Sqrt(25/24): Difference between revisions

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Sqrt(25/24) is ~~an interval that allows to pass~~ from ~~a just major third~~ (5/4) to a ~~just~~ minor third (6/5) ~~by equal contrary motion, and vice versa~~.

{{Infobox interval|Ratio=\sqrt{25/24}|Name=classical semichroma, ptolemaic semichroma|Cents=35.336}}Sqrt(25/24), the '''classical semichroma''' or '''ptolemaic semichroma'''<ref group="note">It is not ''diptolemaic'' as it is only flattened by one [[81/80|comma]] from the (hemi-)Pythagorean semichroma of [[sqrt(2187/2048)]].</ref>, is a the difference between a 5-limit major or minor third and a neutral third [[sqrt(3/2)]] dividing the [[3/2|perfect fifth]] in two.

Let be two voices forming a 5/4 interval. If the lower voice goes up by a sqrt(25/24) and the upper voice goes down by the same interval, the next interval formed by the two voices will be a 6/5 interval.

This is an interval that allows to pass from a just major third (5/4) to a just minor third (6/5) by [[equal contrary motion]], and vice versa. Let be two voices forming a 5/4 interval. If the lower voice goes up by a sqrt(25/24) and the upper voice goes down by the same interval, the next interval formed by the two voices will be a 6/5 interval. Any [[edo]] that maps [[25/24]] to an even number of steps, and thus any edo that contains a true neutral third, contains a representation of this interval.

[~~https:~~//~~www~~.~~yacavone.net~~/~~xen~~-~~calc/?q=sqrt~~(~~25/24~~) ~~xen-calc~~]

Rational intervals that are close to this interval include [[49/48]] and [[50/49]], which stack to 25/24. The difference between 49/48 and 50/49 is [[2401/2400]], and tempering it out leads to the super-accurate [[Breed (temperament)|breed]] temperament.

This interval, when stacked, yields a tuning system close to [[34edo]], which consistently represents it as one step.

== Listen ==

{| class="wikitable"

|[[File:Sqrt(25 24) counterpoint thirds.mp3|left|thumb|Just major third and just minor third alternating by equal contrary motion]]

|[[File:Sqrt(25 24) counterpoint seventh chords.mp3|left|thumb|Just major seventh chord and just minor seventh chord alternating by equal contrary motion]]

|}

== Approximations ==

[[~~34edo|34~~-~~edo~~]] ~~has such an excellent~~ sqrt(25/24) ~~that the next EDO to have a better one is~~ [[~~441edo~~|~~441~~]].

EDOs that have both a good [[5-odd-limit]] and a sqrt(25/24) distinct from [[25/24]] include (among others) [[24edo#Counterpoint|24]], [[27edo|27]], [[31edo|31]] and [[34edo|34]].

~~EDOs that have both a good~~ [[~~5-odd-limit~~]] ~~and a~~ sqrt(25/24) ~~distinct from [[25/24]] include (among others)~~ [[~~24edo#Counterpoint~~|~~24]], [[27edo|27]], [[31edo|31]] and [[34edo|34~~]].

[[34edo]] has such an excellent sqrt(25/24) that the next EDO to have a better one is [[441edo|441]].

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-Sqrt(25/24) is an interval that allows to pass from a just major third (5/4) to a just minor third (6/5) by equal contrary motion, and vice versa.
+{{Infobox interval|Ratio=\sqrt{25/24}|Name=classical semichroma, ptolemaic semichroma|Cents=35.336}}Sqrt(25/24), the '''classical semichroma''' or '''ptolemaic semichroma'''<ref group="note">It is not ''diptolemaic'' as it is only flattened by one [[81/80|comma]] from the (hemi-)Pythagorean semichroma of [[sqrt(2187/2048)]].</ref>, is a the difference between a 5-limit major or minor third and a neutral third [[sqrt(3/2)]] dividing the [[3/2|perfect fifth]] in two.
-Let be two voices forming a 5/4 interval. If the lower voice goes up by a sqrt(25/24) and the upper voice goes down by the same interval, the next interval formed by the two voices will be a 6/5 interval.
+This is an interval that allows to pass from a just major third (5/4) to a just minor third (6/5) by [[equal contrary motion]], and vice versa. Let be two voices forming a 5/4 interval. If the lower voice goes up by a sqrt(25/24) and the upper voice goes down by the same interval, the next interval formed by the two voices will be a 6/5 interval. Any [[edo]] that maps [[25/24]] to an even number of steps, and thus any edo that contains a true neutral third, contains a representation of this interval.
-[https://www.yacavone.net/xen-calc/?q=sqrt(25/24) xen-calc]
+Rational intervals that are close to this interval include [[49/48]] and [[50/49]], which stack to 25/24. The difference between 49/48 and 50/49 is [[2401/2400]], and tempering it out leads to the super-accurate [[Breed (temperament)|breed]] temperament.
+This interval, when stacked, yields a tuning system close to [[34edo]], which consistently represents it as one step.
 == Listen ==
-[[File:Sqrt(25 24) counterpoint thirds.mp3|left|thumb|Just major third and just minor third alternating by equal contrary motion]] [[File:Sqrt(25 24) counterpoint seventh chords.mp3|left|thumb|Just major seventh chord and just minor seventh chord alternating by equal contrary motion]]
+{| class="wikitable"
+|[[File:Sqrt(25 24) counterpoint thirds.mp3|left|thumb|Just major third and just minor third alternating by equal contrary motion]]
+|[[File:Sqrt(25 24) counterpoint seventh chords.mp3|left|thumb|Just major seventh chord and just minor seventh chord alternating by equal contrary motion]]
+|}
 == Approximations ==
-[[34edo|34-edo]] has such an excellent sqrt(25/24) that the next EDO to have a better one is [[441edo|441]].
+EDOs that have both a good [[5-odd-limit]] and a sqrt(25/24) distinct from [[25/24]] include (among others) [[24edo#Counterpoint|24]], [[27edo|27]], [[31edo|31]] and [[34edo|34]].
-EDOs that have both a good [[5-odd-limit]] and a sqrt(25/24) distinct from [[25/24]] include (among others) [[24edo#Counterpoint|24]], [[27edo|27]], [[31edo|31]] and [[34edo|34]].
+[[34edo]] has such an excellent sqrt(25/24) that the next EDO to have a better one is [[441edo|441]].
+<references group="note" />