13/10: Difference between revisions

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| Sound = jid_13_10_pluck_adu_dr220.mp3

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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 [[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 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 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.

{| class="wikitable"

|+ [[Interval chain]] generated by 13/10

! #

! [[Cent]]s

! Approximated [[ratio]]s

! Associated [[comma]]s

|-

| 1

| 454.2

| 13/10 [[17/13]] (+10.2{{cent}})

| [[170/169]] (major naiadma)

|-

| 2

| 908.4

| [[27/16]] (-2.6{{cent}}) [[22/13]] (+2.4{{cent}}) [[17/10]] (+10.2{{cent}})

| [[676/675]] (island comma) [[2200/2197]] (petrma) [[170/169]] (major naiadma)

|-

| 3

| 1362.6

| [[11/5]] (+2.4{{cent}})

| [[2200/2197]] (petrma)

|-

| 4

| 1816.9

| [[20/7]] (+0.6{{cent}})

| [[200000/199927]]

|-

| 5

| 2271.1

| [[26/7]] (+0.6{{cent}})

| [[200000/199927]]

|}

== Approximation ==

== See also ==

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[[Category:Third]]

[[Category:Supermajor third]]

[[Category:Over-5]]

[[Category:Over-5 intervals]]

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 | Sound = jid_13_10_pluck_adu_dr220.mp3
 }}
+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 [[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 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> 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.
+{| class="wikitable"
+|+ [[Interval chain]] generated by 13/10
+! #
+! [[Cent]]s
+! Approximated [[ratio]]s
+! Associated [[comma]]s
+|-
+| 1
+| 454.2
+| 13/10<br>[[17/13]] (+10.2{{cent}})
+| <br>[[170/169]] (major naiadma)
+|-
+| 2
+| 908.4
+| [[27/16]] (-2.6{{cent}})<br>[[22/13]] (+2.4{{cent}})<br>[[17/10]] (+10.2{{cent}})
+| [[676/675]] (island comma)<br>[[2200/2197]] (petrma)<br>[[170/169]] (major naiadma)
+|-
+| 3
+| 1362.6
+| [[11/5]] (+2.4{{cent}})
+| [[2200/2197]] (petrma)
+|-
+| 4
+| 1816.9
+| [[20/7]] (+0.6{{cent}})
+| [[200000/199927]]
+|-
+| 5
+| 2271.1
+| [[26/7]] (+0.6{{cent}})
+| [[200000/199927]]
+|}
+== Approximation ==
+{{Interval edo approximation|13/10}}
 == See also ==
@@ Line 20: / Line 64: @@
 [[Category:Third]]
 [[Category:Supermajor third]]
-[[Category:Over-5]]
+[[Category:Over-5 intervals]]