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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.  
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.  


== Introduction to tritave equivalence ==
EDTs are often associated with tritave equivalence, an approach where 3/1 is considered the [[interval of equivalence]] rather than 2/1. This is very [[xenharmonic]] since [[1/1]], 3/1, [[9/1]], … are considered the same pitch class. Typically tritave-equivalent systems base harmony off of only [[odd harmonic]]s, for example with the [[3:5:7]] triad as analogous to [[4:5:6]].  
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.
Another common use or conceptualization of EDTs is as [[stretched and compressed tuning|stretched or squashed]] EDOs. For instance, [[19edt]] (a.k.a. the Bernhard–Stopper tuning), might be useful in tuning the [[piano]], being equivalent to 12edo except for a 1.2-cent-sharp octave which is relevant to compensating for the inharmonicity.


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.
For an introduction to tritave equivalence, see [[3/1 #As an interval of equivalence]].  


== As generator chains for temperaments ==
== 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.
Taking the ''n''-th root of 3 is itself an approach to finding or defining temperaments like [[magic]], [[hanson]], etc. Below is a list of such EDTs:
 
* [[3edt]] (liese generator)
Below is a large list of EDTs; additionally, some equal divisions of the tritave are known by alternate names or have special interest:
* [[4edt]] (vulture generator)
 
* [[5edt]] (magic generator)
* [[3edt]] (Liese generator)
* [[6edt]] (hanson generator)
* [[4edt]] (Vulture generator)
* [[7edt]] (orwell generator)
* [[5edt]] (Tritave counterpart of Magic)
* [[11edt]] ("Euler Temperament"{{cn}})
* [[6edt]] (Tritave counterpart of Hanson)
* [[13edt]] ([[Bohlen–Pierce]])
* [[7edt]] (Tritave counterpart of Orwell)
* [[15edt]] (mowgli generator)
* [[8edt]] (Tritave counterpart of Vulture)
* [[19edt]] (Bernhard–Stopper)
* [[11edt]] "Euler Temperament"
* [[39edt]] (Triple Bohlen–Pierce)
* [[BP|"Bohlen–Pierce" or "BP"]]
* [[15edt]] (Mowgli generator)
* [[19edt|"Bernhard Stopper"]]
* [[39edt]] Triple Bohlen–Pierce (Erlich)


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


== EDT-EDO correspondences ==
== 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).  
It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. As discussed above, the former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-2's 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.  
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.  
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