2ed13/10: Difference between revisions

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==13:10==

'''2 equal divisions of 13/10''', when viewed from a regular temperament perspective, is a nonoctave tuning system created by diving the interval of 13/10 into two steps of about 227.1 cents each. It is equivalent to about 5.284edo.

[[13/10|13:10]], as a frequency ratio, measures approximately 454.2 cents. It lies in the extremely xenharmonic and ambiguous territory between the perceptual category of a "third" and that of a "fourth" (see [[Interseptimal|interseptimal]]). It appears in the [[OverToneSeries|overtone series]] between the tenth and thirteen overtones.

==Theory==

~~=sqrt~~ 13:10 as an interval=

[[13/10|13:10]], as a frequency ratio, measures approximately 454.2 cents. It lies in the extremely xenharmonic and ambiguous territory between the perceptual category of a "third" and that of a "fourth" (see [[interseptimal]]). It appears in the [[OverToneSeries|overtone series]] between the tenth and thirteen overtones. The "square root of 13:10", then, means an interval which logarithmically bisects 13:10. It's an irrational number which measures, in cents, about 227.1.

The "square root of 13:10"~~, then, means an~~ interval ~~which logarithmically bisects~~ 13:10. It's an ~~irrational number which measures~~, ~~in cents~~, ~~about~~ 227.1.

"The square root of 13:10" can also refer to the scale that is produced by repeatedly stacking the interval "the square root of 13:10". It is an [[edonoi|EDONOI]], or "equal division of a nonoctave interval," and as such, it does not contain a perfect octave (2:1). It is also [[macrotonal|macrotonal]], since the smallest step, at 227.1 cents, is larger than a semitone.

~~==2ed13/10 as a scale==~~

In terms of [[EDO|equal divisions of the octave]], it fits between [[5edo|5edo]] and [[6edo|6edo]]. Melodically, it can sound somewhat "pentatonic," but harmonically it is very different.

"The square root of 13:10" can also refer to the scale that is produced by repeatedly stacking the interval "the square root of 13:10". It is an [[edonoi|EDONOI]], or "equal division of a nonoctave interval," and as such, it does not contain a perfect octave (2:1). It is also [[macrotonal|macrotonal]], since the smallest step, at 227.1 cents, is larger than a semitone.

===Harmonic content===

~~In terms of [[EDO~~|~~equal divisions of the octave]], it fits between [[5edo~~|~~5edo]] and [[6edo~~|~~6edo]] (about 5.284edo). Melodically, it can sound somewhat "pentatonic," but harmonically it is very different.~~

~~==harmonic content==~~

Sqrt 13:10 (the scale) contains no octaves, and also no close approximation of the third harmonic (the perfect fifth). However, it comes very close to certain just intervals involving the numbers 5, 7, 11 and 13: in particular: 8/7, 13/10, 22/13, 11/5, 5/2, 20/7, 13/4, 26/7, 11/2, and 44/7. These near-just intervals can be combined so as to make available a set of 20 harmonic and subharmonic chords.

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[[File:sqrt13_10_harmonic_contents.jpg|alt=sqrt13_10_harmonic_contents.jpg|sqrt13_10_harmonic_contents.jpg]]

~~=== ===~~

== Music ==

==Music==

~~[http://soundcloud.com/andrew_heathwaite/truffles Truffles?] by Andrew Heathwaite and Michael Gaiuranos~~

~~===Scala file===~~

* [http://soundcloud.com/andrew_heathwaite/truffles Truffles?] by [[Andrew Heathwaite]] and [[Michael Gaiuranos]]

== Scala file ==

<pre>

! sqrt13over10.scl

The square root of 13:10 nonoctave temperament.

2

!

227.106973952238

13/10

</pre>

@@ Line 1: / Line 1: @@
-==13:10==
+{{Infobox ET}}
+'''2 equal divisions of 13/10''', when viewed from a regular temperament perspective, is a nonoctave tuning system created by diving the interval of 13/10 into two steps of about 227.1 cents each. It is equivalent to about 5.284edo.
-[[13/10|13:10]], as a frequency ratio, measures approximately 454.2 cents. It lies in the extremely xenharmonic and ambiguous territory between the perceptual category of a "third" and that of a "fourth" (see [[Interseptimal|interseptimal]]). It appears in the [[OverToneSeries|overtone series]] between the tenth and thirteen overtones.
+==Theory==
-=sqrt 13:10 as an interval=
+[[13/10|13:10]], as a frequency ratio, measures approximately 454.2 cents. It lies in the extremely xenharmonic and ambiguous territory between the perceptual category of a "third" and that of a "fourth" (see [[interseptimal]]). It appears in the [[OverToneSeries|overtone series]] between the tenth and thirteen overtones. The "square root of 13:10", then, means an interval which logarithmically bisects 13:10. It's an irrational number which measures, in cents, about 227.1.
-The "square root of 13:10", then, means an interval which logarithmically bisects 13:10. It's an irrational number which measures, in cents, about 227.1.
+"The square root of 13:10" can also refer to the scale that is produced by repeatedly stacking the interval "the square root of 13:10". It is an [[edonoi|EDONOI]], or "equal division of a nonoctave interval," and as such, it does not contain a perfect octave (2:1). It is also [[macrotonal|macrotonal]], since the smallest step, at 227.1 cents, is larger than a semitone.
-==2ed13/10 as a scale==
+In terms of [[EDO|equal divisions of the octave]], it fits between [[5edo|5edo]] and [[6edo|6edo]]. Melodically, it can sound somewhat "pentatonic," but harmonically it is very different.
-"The square root of 13:10" can also refer to the scale that is produced by repeatedly stacking the interval "the square root of 13:10". It is an [[edonoi|EDONOI]], or "equal division of a nonoctave interval," and as such, it does not contain a perfect octave (2:1). It is also [[macrotonal|macrotonal]], since the smallest step, at 227.1 cents, is larger than a semitone.
+===Harmonic content===
-In terms of [[EDO|equal divisions of the octave]], it fits between [[5edo|5edo]] and [[6edo|6edo]] (about 5.284edo). Melodically, it can sound somewhat "pentatonic," but harmonically it is very different.
+{{Harmonics in equal|2|13|10}}
-==harmonic content==
 Sqrt 13:10 (the scale) contains no octaves, and also no close approximation of the third harmonic (the perfect fifth). However, it comes very close to certain just intervals involving the numbers 5, 7, 11 and 13: in particular: 8/7, 13/10, 22/13, 11/5, 5/2, 20/7, 13/4, 26/7, 11/2, and 44/7. These near-just intervals can be combined so as to make available a set of 20 harmonic and subharmonic chords.
@@ Line 21: / Line 20: @@
 [[File:sqrt13_10_harmonic_contents.jpg|alt=sqrt13_10_harmonic_contents.jpg|sqrt13_10_harmonic_contents.jpg]]
-=== ===
+== Music ==
-==Music==
-[http://soundcloud.com/andrew_heathwaite/truffles Truffles?] by Andrew Heathwaite and Michael Gaiuranos
-===Scala file===
+* [http://soundcloud.com/andrew_heathwaite/truffles Truffles?] by [[Andrew Heathwaite]] and [[Michael Gaiuranos]]
+== Scala file ==
+<pre>
 ! sqrt13over10.scl
 The square root of 13:10 nonoctave temperament.
 !
 .106973952238
 /10
+</pre>