EDT: Difference between revisions

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 using timbres restricted to odd harmonics such as clarinets. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain 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 second such arrangement to be seriously studied and made into music, the first being the [[Obikhod]] pitch set of the Russian Orthodox Church which seems to have been by extension of the diatonic scale. 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.

{{Todo|cleanup|inline=1|text=Rewrite for clarity}}

The standard [[Bohlen-Pierce]] theory takes 3:5:7 to be the fundamental triad, and therefore naturally goes together with scales generated by 7/3, or equivalently 9/7 (the latter being convention), against the tritave. 7/3 generates pentatonic ({{sl|4L 1s}}) and enneatonic ({{sl|4L 5s}}) [[MOS]] scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from [[9edt|7\9]] to [[4edt|3\4]], [[13edt]] is the smallest equal temperament supporting it, and can be seen as an equivalent of [[12edo]]. However, 13edt's accuracy in the [[3.5.7 subgroup]] is much better than 12edo's in the 5-limit, more comparable to that of [[31edo]]. Therefore, higher multiples of 13edt remain excellent 3.5.7 subgroup tunings as well, and can be used to introduce higher harmonics ([[39edt]] is especially notable in this regard, with a good representation of both the [[11/9|11th]] and [[13/9|13th]] harmonics).

The linear temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is [[Bohlen-Pierce-Stearns]], which tempers out the comma [[245/243]] and thereby equates the interval [[5/3]] to two generators down ([[81/49]] considering tritave-reduction)—therefore flattening 7/3 by a fraction of this comma. It is also the {{nowrap|4 & 9}} temperament in the 3.5.7 subgroup, and for these reasons serves a function very analogous to that of [[meantone]] in the 5-limit.

If we instead take [[9/5]], or more simply [[5/3]], as a generator, the temperament supported by 13edt is [[Arcturus]], which equates 7/3, two tritaves up (i.e. [[21/1]]) to six steps of 5/3. Naively, 5/3 as generator would be the most natural application of the [[Pythagorean tuning|Pythagorean]] principle of using the next higher prime harmonic (5) as a generator against the tritave. However, a larger MOS scale is needed to get full use out of the 7th harmonic, and due to the proximity of 5/3 to half the tritave, most simple MOS scales of Arcturus are quite hard. It is advisable to use ({{sl|2L 9s}}) or ({{sl|2L 11s}}) scales—and therefore, higher EDTs such as [[28edt]] or [[41edt]].

The named but not necessarily no twos rank two temperament which 13EDT "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.

At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.

The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s (tritave-equivalent)|5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen–Pierce temperament is its index-2 subtemperament.

Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale.

Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDT supporting the BP nonatonic scale—13EDT, the traditional tempered BP scale—is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.

For example, [[User:CompactStar|CompactStar]] suggested the alternative fundamental chord 11:13:15 to avoid the highly-dissonant [[7/5]] tritone present in the simpler 3:5:7 chord, with the best temperament for this being [[Electra]] temperament. 15EDT very well approximates the 5th and 13th harmonics, and 12EDT, the 13th and 17th. 39EDT makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the [{{val|13 19 23 0 2}}, {{val|0 0 0 1 1}}] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc.  

'''One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional pitch class terminology.

* [[BP|"Bohlen-Pierce" or "BP"]]

* [[19ED3|"Bernhard Stopper"]]

* [[39edt]] Triple Bohlen-Pierce (Erlich)

 

{| class="wikitable center-all"

| [[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]]

| [[40edt|40]]

| [[41edt|41]]

| [[42edt|42]]

| [[43edt|43]]

| [[44edt|44]]

| [[45edt|45]]

| [[46edt|46]]

| [[47edt|47]]

| [[48edt|48]]

| [[49edt|49]]

| [[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]]

| [[80edt|80]]

| [[81edt|81]]

| [[82edt|82]]

| [[83edt|83]]

| [[84edt|84]]

| [[85edt|85]]

| [[86edt|86]]

| [[87edt|87]]

| [[88edt|88]]

| [[89edt|89]]

| [[90edt|90]]

| [[91edt|91]]

| [[92edt|92]]

| [[93edt|93]]

| [[94edt|94]]

| [[95edt|95]]

| [[96edt|96]]

| [[97edt|97]]

| [[98edt|98]]

| [[99edt|99]]

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

| [[100edt|100]]

| [[101edt|101]]

| [[102edt|102]]

| [[103edt|103]]

| [[104edt|104]]

| [[105edt|105]]

| [[106edt|106]]

| [[107edt|107]]

| [[108edt|108]]

| [[109edt|109]]

| [[110edt|110]]

| [[111edt|111]]

| [[112edt|112]]

| [[113edt|113]]

| [[114edt|114]]

| [[115edt|115]]

| [[116edt|116]]

| [[117edt|117]]

| [[118edt|118]]

| [[119edt|119]]

| [[120edt|120]]

| [[121edt|121]]

| [[122edt|122]]

| [[123edt|123]]

| [[124edt|124]]

| [[125edt|125]]

| [[126edt|126]]

| [[127edt|127]]

| [[128edt|128]]

| [[129edt|129]]

| [[130edt|130]]

| [[131edt|131]]

| [[132edt|132]]

| [[133edt|133]]

| [[134edt|134]]

| [[135edt|135]]

| [[136edt|136]]

| [[137edt|137]]

| [[138edt|138]]

| [[139edt|139]]

| [[140edt|140]]

| [[141edt|141]]

| [[142edt|142]]

| [[143edt|143]]

| [[144edt|144]]

| [[145edt|145]]

| [[146edt|146]]

| [[147edt|147]]

| [[148edt|148]]

| [[149edt|149]]

| [[150edt|150]]

| [[151edt|151]]

| [[152edt|152]]

| [[153edt|153]]

| [[154edt|154]]

| [[155edt|155]]

| [[156edt|156]]

| [[157edt|157]]

| [[158edt|158]]

| [[159edt|159]]

| [[160edt|160]]

| [[161edt|161]]

| [[162edt|162]]

| [[163edt|163]]

| [[164edt|164]]

| [[165edt|165]]

| [[166edt|166]]

| [[167edt|167]]

| [[168edt|168]]

| [[169edt|169]]

| [[170edt|170]]

| [[171edt|171]]

| [[172edt|172]]

| [[173edt|173]]

| [[174edt|174]]

| [[175edt|175]]

| [[176edt|176]]

| [[177edt|177]]

| [[178edt|178]]

| [[179edt|179]]

| [[180edt|180]]

| [[181edt|181]]

| [[182edt|182]]

| [[183edt|183]]

| [[184edt|184]]

| [[185edt|185]]

| [[186edt|186]]

| [[187edt|187]]

| [[188edt|188]]

| [[189edt|189]]

| [[190edt|190]]

| [[191edt|191]]

| [[192edt|192]]

| [[193edt|193]]

| [[194edt|194]]

| [[195edt|195]]

| [[196edt|196]]

| [[197edt|197]]

| [[198edt|198]]

| [[199edt|199]]

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

| [[200edt|200]]

| [[201edt|201]]

| [[202edt|202]]

| [[203edt|203]]

| [[204edt|204]]

| [[205edt|205]]

| [[206edt|206]]

| [[207edt|207]]

| [[208edt|208]]

| [[209edt|209]]

| [[210edt|210]]

| [[211edt|211]]

| [[212edt|212]]

| [[213edt|213]]

| [[214edt|214]]

| [[215edt|215]]

| [[216edt|216]]

| [[217edt|217]]

| [[218edt|218]]

| [[219edt|219]]

| [[220edt|220]]

| [[221edt|221]]

| [[222edt|222]]

| [[223edt|223]]

| [[224edt|224]]

| [[225edt|225]]

| [[226edt|226]]

| [[227edt|227]]

| [[228edt|228]]

| [[229edt|229]]

| [[230edt|230]]

| [[231edt|231]]

| [[232edt|232]]

| [[233edt|233]]

| [[234edt|234]]

| [[235edt|235]]

| [[236edt|236]]

| [[237edt|237]]

| [[238edt|238]]

| [[239edt|239]]

| [[240edt|240]]

| [[241edt|241]]

| [[242edt|242]]

| [[243edt|243]]

| [[244edt|244]]

| [[245edt|245]]

| [[246edt|246]]

| [[247edt|247]]

| [[248edt|248]]

| [[249edt|249]]

| [[250edt|250]]

| [[251edt|251]]

| [[252edt|252]]

| [[253edt|253]]

| [[254edt|254]]

| [[255edt|255]]

| [[256edt|256]]

| [[257edt|257]]

| [[258edt|258]]

| [[259edt|259]]

| [[260edt|260]]

| [[261edt|261]]

| [[262edt|262]]

| [[263edt|263]]

| [[264edt|264]]

| [[265edt|265]]

| [[266edt|266]]

| [[267edt|267]]

| [[268edt|268]]

| [[269edt|269]]

| [[270edt|270]]

| [[271edt|271]]

| [[272edt|272]]

| [[273edt|273]]

| [[274edt|274]]

| [[275edt|275]]

| [[276edt|276]]

| [[277edt|277]]

| [[278edt|278]]

| [[279edt|279]]

| [[280edt|280]]

| [[281edt|281]]

| [[282edt|282]]

| [[283edt|283]]

| [[284edt|284]]

| [[285edt|285]]

| [[286edt|286]]

| [[287edt|287]]

| [[288edt|288]]

| [[289edt|289]]

| [[290edt|290]]

| [[291edt|291]]

| [[292edt|292]]

| [[293edt|293]]

| [[294edt|294]]

| [[295edt|295]]

| [[296edt|296]]

| [[297edt|297]]

| [[298edt|298]]

| [[299edt|299]]

* [[316edt]], [[372edt]], [[415edt]], [[527edt]], [[729edt]], [[1342edt]], [[3401edt]], [[27208edt]]

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 {{sl|8L 3s}} scale against the tritave, therefore being between 5\8edt and 7\11edt.  

|[[164edt]]

| -

| -

|239edt is 151edo with ~1.65 cent stretched octaves.

| -

* [[Tritave Reduced Harmonics]]

* [[No-twos 31-limit]]