IFDO: Difference between revisions

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<math>\displaystyle c = (2n)/(2n - k)</math>

== Equal divisions of length ==

The '''equal division of length''' ('''EDL''') is equivalent to the IFDO. However, ''n''-edl corresponds to (''n''/2)-ifdo for any even number ''n''. Therefore, EDL cannot be used to represent an odd-numbered IFDO.

''n''-edl divides a string length to ''n'' equal divisions, so we have ''n''/2 divisions per octave. If the first division is ''l''1 and the last, ''l''''n'', we have:

: ''l''1 = ''l''2 = ''l''3 = … = ''l''''n''

So the sum of divisions is ''l'' or the string length. Note that the number of divisions in octave is half of the string length. By dividing a string length of ''l'' to ''n'' divisions we have:

: ''n'':(''n'' - 1):(''n'' - 2):(''n'' - 3):…:(''n'' - ''m''):…:1

where ''n'' - ''m'' is ''n''/2.

For example, by dividing string length to 12 equal divisions we have a series as:

: 12:11:10:9:8:7:6:5:4:3:2:1

which shows 12-edl:

[[file:edl1.jpg]]

12:12 means 12 from 12 divisions, 12:11 means 11 from 12 divisions and so on. Ratios as 12:11 shows active string length for each degree, which is vibrating. EDL system shows ascending trend of divisions sizes due to its inner structure and if compared with [[EDO]]:

[[file:Edl2.JPG]]

== Relation to superparticular ratios ==

An IFDO has step sizes of [[superparticular ratio]]s with decreasing numerators. For example, [[5ifdo]] has step sizes [[10/9]], [[9/8]], [[8/7]], [[7/6]], and [[6/5]].

== Relation to utonality and subharmonic series ==

We can consider an IFDO system as a [[Otonality and utonality|utonal system]]. ''Utonality'' is a term introduced by [[Harry Partch]] to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "[http://tonalsoft.com/enc/n/nexus.aspx Numerary nexus]".

== Individual pages for IFDOs ==

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** [[AFDO]] – arithmetic frequency division of the octave

** [[EDO]] – equal division of the octave

* [[UD|UD, or utonal division]]

== External links ==

* [https://web.archive.org/web/20220630142502/https://sites.google.com/site/240edo/equaldivisionsoflength(edl) 96~EDO | ''Equal divisions of length'']

== Notes ==

@@ Line 11: / Line 11: @@
 <math>\displaystyle c = (2n)/(2n - k)</math>
+== Equal divisions of length ==
+The '''equal division of length''' ('''EDL''') is equivalent to the IFDO. However, ''n''-edl corresponds to (''n''/2)-ifdo for any even number ''n''. Therefore, EDL cannot be used to represent an odd-numbered IFDO.
+''n''-edl divides a string length to ''n'' equal divisions, so we have ''n''/2 divisions per octave. If the first division is ''l''<sub>1</sub> and the last, ''l''<sub>''n''</sub>, we have:
+: ''l''<sub>1</sub> = ''l''<sub>2</sub> = ''l''<sub>3</sub> = … = ''l''<sub>''n''</sub>
+So the sum of divisions is ''l'' or the string length.  Note that the number of divisions in octave is half of the string length. By dividing a string length of ''l'' to ''n'' divisions we have:
+: ''n'':(''n'' - 1):(''n'' - 2):(''n'' - 3):…:(''n'' - ''m''):…:1
+where ''n'' - ''m'' is ''n''/2.
+For example, by dividing string length to 12 equal divisions we have a series as:
+: 12:11:10:9:8:7:6:5:4:3:2:1
+which shows 12-edl:
+[[file:edl1.jpg]]
+:12 means 12 from 12 divisions, 12:11 means 11 from 12 divisions and so on. Ratios as 12:11 shows active string length for each degree, which is vibrating. EDL system shows ascending trend of divisions sizes due to its inner structure and if compared with [[EDO]]:
+[[file:Edl2.JPG]]
+== Relation to superparticular ratios ==
+An IFDO has step sizes of [[superparticular ratio]]s with decreasing numerators. For example, [[5ifdo]] has step sizes [[10/9]], [[9/8]], [[8/7]], [[7/6]], and [[6/5]].
+== Relation to utonality and subharmonic series ==
+We can consider an IFDO system as a [[Otonality and utonality|utonal system]]. ''Utonality'' is a term introduced by [[Harry Partch]] to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "[http://tonalsoft.com/enc/n/nexus.aspx Numerary nexus]".
 == Individual pages for IFDOs ==
@@ Line 24: / Line 56: @@
 ** [[AFDO]] – arithmetic frequency division of the octave
 ** [[EDO]] – equal division of the octave
+* [[UD|UD, or utonal division]]
+== External links ==
+* [https://web.archive.org/web/20220630142502/https://sites.google.com/site/240edo/equaldivisionsoflength(edl) 96~EDO | ''Equal divisions of length'']
 == Notes ==