/6, 23/12, 2/1<br />
in cents:<br />
0, 139, 267, 386, 498, 603, 702, 796, 884, 969, 1049, 1126, 1200<br />
<br />
Intervals between adjacent steps gradually get smaller, as is usual for overtone chunks like this:<br />
<br />
13:12, 14:13, 15:14, 16:15, 17:16, 18:17, 19:18, 21:20, 22:21, 23:22, 24:23<br />
in cents:<br />
139, 128, 119, 112, 105, 99, 94, 89, 84, 81, 77, 74<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-modes"></a><!-- ws:end:WikiTextHeadingRule:0 -->modes</h2>
 <br />
For your tuning pleasure, all 12 modes, arranged in a handy-dandy table. The following matrix uses a keyboard mapping that starts the scale on C. Thus, C=1/1, C#=13/12, etc. To find an interval, say the interval from C to F#, first find the lower pitch on the left, C, &amp; follow it across the row to the column of the higher pitch, F# to find 702 cents. To find the name of the interval in just intonation, use the number of the higher pitch as the numerator (18) and the number of the lower pitch (12) as the denominator, then reduce (3/2).<br />
<br />


<table class="wiki_table">
    <tr>
        <td><br />
</td>
        <td>C-12<br />
</td>
        <td>C#-13<br />
</td>
        <td>D-14<br />
</td>
        <td>D#-15<br />
</td>
        <td>E-16<br />
</td>
        <td>F-17<br />
</td>
        <td>F#-18<br />
</td>
        <td>G-19<br />
</td>
        <td>G#-20<br />
</td>
        <td>A-21<br />
</td>
        <td>A#-22<br />
</td>
        <td>B-23<br />
</td>
    </tr>
    <tr>
        <td>C-12<br />
</td>
        <td>0<br />
</td>
        <td>139<br />
</td>
        <td>267<br />
</td>
        <td>386<br />
</td>
        <td>498<br />
</td>
        <td>603<br />
</td>
        <td>702<br />
</td>
        <td>796<br />
</td>
        <td>884<br />
</td>
        <td>969<br />
</td>
        <td>1049<br />
</td>
        <td>1126<br />
</td>
    </tr>
    <tr>
        <td>C#-13<br />
</td>
        <td>1061<br />
</td>
        <td>0<br />
</td>
        <td>128<br />
</td>
        <td>248<br />
</td>
        <td>359<br />
</td>
        <td>464<br />
</td>
        <td>563<br />
</td>
        <td>657<br />
</td>
        <td>746<br />
</td>
        <td>830<br />
</td>
        <td>911<br />
</td>
        <td>988<br />
</td>
    </tr>
    <tr>
        <td>D-14<br />
</td>
        <td>933<br />
</td>
        <td>1072<br />
</td>
        <td>0<br />
</td>
        <td>119<br />
</td>
        <td>231<br />
</td>
        <td>336<br />
</td>
        <td>435<br />
</td>
        <td>529<br />
</td>
        <td>617<br />
</td>
        <td>702<br />
</td>
        <td>782<br />
</td>
        <td>859<br />
</td>
    </tr>
    <tr>
        <td>D#-15<br />
</td>
        <td>814<br />
</td>
        <td>952<br />
</td>
        <td>1081<br />
</td>
        <td>0<br />
</td>
        <td>112<br />
</td>
        <td>217<br />
</td>
        <td>316<br />
</td>
        <td>409<br />
</td>
        <td>498<br />
</td>
        <td>583<br />
</td>
        <td>663<br />
</td>
        <td>740<br />
</td>
    </tr>
    <tr>
        <td>E-16<br />
</td>
        <td>702<br />
</td>
        <td>841<br />
</td>
        <td>969<br />
</td>
        <td>1088<br />
</td>
        <td>0<br />
</td>
        <td>105<br />
</td>
        <td>204<br />
</td>
        <td>298<br />
</td>
        <td>386<br />
</td>
        <td>471<br />
</td>
        <td>551<br />
</td>
        <td>628<br />
</td>
    </tr>
    <tr>
        <td>F-17<br />
</td>
        <td>597<br />
</td>
        <td>736<br />
</td>
        <td>864<br />
</td>
        <td>983<br />
</td>
        <td>1095<br />
</td>
        <td>0<br />
</td>
        <td>99<br />
</td>
        <td>193<br />
</td>
        <td>281<br />
</td>
        <td>366<br />
</td>
        <td>446<br />
</td>
        <td>523<br />
</td>
    </tr>
    <tr>
        <td>F#-18<br />
</td>
        <td>498<br />
</td>
        <td>637<br />
</td>
        <td>765<br />
</td>
        <td>884<br />
</td>
        <td>996<br />
</td>
        <td>1101<br />
</td>
        <td>0<br />
</td>
        <td>94<br />
</td>
        <td>182<br />
</td>
        <td>267<br />
</td>
        <td>347<br />
</td>
        <td>424<br />
</td>
    </tr>
    <tr>
        <td>G-19<br />
</td>
        <td>404<br />
</td>
        <td>543<br />
</td>
        <td>671<br />
</td>
        <td>791<br />
</td>
        <td>902<br />
</td>
        <td>1007<br />
</td>
        <td>1106<br />
</td>
        <td>0<br />
</td>
        <td>89<br />
</td>
        <td>173<br />
</td>
        <td>254<br />
</td>
        <td>331<br />
</td>
    </tr>
    <tr>
        <td>G#-20<br />
</td>
        <td>316<br />
</td>
        <td>454<br />
</td>
        <td>583<br />
</td>
        <td>702<br />
</td>
        <td>814<br />
</td>
        <td>919<br />
</td>
        <td>1018<br />
</td>
        <td>1111<br />
</td>
        <td>0<br />
</td>
        <td>84<br />
</td>
        <td>165<br />
</td>
        <td>242<br />
</td>
    </tr>
    <tr>
        <td>A-21<br />
</td>
        <td>231<br />
</td>
        <td>370<br />
</td>
        <td>498<br />
</td>
        <td>617<br />
</td>
        <td>729<br />
</td>
        <td>834<br />
</td>
        <td>933<br />
</td>
        <td>1027<br />
</td>
        <td>1116<br />
</td>
        <td>0<br />
</td>
        <td>81<br />
</td>
        <td>157<br />
</td>
    </tr>
    <tr>
        <td>A#-22<br />
</td>
        <td>151<br />
</td>
        <td>289<br />
</td>
        <td>418<br />
</td>
        <td>537<br />
</td>
        <td>649<br />
</td>
        <td>754<br />
</td>
        <td>853<br />
</td>
        <td>946<br />
</td>
        <td>1035<br />
</td>
        <td>1119<br />
</td>
        <td>0<br />
</td>
        <td>77<br />
</td>
    </tr>
    <tr>
        <td>B-23<br />
</td>
        <td>74<br />
</td>
        <td>212<br />
</td>
        <td>341<br />
</td>
        <td>460<br />
</td>
        <td>572<br />
</td>
        <td>677<br />
</td>
        <td>776<br />
</td>
        <td>869<br />
</td>
        <td>958<br />
</td>
        <td>1043<br />
</td>
        <td>1123<br />
</td>
        <td>0<br />
</td>
    </tr>
</table>

<br />
You can see that, due to the varying step sizes, this relatively small scale contains a large number of unique rational intervals up to the 23-limit.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-modes-Inventory of intervals from 0 to 1200 cents:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Inventory of intervals from 0 to 1200 cents:</h3>
 <br />
0 - 1/1 - ...<br />
74 - 24/23 - B-C<br />
77 - 23/22 - A#-B<br />
81 - 22/21 - A-A#<br />
84 - 21/20 - G#-A<br />
89 - 20/19 - G-G#<br />
94 - 19/18 - F#-G<br />
99 - 18/17 - F-F#<br />
105 - 17/16 - E-F<br />
112 - 16/15 - D#-E<br />
119 - 15/14 - D-D#<br />
128 - 14/13 - C#-D<br />
139 - 13/12 - C-C#<br />
151 - 12/11 - A#-C<br />
157 - 23/21 - A-B<br />
165 - 11/10 - A#-B#<br />
173 - 21/19 - G-A<br />
182 - 10/9 - F#-G#<br />
193 - 19/17 - F-G<br />
204 - 9/8 - E-F#<br />
212 - 26/23 - B-C#<br />
217 - 17/15 - D#-F<br />
231 - 8/7 - D-E ; A-C<br />
242 - 23/20 - G#-B<br />
248 - 15/13 - C#-D#<br />
254 - 22/19 - G-A#<br />
267 - 7/6 - C-D ; F#-A<br />
281 - 20/17 - F-G#<br />
289 - 13/11 - A#-C#<br />
316 - 6/5 - D#-F# ; G#-C<br />
331 - 23/19 - G-B<br />
336 - 17/14 - D-F<br />
347 - 11/9 - F#-A#<br />
359 - 16/13 - C#-E<br />
366 - 21/17 - F-A<br />
370 - 26/21 - A-C#<br />
386 - 5/4 - C-D# ; E-G#<br />
404 - 24/19 - G-C<br />
409 - 19/15 - D#-G<br />
418 - 14/11 - A#-D<br />
424 - 23/18 - F#-B<br />
435 - 9/7 - D-F#<br />
446 - 22/17 - F-A#<br />
454 - 13/10 - G#-C#<br />
460 - 30/23 - B-D#<br />
464 - 17/13 - C#-F<br />
471 - 21/16 - E-A<br />
498 - 4/3 - C-E ; D#-G# ; F#-C ; A-D<br />
523 - 23/17 - F-B<br />
529 - 19/14 - D-G<br />
537 - 15/11 - A#-D#<br />
543 - 26/19 - G-C#<br />
551 - 11/8 - E-A<br />
563 - 18/13 - C#-F#<br />
572 - 32/23 - B-E<br />
583 - 7/5 - D#-A ; G#-D<br />
597 - 24/17 - F-C<br />
<br />
603 - 17/12 - C-F<br />
617 - 10/7 - D-G# ; A-D#<br />
628 - 23/16 - E-B<br />
637 - 13/9 - F#-C#<br />
649 - 16/11 - A-E<br />
657 - 19/13 - C#-G<br />
663 - 22/15 - D#-A#<br />
671 - 28/19 - G-D<br />
677 - 34/23 - B-F<br />
...</body></html>

Otones12-24: Difference between revisions

Revision as of 19:33, 19 April 2009

IMPORTED REVISION FROM WIKISPACES

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