AFDO: Difference between revisions

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An '''ADO''' ('''arithmetic divisions of the octave''') or '''EFDO''' (~~'''~~[[equal frequency division]] of the octave~~'''~~), also known as a '''chord of nature''', is a [[period]]ic [[tuning system]] which divides the [[octave]] arithmetically rather than logarithmically.

An '''ADO''' ('''arithmetic divisions of the octave''') or '''EFDO''' ([[equal frequency division]] of the octave), also known as a '''chord of nature''', is a [[period]]ic [[tuning system]] which divides the [[octave]] [[equal frequency division|arithmetically]] rather than logarithmically.

~~For any ''C''-ADO system, the ''m''-th degree is equal to the ratio (''C'' + ''m'')/''C''.~~ For example, in [[12ado]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an ADO system, the distance between interval ratios is equal, rather than the distance between their logarithms like in EDO systems. All ADOs are subsets of [[just intonation]]. ADOs with more divisors such as [[Highly_composite_equal_division|highly composite]] ADOs generally have more useful just intervals.

For example, in [[12ado]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an ADO system, the distance between interval ratios is equal, rather than the distance between their logarithms like in EDO systems. All ADOs are subsets of [[just intonation]]. ADOs with more divisors such as [[Highly_composite_equal_division|highly composite]] ADOs generally have more useful just intervals.

If the ~~first division is <math>R_1</math> (which is~~ ratio of ~~C/C) and~~ the ~~last , <math>R_n</math> (which~~ is ~~ratio of 2C/C), with common difference of d~~

== Formula ==

Within each period of any ''n''-ado system, the [[frequency ratio]] ''r'' of the ''m''-th degree is

(which is 1/C), we ~~have~~ :

<math>\displaystyle r = (n + m)/n</math>

If the first division is ''r''<sub>0</sub> (which is ratio of (''n'' + 0)/''n'' = 1) and the last, ''r''<sub>''n''</sub> (which is ratio of (''n'' + ''n'')/''n'' = 2), with common difference of ''d'' (which is 1/''n''), we have:

<math>

~~R_2~~ = ~~R_1~~ + d \\

r_1 = r_0 + d \\

~~R_3~~= ~~R_1~~ + 2d \\

r_2 = r_0 + 2d \\

~~R_4~~ = ~~R_1~~ + 3d \\

r_3 = r_0 + 3d \\

\vdots \\

~~R_n~~ = ~~R_1~~ + ~~(n-1)d~~

r_m = r_0 + md

</math>

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-An '''ADO''' ('''arithmetic divisions of the octave''') or '''EFDO''' ('''[[equal frequency division]] of the octave'''), also known as a '''chord of nature''', is a [[period]]ic [[tuning system]] which divides the [[octave]] arithmetically rather than logarithmically.
+An '''ADO''' ('''arithmetic divisions of the octave''') or '''EFDO''' ([[equal frequency division]] of the octave), also known as a '''chord of nature''', is a [[period]]ic [[tuning system]] which divides the [[octave]] [[equal frequency division|arithmetically]] rather than logarithmically.
-For any ''C''-ADO system, the ''m''-th degree is equal to the ratio (''C'' + ''m'')/''C''. For example, in [[12ado]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an ADO system, the distance between interval ratios is equal, rather than the distance between their logarithms like in EDO systems. All ADOs are subsets of [[just intonation]]. ADOs with more divisors such as [[Highly_composite_equal_division|highly composite]] ADOs generally have more useful just intervals.
+For example, in [[12ado]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an ADO system, the distance between interval ratios is equal, rather than the distance between their logarithms like in EDO systems. All ADOs are subsets of [[just intonation]]. ADOs with more divisors such as [[Highly_composite_equal_division|highly composite]] ADOs generally have more useful just intervals.
-If the first division is <math>R_1</math> (which is ratio of C/C) and the last , <math>R_n</math> (which is ratio of 2C/C), with common difference of d
+== Formula ==
+Within each period of any ''n''-ado system, the [[frequency ratio]] ''r'' of the ''m''-th degree is
-(which is 1/C), we have :
+<math>\displaystyle r = (n + m)/n</math>
+If the first division is ''r''<sub>0</sub> (which is ratio of (''n'' + 0)/''n'' = 1) and the last, ''r''<sub>''n''</sub> (which is ratio of (''n'' + ''n'')/''n'' = 2), with common difference of ''d'' (which is 1/''n''), we have:
 <math>
-R_2 = R_1 + d \\
+r_1 = r_0 + d \\
-R_3= R_1 + 2d \\
+r_2 = r_0 + 2d \\
-R_4 = R_1 + 3d \\
+r_3 = r_0 + 3d \\
 \vdots \\
-R_n = R_1 + (n-1)d
+r_m = r_0 + md
 </math>