13/10: Difference between revisions

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In [[13-limit]] [[just intonation]], '''13/10''', the '''tridecimal semisixth''' is an [[interseptimal]] interval measuring about 454.2¢. It falls in an ambiguous zone between a wide major third such as [[9/7]] and a flat perfect fourth such as [[21/16]]. The descriptor "interseptimal" comes from [[Margo Schulter]], and indicates its position between those two septimal (7-based) extremes. 13/10 ~~appears between the 10th~~ and ~~13th overtones~~ of the [[~~harmonic series~~]] ~~and appears~~ in ~~such chords as 8~~:10:13, ~~a quasi-augmented triad.~~ 13/10 also appears in the relatively-simple 10:13:15 triad, which consists of ~~an interseptimal ultramajor third (~~13/10) and ~~an interseptimal inframinor third (~~[[15/13]]~~) which~~ stack to make a [[3/2]] perfect fifth. It is well-approximated in [[16edo]], [[21edo]], [[24edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[~~EDO~~]] systems.

In [[13-limit]] [[just intonation]], '''13/10''', the '''tridecimal semisixth''' is an [[interseptimal]] interval measuring about 454.2 [[cent]]s. It falls in an ambiguous zone between a wide major third such as [[9/7]] and a flat perfect fourth such as [[21/16]]. The descriptor "interseptimal" comes from [[Margo Schulter]], and indicates its position between those two septimal (7-based) extremes.

In many notation systems based on the [[5L 2s|diatonic]] [[chain-of-fifths notation]] with commatic alterations (e.g. [[FJS]], [[HEJI]]), 13/10 is a fourth, as it is a [[4/3|perfect fourth (4/3)]] minus an instance of [[40/39]], which is a [[2187/2048|Pythagorean apotome]] minus a stack consisting of an [[81/80|syntonic comma (81/80)]] and a [[1053/1024|tridecimal quartertone (1053/1024)]], none of which changes the [[scale|scale degree]]. It functions as such in the harmonic thirteenth chord, [[4:5:6:7:9:11:13]].

However, 13/10 also appears in the relatively-simple [[10:13:15]] triad, which consists of 13/10 and [[15/13]] that stack to make a [[3/2]] perfect fifth. This makes 13/10 function as an ultramajor third (if the chord is not taken as a suspension). It is well-approximated in [[16edo]], [[21edo]], [[24edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[edo]] systems.

== Interval chain ==

Because 13/10 is an interseptimal interval, stacking it four times will result in a good approximation of a septimal interval. In this case, (13/10)4 approximates 20/7 (compound [[10/7]]) remarkably well, with less than 1{{cent}} error.

Additionally, while it may seem as though (13/10)2 ~~doesn't~~ approximate 17/10 very well at first glance, it allows for an elegant interpretation of the tetrad formed by stacking 13/10 three times on top of itself: [[~]]10:13:17:22.

Additionally, while it may seem as though (13/10)2 does not approximate 17/10 very well at first glance, it allows for an elegant interpretation of the tetrad formed by stacking 13/10 three times on top of itself: [[~]]10:13:17:22.

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== Approximation ==

== See also ==

* [[20/13]] – its [[octave complement]]

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-In [[13-limit]] [[just intonation]], '''13/10''', the '''tridecimal semisixth''' is an [[interseptimal]] interval measuring about 454.2¢. It falls in an ambiguous zone between a wide major third such as [[9/7]] and a flat perfect fourth such as [[21/16]]. The descriptor "interseptimal" comes from [[Margo Schulter]], and indicates its position between those two septimal (7-based) extremes. 13/10 appears between the 10th and 13th overtones of the [[harmonic series]] and appears in such chords as 8:10:13, a quasi-augmented triad. 13/10 also appears in the relatively-simple 10:13:15 triad, which consists of an interseptimal ultramajor third (13/10) and an interseptimal inframinor third ([[15/13]]) which stack to make a [[3/2]] perfect fifth. It is well-approximated in [[16edo]], [[21edo]], [[24edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[EDO]] systems.
+In [[13-limit]] [[just intonation]], '''13/10''', the '''tridecimal semisixth''' is an [[interseptimal]] interval measuring about 454.2 [[cent]]s. It falls in an ambiguous zone between a wide major third such as [[9/7]] and a flat perfect fourth such as [[21/16]]. The descriptor "interseptimal" comes from [[Margo Schulter]], and indicates its position between those two septimal (7-based) extremes.
+In many notation systems based on the [[5L 2s|diatonic]] [[chain-of-fifths notation]] with commatic alterations (e.g. [[FJS]], [[HEJI]]), 13/10 is a fourth, as it is a [[4/3|perfect fourth (4/3)]] minus an instance of [[40/39]], which is a [[2187/2048|Pythagorean apotome]] minus a stack consisting of an [[81/80|syntonic comma (81/80)]] and a [[1053/1024|tridecimal quartertone (1053/1024)]], none of which changes the [[scale|scale degree]]. It functions as such in the harmonic thirteenth chord, [[4:5:6:7:9:11:13]].
+However, 13/10 also appears in the relatively-simple [[10:13:15]] triad, which consists of 13/10 and [[15/13]] that stack to make a [[3/2]] perfect fifth. This makes 13/10 function as an ultramajor third (if the chord is not taken as a suspension). It is well-approximated in [[16edo]], [[21edo]], [[24edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[edo]] systems.
 == Interval chain ==
 Because 13/10 is an interseptimal interval, stacking it four times will result in a good approximation of a septimal interval. In this case, (13/10)<sup>4</sup> approximates 20/7 (compound [[10/7]]) remarkably well, with less than 1{{cent}} error.
-Additionally, while it may seem as though (13/10)<sup>2</sup> doesn't approximate 17/10 very well at first glance, it allows for an elegant interpretation of the tetrad formed by stacking 13/10 three times on top of itself: [[~]]10:13:17:22.
+Additionally, while it may seem as though (13/10)<sup>2</sup> does not approximate 17/10 very well at first glance, it allows for an elegant interpretation of the tetrad formed by stacking 13/10 three times on top of itself: [[~]]10:13:17:22.
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 == Approximation ==
 {{Interval edo approximation|13/10}}
 == See also ==
 * [[20/13]] – its [[octave complement]]