View source for IFDO

An '''IFDO''' ('''inverse-arithmetic frequency division of the octave'''), or '''UDO''' ('''utonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to the inverse-arithmetic progression of frequency.

The inverse-arithmetic progression is known in general mathematics as the {{W|Harmonic progression (mathematics)|harmonic progression}}, but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to ''subharmonic'' sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the {{W|arithmetic progression}}. 

For example, in [[12ifdo]] the first degree is [[24/23]], the second is 24/22 ([[12/11]]), and so on. For an IFDO system, the difference between ''inverse'' interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in [[AFDO]] systems (an arithmetic progression). All IFDOs are subsets of [[just intonation]], and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger IFDO (i.e. ''n''-ifdo is a superset of (''n'' - 1)-ifdo and a subset of (''n'' + 1)-ifdo for any integer ''n'' > 1). 

When treated as a [[scale]], the IFDO is equivalent to the undertone scale, also known as an aliquot scale<ref>''1/1, The Journal of the Just Intonation Network'', Volume 4, Number 1, Winter 1988, p.6, Michael Sloper. </ref>. However, an undertone scale often has an assumption of a tonic whereas an IFDO simply describes all the theoretically available pitch relations. Therefore, a passage built on 1/(12::22) could be said to be in Mode 12, but is technically covered by 11ifdo. 

An IFDO is equivalent to a UDO ([[utonal division]] of the octave). It may also be called an ''n''-ELDO ([[equal length division]] of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into ''n'' equal parts; however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation. 

== Formula ==
Within each period of ''n''-ifdo, the [[frequency ratio]] ''c'' of the ''k''-th step is

<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]]

12:12 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 "[[numerary nexus]]".

== Properties ==
* ''n''-ifdo has [[maximum variety]] ''n''. 
* Except for 1ifdo and 2ifdo, IFDOs are [[chiral]]. The inverse of ''n''-ifdo is ''n''-afdo. 
** 1ifdo is equivalent to 1afdo and 1edo; 
** 2ifdo is equivalent to 2afdo.

== Individual pages for IFDOs == 
=== By size ===
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…99
| [[0ifdo|0]]
| [[1edo|1]]
| [[2ifdo|2]]
| [[3ifdo|3]]
| [[4ifdo|4]]
| [[5ifdo|5]]
| [[6ifdo|6]]
| [[7ifdo|7]]
| [[8ifdo|8]]
| [[9ifdo|9]]
|-
| [[10ifdo|10]]
| [[11ifdo|11]]
| [[12ifdo|12]]
| [[13ifdo|13]]
| [[14ifdo|14]]
| [[15ifdo|15]]
| [[16ifdo|16]]
| [[17ifdo|17]]
| [[18ifdo|18]]
| [[19ifdo|19]]
|-
| [[20ifdo|20]]
| [[21ifdo|21]]
| [[22ifdo|22]]
| [[23ifdo|23]]
| [[24ifdo|24]]
| [[25ifdo|25]]
| [[26ifdo|26]]
| [[27ifdo|27]]
| [[28ifdo|28]]
| [[29ifdo|29]]
|-
| [[30ifdo|30]]
| [[31ifdo|31]]
| [[32ifdo|32]]
| [[33ifdo|33]]
| [[34ifdo|34]]
| [[35ifdo|35]]
| [[36ifdo|36]]
| [[37ifdo|37]]
| [[38ifdo|38]]
| [[39ifdo|39]]
|-
| [[40ifdo|40]]
| [[41ifdo|41]]
| [[42ifdo|42]]
| [[43ifdo|43]]
| [[44ifdo|44]]
| [[45ifdo|45]]
| [[46ifdo|46]]
| [[47ifdo|47]]
| [[48ifdo|48]]
| [[49ifdo|49]]
|-
| [[50ifdo|50]]
| [[51ifdo|51]]
| [[52ifdo|52]]
| [[53ifdo|53]]
| [[54ifdo|54]]
| [[55ifdo|55]]
| [[56ifdo|56]]
| [[57ifdo|57]]
| [[58ifdo|58]]
| [[59ifdo|59]]
|-
| [[60ifdo|60]]
| [[61ifdo|61]]
| [[62ifdo|62]]
| [[63ifdo|63]]
| [[64ifdo|64]]
| [[65ifdo|65]]
| [[66ifdo|66]]
| [[67ifdo|67]]
| [[68ifdo|68]]
| [[69ifdo|69]]
|-
| [[70ifdo|70]]
| [[71ifdo|71]]
| [[72ifdo|72]]
| [[73ifdo|73]]
| [[74ifdo|74]]
| [[75ifdo|75]]
| [[76ifdo|76]]
| [[77ifdo|77]]
| [[78ifdo|78]]
| [[79ifdo|79]]
|-
| [[80ifdo|80]]
| [[81ifdo|81]]
| [[82ifdo|82]]
| [[83ifdo|83]]
| [[84ifdo|84]]
| [[85ifdo|85]]
| [[86ifdo|86]]
| [[87ifdo|87]]
| [[88ifdo|88]]
| [[89ifdo|89]]
|-
| [[90ifdo|90]]
| [[91ifdo|91]]
| [[92ifdo|92]]
| [[93ifdo|93]]
| [[94ifdo|94]]
| [[95ifdo|95]]
| [[96ifdo|96]]
| [[97ifdo|97]]
| [[98ifdo|98]]
| [[99ifdo|99]]
|}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | 100…199
| [[100ifdo|100]]
| [[101ifdo|101]]
| [[102ifdo|102]]
| [[103ifdo|103]]
| [[104ifdo|104]]
| [[105ifdo|105]]
| [[106ifdo|106]]
| [[107ifdo|107]]
| [[108ifdo|108]]
| [[109ifdo|109]]
|-
| [[110ifdo|110]]
| [[111ifdo|111]]
| [[112ifdo|112]]
| [[113ifdo|113]]
| [[114ifdo|114]]
| [[115ifdo|115]]
| [[116ifdo|116]]
| [[117ifdo|117]]
| [[118ifdo|118]]
| [[119ifdo|119]]
|-
| [[120ifdo|120]]
| [[121ifdo|121]]
| [[122ifdo|122]]
| [[123ifdo|123]]
| [[124ifdo|124]]
| [[125ifdo|125]]
| [[126ifdo|126]]
| [[127ifdo|127]]
| [[128ifdo|128]]
| [[129ifdo|129]]
|-
| [[130ifdo|130]]
| [[131ifdo|131]]
| [[132ifdo|132]]
| [[133ifdo|133]]
| [[134ifdo|134]]
| [[135ifdo|135]]
| [[136ifdo|136]]
| [[137ifdo|137]]
| [[138ifdo|138]]
| [[139ifdo|139]]
|-
| [[140ifdo|140]]
| [[141ifdo|141]]
| [[142ifdo|142]]
| [[143ifdo|143]]
| [[144ifdo|144]]
| [[145ifdo|145]]
| [[146ifdo|146]]
| [[147ifdo|147]]
| [[148ifdo|148]]
| [[149ifdo|149]]
|-
| [[150ifdo|150]]
| [[151ifdo|151]]
| [[152ifdo|152]]
| [[153ifdo|153]]
| [[154ifdo|154]]
| [[155ifdo|155]]
| [[156ifdo|156]]
| [[157ifdo|157]]
| [[158ifdo|158]]
| [[159ifdo|159]]
|-
| [[160ifdo|160]]
| [[161ifdo|161]]
| [[162ifdo|162]]
| [[163ifdo|163]]
| [[164ifdo|164]]
| [[165ifdo|165]]
| [[166ifdo|166]]
| [[167ifdo|167]]
| [[168ifdo|168]]
| [[169ifdo|169]]
|-
| [[170ifdo|170]]
| [[171ifdo|171]]
| [[172ifdo|172]]
| [[173ifdo|173]]
| [[174ifdo|174]]
| [[175ifdo|175]]
| [[176ifdo|176]]
| [[177ifdo|177]]
| [[178ifdo|178]]
| [[179ifdo|179]]
|-
| [[180ifdo|180]]
| [[181ifdo|181]]
| [[182ifdo|182]]
| [[183ifdo|183]]
| [[184ifdo|184]]
| [[185ifdo|185]]
| [[186ifdo|186]]
| [[187ifdo|187]]
| [[188ifdo|188]]
| [[189ifdo|189]]
|-
| [[190ifdo|190]]
| [[191ifdo|191]]
| [[192ifdo|192]]
| [[193ifdo|193]]
| [[194ifdo|194]]
| [[195ifdo|195]]
| [[196ifdo|196]]
| [[197ifdo|197]]
| [[198ifdo|198]]
| [[199ifdo|199]]
|}

=== By prime family ===


'''Under-2''': {{IFDOs|2, 4, 8, 16, 32, 64, 128, 256, 512}}


'''Under-3''': {{IFDOs|3, 6, 12, 24, 48, 96, 192, 384}}


'''Under-5''': {{IFDOs|5, 10, 20, 40, 80, 160, 320}}


'''Under-7''': {{IFDOs|7, 14, 28, 56, 112, 224, 448}}


'''Under-11''': {{IFDOs|11, 22, 44, 88, 176, 352}}


'''Under-13''': {{IFDOs|13, 26, 52, 104, 208, 416}}


'''Under-17''': {{IFDOs|17, 34, 68, 136, 272, 544}}


'''Under-19''': {{IFDOs|19, 38, 76, 152, 304}}


'''Under-23''': {{IFDOs|23, 46, 92, 184, 368}}


'''Under-29''': {{IFDOs|29, 58, 116, 232, 464}}


'''Under-31''': {{IFDOs|31, 62, 124, 248, 496}}

=== By other properties ===


'''Prime''': {{IFDOs|2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271}}


'''Semiprime''': {{IFDOs|4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187}}


'''Odd squarefree semiprime''': {{IFDOs|15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303}}


'''Prime powers (without primes)''': {{IFDOs|4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048}}

== See also ==
* Through other [[Pythagorean means]]: 
** [[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 ==

[[Category:IFDO| ]] <!-- main article -->
[[Category:Acronyms]]
[[Category:Lists of scales]]
[[Category:Just intonation]]
[[Category:Utonality]]
[[Category:Subharmonic series]]