34edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|34}} | {{EDO intro|34}} | ||
{{Wikipedia|34 equal temperament}} | {{Wikipedia| 34 equal temperament }} | ||
== Theory == | == Theory == | ||
34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. It excels in | 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. It excels in approximating harmonics 3, 5, 13, 17, and 23 (2.3.5.13.17.23 [[subgroup]] a.k.a. the no-7's no-11's no-19's 23-limit), with tuning even more accurate than [[31edo]] in the 5-limit, but with a sharp tendency and fifth rather than a flat one, and ''not'' tempering out [[81/80]] unlike 31edo. | ||
=== Odd harmonics === | === Odd harmonics === | ||
{{ | {{Harmonics in equal|34|intervals=odd}} | ||
== Intervals == | == Intervals == | ||
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In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%. | In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%. | ||
== | == Approximation to logarithmic phi == | ||
As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[ | As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[Moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].) | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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* [[List of edo-distinct 34d rank two temperaments]] | * [[List of edo-distinct 34d rank two temperaments]] | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ Rank-2 temperaments by period and generator | |+ Rank-2 temperaments by period and generator | ||
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! Generator | ! Generator | ||
! Cents | ! Cents | ||
! | ! Mosses | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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=== Commas === | === Commas === | ||
34edo [[tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 34 54 79 95 118 126 }}. | |||
{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
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<references/> | <references/> | ||
== | == Notation == | ||
=== Kosmorsky's thoughts === | === Kosmorsky's thoughts === | ||
The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away – the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which ''may not be most efficient expressions of some musical purposes.'' Users can construct their own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C … F, instead of seven, might be waste – of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended. | The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away – the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which ''may not be most efficient expressions of some musical purposes.'' Users can construct their own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C … F, instead of seven, might be waste – of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended. | ||
== Counterpoint == | == Counterpoint == | ||
34edo has such an excellent [[sqrt(25/24)]] that the next | 34edo has such an excellent [[sqrt(25/24)]] that the next edo to have a better one is [[441edo|441]]. | ||
Every sequence of intervals available in [[17edo]] are reachable by [[strict contrary motion]] in 34edo. | Every sequence of intervals available in [[17edo]] are reachable by [[strict contrary motion]] in 34edo. | ||
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== External links == | == External links == | ||
* [http://microstick.net/products/34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry Hanson]] {{dead link}} | * [http://microstick.net/products/34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry Hanson]] {{dead link}} | ||
* [https://microstick.net | * [https://microstick.net Websites of Neil Haverstick] | ||
* https://myspace.com/microstick | * [https://myspace.com/microstick] – somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything) | ||
[[Category:Diaschismic]] | [[Category:Diaschismic]] | ||
[[Category:Keemun]] | [[Category:Keemun]] | ||