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 [[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 [[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]]. | 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 | 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 == | == Interval chain == | ||
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| 1 | | 1 | ||
| 454.2 | | 454.2 | ||
| 13/10<br>[[17/13]] (+10.2{{ | | 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{{ | | [[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{{ | | [[11/5]] (+2.4{{c}}) | ||
| [[2200/2197]] (petrma) | | [[2200/2197]] (petrma) | ||
|- | |- | ||
| 4 | | 4 | ||
| 1816.9 | | 1816.9 | ||
| [[20/7]] (+0.6{{ | | [[20/7]] (+0.6{{c}}) | ||
| [[200000/199927]] | | [[200000/199927]] | ||
|- | |- | ||
| 5 | | 5 | ||
| 2271.1 | | 2271.1 | ||
| [[26/7]] (+0.6{{ | | [[26/7]] (+0.6{{c}}) | ||
| [[200000/199927]] | | [[200000/199927]] | ||
|} | |} | ||