13/10: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = Barbados third, tridecimal semisixth
| Name = Barbados third, tridecimal semisixth, tridecimal third-fourth
| Color name = 3og4, thogu 4th
| Color name = 3og4, thogu 4th
| Sound = jid_13_10_pluck_adu_dr220.mp3
| 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).
 
== Approximation ==
13/10 is well-approximated in [[8edo]] (as well as all multiples of it such as [[16edo]], [[24edo]] and [[32edo]]), [[21edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[edo]] systems.
{{Interval edo approximation|13/10}}


== Interval chain ==
== 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.
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.


{| class="wikitable"
{| class="wikitable"
Line 21: Line 28:
| 1
| 1
| 454.2
| 454.2
| 13/10<br>[[17/13]] (+10.2{{cent}})
| 13/10<br>[[17/13]] (+10.2{{c}})
| <br>[[170/169]] (major naiadma)
| <br>[[170/169]] (major naiadma)
|-
|-
| 2
| 2
| 908.4
| 908.4
| [[27/16]] (-2.6{{cent}})<br>[[22/13]] (+2.4{{cent}})<br>[[17/10]] (+10.2{{cent}})
| [[27/16]] (-2.6{{c}})<br>[[22/13]] (+2.4{{c}})<br>[[17/10]] (+10.2{{c}})
| [[676/675]] (island comma)<br>[[2200/2197]] (petrma)<br>[[170/169]] (major naiadma)
| [[676/675]] (island comma)<br>[[2200/2197]] (petrma)<br>[[170/169]] (major naiadma)
|-
|-
| 3
| 3
| 1362.6
| 1362.6
| [[11/5]] (+2.4{{cent}})
| [[11/5]] (+2.4{{c}})
| [[2200/2197]] (petrma)
| [[2200/2197]] (petrma)
|-
|-
| 4
| 4
| 1816.9
| 1816.9
| [[20/7]] (+0.6{{cent}})
| [[20/7]] (+0.6{{c}})
| [[200000/199927]]
| [[200000/199927]]
|-
|-
| 5
| 5
| 2271.1
| 2271.1
| [[26/7]] (+0.6{{cent}})
| [[26/7]] (+0.6{{c}})
| [[200000/199927]]
| [[200000/199927]]
|}
|}
== Approximation ==
 
{{Interval edo approximation|13/10}}
== See also ==
== See also ==
* [[20/13]] – its [[octave complement]]
* [[20/13]] – its [[octave complement]]

Latest revision as of 17:55, 14 November 2026

Interval information
Ratio 13/10
Factorization 2-1 × 5-1 × 13
Monzo [-1 0 -1 0 0 1⟩
Size in cents 454.2139 ¢
Names Barbados third,
tridecimal semisixth,
tridecimal third-fourth
Color name 3og4, thogu 4th
FJS name [math]\displaystyle{ \text{d4}^{13}_{5} }[/math]
Special properties reduced
Tenney norm (log2 nd) 7.02237
Weil norm (log2 max(n, d)) 7.40088
Wilson norm (sopfr(nd)) 20

[sound info]
Open this interval in xen-calc

In 13-limit just intonation, 13/10, the tridecimal semisixth is an interseptimal interval measuring about 454.2 cents. 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 diatonic chain-of-fifths notation with commatic alterations (e.g. FJS, HEJI), 13/10 is a fourth, as it is a perfect fourth (4/3) minus an instance of 40/39, which is a Pythagorean apotome minus a stack consisting of an syntonic comma (81/80) and a tridecimal quartertone (1053/1024), none of which changes the 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).

Approximation

13/10 is well-approximated in 8edo (as well as all multiples of it such as 16edo, 24edo and 32edo), 21edo, 29edo, 37edo, and of course, infinitely many other edo systems.

Edo approximations for 13/10 (454.21 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
8 3\8 450.00 -4.21 -2.81
13 5\13 461.54 +7.32 +7.93
16 6\16 450.00 -4.21 -5.62
21 8\21 457.14 +2.93 +5.13
24 9\24 450.00 -4.21 -8.43
29 11\29 455.17 +0.96 +2.32
37 14\37 454.05 -0.16 -0.49
45 17\45 453.33 -0.88 -3.30
50 19\50 456.00 +1.79 +7.44
53 20\53 452.83 -1.38 -6.11
58 22\58 455.17 +0.96 +4.63
61 23\61 452.46 -1.75 -8.92
66 25\66 454.55 +0.33 +1.82
74 28\74 454.05 -0.16 -0.99
79 30\79 455.70 +1.48 +9.76

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 ¢ 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.

Interval chain generated by 13/10
# Cents Approximated ratios Associated commas
1 454.2 13/10
17/13 (+10.2 ¢)
170/169 (major naiadma)
2 908.4 27/16 (-2.6 ¢)
22/13 (+2.4 ¢)
17/10 (+10.2 ¢) 		676/675 (island comma)
2200/2197 (petrma)
170/169 (major naiadma)
3 1362.6 11/5 (+2.4 ¢) 2200/2197 (petrma)
4 1816.9 20/7 (+0.6 ¢) 200000/199927
5 2271.1 26/7 (+0.6 ¢) 200000/199927

See also

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