EDT: Difference between revisions

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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".

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 2c sharp octave which is relevant to inharmonicity.

{| class="wikitable"

Also may be found convenient: [http://www.nonoctave.com/tuning/twelfth.html http://www.nonoctave.com/tuning/twelfth.html]

{| class="wikitable"

| | [[8edt|8edt]]

| | 8edt is equivalent to 5edo with ~11 cent octave compression.

Equivalently, 5edo is 8edt with ~18 cent stretched tritaves.

[[Patent_val|Patent vals]] match through the 13 limit.

|-

| | [[11edt|11edt]]

| | 11edt is equivalent to 7edo with ~10 cent stretched octaves.

Patent vals differ in the 7 limit, but neither can really be said

to represent the 7th harmonic with a straight face.

|-

| | [[14edt|14edt]]

| | There is a lot of mismatch between the pure-octave and pure-tritave tunings,

but the patent vals match through the 13 limit. Great for stretched-octave pelog!

|-

| | [[19edt|19edt]]

| | 19edt is 12edo with ~1.2 cent octave stretch. Patent vals match

only through the 7 limit, but neither can be said to include 11 at all.

|-

| | [[24edt|24edt]]

| | This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence

begins to break down. The patent vals match only through the 5 limit.

|-

| | [[25edt|25edt]]

| | Also only a rough correspondence; 25edt corresponds to 16edo

with ~17 cent octave stretch. Patent vals match through the 5 limit.

|-

| | [[27edt|27edt]]

| | 27edt is 17edo with ~2.5 cent compressed octaves. With the exception of

5 (which neither represents well), patent vals match through the 13 limit.

|-

| | [[30edt|30edt]]

| | 30edt is 19edo with ~5 cent stretched octaves.

Patent vals match through the 7 limit.

|-

| | [[35edt|35edt]]

| | 35edt is 22edo with ~4 cent compressed octaves.

Patent vals match through the 11 limit.

|-

| | [[38edt|38edt]]

| | Same ~1.2 cent octave stretch as 12edo~19edt.

Patent vals match through the 19 limit.

|-

| | [[41edt|41edt]]

| | 41edt is 26edo with ~6 cent stretched octaves.

Patent vals match through the 7 limit.

|-

| | [[43edt|43edt]]

| | 43edt is 27edo with ~6 cent compressed octaves.

Patent vals match through the 7 limit.

|-

| | [[46edt|46edt]]

| | 46edt is 29edo with ~0.9 cent compressed octaves.

Patent vals match through the 89 limit. (Really! I checked!)

|}

=Multiples of 13EDT which approximate EDO=

[[Category:3th_harmonic]]

[[Category:edonoi]]