16/15: Difference between revisions

(10 intermediate revisions by 4 users not shown)

Line 1:

{{Infobox Interval

| Name = just diatonic semitone, classic(al) diatonic semitone, ptolemaic diatonic semitone

| Comma = yes

| Color name = g2, gu 2nd

| Sound = jid_16_15_pluck_adu_dr220.mp3

}}

The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just diatonic semitone''', '''classic(al) diatonic semitone''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5-limit]]. </ref>.

~~The~~ [[~~5-limit~~]] [[~~superparticular]] interval '''16~~/~~15''' is the '''just''', '''classic(al)''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[~~5/4]]~~. </ref> – the difference between~~ the major third [[5/4]] and the fourth [[4/3]]~~, and between~~ [[3/2]] and [[8/5]].

It is the difference between:

* the major second [[9/8]] and the minor third [[6/5]];

* the major third [[5/4]] and the fourth [[4/3]];

* the perfect fifth [[3/2]] and the minor sixth [[8/5]];

* the major sixth [[5/3]] and the minor seventh [[16/9]];

* the major seventh [[15/8]] and the perfect octave [[2/1]].

== Approximation ==

16/15 is very accurately approximated by [[43edo]] (4\43).

== Temperaments ==

When this ratio is taken as a comma to be tempered, it produces [[father]] temperament, ~~where~~ 4/3 and 5/4 are equated~~. In this temperament~~, major thirds and fifths become [[octave complement]]s of each other.

When this ratio is taken as a [[comma]] to be [[tempering out|tempered out]], it produces [[father]] temperament, and lends itself the name '''father comma'''. In this [[exotemperament]], 4/3 and 5/4 are equated, and major thirds and fifths become [[octave complement]]s of each other. It is a [[Mersenne comma]].

The following [[linear temperament]]s are [[generate]]d by a [[~]]16/15:

* [[Vavoom]]

* [[Stockhausenic]]

In addition, this [[fractional-octave temperaments]] are generated by a ~16/15:

* [[Quintosec]]

Some [[11th-octave temperaments]] treat ~16/15 as the period, including [[hendecatonic]].

== See also ==

Line 17:

Line 38:

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

* [[16/15 equal-step tuning]] – equal multiplication of this interval

== Notes ==

Line 24:

Line 44:

[[Category:Second]]

[[Category:Semitone]]

[[Category:Father]]

[[Category:Commas named after their interval size]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Name = just diatonic semitone, classic(al) diatonic semitone, ptolemaic diatonic semitone
+| Comma = yes
 | Color name = g2, gu 2nd
 | Sound = jid_16_15_pluck_adu_dr220.mp3
 }}
+{{Wikipedia|Semitone}}
+The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just diatonic semitone''', '''classic(al) diatonic semitone''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5-limit]]. </ref>.
-The [[5-limit]] [[superparticular]] interval '''16/15''' is the '''just''', '''classic(al)''' or '''ptolemaic diatonic semitone'''<ref>For reference, see [[5/4]]. </ref> – the difference between the major third [[5/4]] and the fourth [[4/3]], and between [[3/2]] and [[8/5]].
+It is the difference between:
+* the major second [[9/8]] and the minor third [[6/5]];
+* the major third [[5/4]] and the fourth [[4/3]];
+* the perfect fifth [[3/2]] and the minor sixth [[8/5]];
+* the major sixth [[5/3]] and the minor seventh [[16/9]];
+* the major seventh [[15/8]] and the perfect octave [[2/1]].
+== Approximation ==
+/15 is very accurately approximated by [[43edo]] (4\43).
 == Temperaments ==
-When this ratio is taken as a comma to be tempered, it produces [[father]] temperament, where 4/3 and 5/4 are equated. In this temperament, major thirds and fifths become [[octave complement]]s of each other.
+When this ratio is taken as a [[comma]] to be [[tempering out|tempered out]], it produces [[father]] temperament, and lends itself the name '''father comma'''. In this [[exotemperament]], 4/3 and 5/4 are equated, and major thirds and fifths become [[octave complement]]s of each other. It is a [[Mersenne comma]].
+The following [[linear temperament]]s are [[generate]]d by a [[~]]16/15:
+* [[Vavoom]]
+* [[Stockhausenic]]
+In addition, this [[fractional-octave temperaments]] are generated by a ~16/15:
+* [[Quintosec]]
+{{todo|complete list}}
+Some [[11th-octave temperaments]] treat ~16/15 as the period, including [[hendecatonic]].
 == See also ==
@@ Line 17: / Line 38: @@
 * [[Gallery of just intervals]]
 * [[List of superparticular intervals]]
-* [[16/15 equal-step tuning]] – equal multiplication of this interval
 == Notes ==
@@ Line 24: / Line 44: @@
 [[Category:Second]]
 [[Category:Semitone]]
+[[Category:Father]]
+[[Category:Commas named after their interval size]]