EDT: Difference between revisions

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The '''equal division of the tritave''' or '''twelfth''' ('''EDT''') or '''3rd harmonic''' ('''ED3''') is a [[tuning]] obtained by dividing the [[3/1|3rd harmonic]] in a certain number of [[equal]] steps.

Western music generally revolves around the principle of [[octave equivalence]]: notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]".

It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system]{{dead link}} is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence, especially when using timbres whose overtones consist of primarily or only odd harmonics such as clarinets, square waves, or triangle waves. While is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence, it is known that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.

The [[Bohlen–Pierce scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.

== As generator chains for temperaments ==

There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.

Below is a large list of EDTs; additionally, some equal divisions of the tritave are known by alternate names or have special interest:

* [[5edt]] (Tritave counterpart of Magic)

* [[6edt]] (Tritave counterpart of Hanson)

* [[7edt]] (Tritave counterpart of Orwell)

* [[8edt]] (Tritave counterpart of Vulture)

== Individual pages for EDTs ==

| [[0edt|0]] || [[1edt|1]] || [[2edt|2]] || [[3edt|3]] || [[4edt|4]] || [[5edt|5]] || [[6edt|6]] || [[7edt|7]] || [[8edt|8]] || [[9edt|9]]

| [[10edt|10]] || [[11edt|11]] || [[12edt|12]] || [[13edt|13]] || [[14edt|14]] || [[15edt|15]] || [[16edt|16]] || [[17edt|17]] || [[18edt|18]] || [[19edt|19]]

| [[20edt|20]] || [[21edt|21]] || [[22edt|22]] || [[23edt|23]] || [[24edt|24]] || [[25edt|25]] || [[26edt|26]] || [[27edt|27]] || [[28edt|28]] || [[29edt|29]]

| [[30edt|30]] || [[31edt|31]] || [[32edt|32]] || [[33edt|33]] || [[34edt|34]] || [[35edt|35]] || [[36edt|36]] || [[37edt|37]] || [[38edt|38]] || [[39edt|39]]

| [[50edt|50]] || [[51edt|51]] || [[52edt|52]] || [[53edt|53]] || [[54edt|54]] || [[55edt|55]] || [[56edt|56]] || [[57edt|57]] || [[58edt|58]] || [[59edt|59]]

| [[60edt|60]] || [[61edt|61]] || [[62edt|62]] || [[63edt|63]] || [[64edt|64]] || [[65edt|65]] || [[66edt|66]] || [[67edt|67]] || [[68edt|68]] || [[69edt|69]]

| [[70edt|70]] || [[71edt|71]] || [[72edt|72]] || [[73edt|73]] || [[74edt|74]] || [[75edt|75]] || [[76edt|76]] || [[77edt|77]] || [[78edt|78]] || [[79edt|79]]

|+ style="font-size: 105%; white-space: nowrap;" | [[100edt|100]]…199

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | [[200edt|200]]…299

|-

|-

|-

; 300 and beyond

* [[314edt|314]], [[316edt|316]], [[336edt|336]], [[372edt|372]], [[415edt|415]], [[428edt|428]], [[499edt|499]], [[527edt|527]], [[613edt|613]], [[729edt|729]], [[800edt|800]], [[953edt|953]], [[1213edt|1213]], [[1342edt|1342]], [[3401edt|3401]], [[6181edt|6181]], [[27208edt|27208]]

* A [[list of tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems.

* Also may be found convenient: [http://www.nonoctave.com/tuning/twelfth.html Nonoctave.com | Tuning | Equal Divisions of the Twelfth]

== EDT-EDO correspondences ==

It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).

Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the {{mos scalesig|8L 3s<3/1>|link=1}} scale against the tritave, therefore being between 5\8edt and 7\11edt.  

EDTs with this property include {{EDTs| 19, 27, 30, 35, 38, 41, 43, 46, 49, 51, 52, 54, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73 to 76, 78, 79, 81 to 87, and all greater than 88.}}

EDTs ''without'' a diatonic val are 1 to {{EDTs| 7, 9, 10, 12 to 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 34, 36, 37, 39, 42, 45, 47, 50, 53, 58, 61, and 69.}}

Borderline cases (i.e. EDTs corresponding to a heptatonic or pentatonic fifth) are {{EDTs| 8, 11, 16, 22, 24, 32, 33, 40, 44, 48, 55, 56, 64, 66, 72, 77, 80, and 88.}}

Correspondences are explained in more detail in the table below.

==== Multiples of 13EDT which approximate EDO ====

On the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo]] and 78EDT is very nearly [[49edo]], while 65EDT is practically identical to [[41edo]].

== See also ==

* [[Consistency levels of small EDTs]]

* [[Relative errors of small EDTs]]

* [[List of tritave reduced harmonics]]

* [[List of no-twos chords in JI]]

* Heinz Bohlen's work: [http://www.huygens-fokker.org/bpsite/otherscales.html ''The Bohlen-Pierce Site: Other Unusual Scales'']

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[[Category:Tritave]]

[[Category:Acronyms]]