25/16: Difference between revisions

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'''25/16''', the '''classic(al) augmented fifth''' or '''diptolemaic augmented fifth''' is the interval obtained by stacking two [[5/4]] major thirds, however, it gains additional [[isoharmonic chord|isoharmonic]] identity from its position between [[11/8]] and [[7/4]], so it can frequently be used in conjunction with those, even in chords.  It is [[100/99]] sharp of [[99/64]], and tempering out 100/99 makes 1-5/4-25/16 into the [[ptolemismic chords|ptolemismic augmented triad]].
'''25/16''', the '''classic(al) augmented fifth''' or '''diptolemaic augmented fifth''' is the interval obtained by stacking two [[5/4]] major thirds, however, it gains additional [[isoharmonic chord|isoharmonic]] identity from its position between [[11/8]] and [[7/4]], so it can frequently be used in conjunction with those, even in chords.  It is [[100/99]] sharp of [[99/64]], and tempering out 100/99 makes 1-5/4-25/16 into the [[ptolemismic chords|ptolemismic augmented triad]].
== Names ==
While this interval has been referred to as the ''classic'' or ''classical'' augmented fifth for some time, the term "diptolemaic" [https://discord.com/channels/332357996569034752/516067802864549890/912167264789364736 was coined on Discord] by [[Flora Canou]] while discussing a proposal for a consistent naming scheme for different 5-limit intervals with [[User:Aura|Aura]].  The concept of "diptolemaic" intervals builds on the concept of "ptolemaic"<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref> intervals, with "diptolemaic" intervals being those that have two factors of 5 in their [[monzo]] and differ from their Pythagorean counterparts by two instances of [[81/80]] in the direction of ratio simplicity— for instance, 25/16 differing from [[6561/4096]]—, while 5-limit intervals that differ from their Pythagorean counterparts by a single instance of 81/80 in the same direction are simply called "ptolemaic" intervals.


== See also ==
== See also ==
* [[32/25]] – its [[octave complement]]
* [[32/25]] – its [[octave complement]]
== References ==
<references/>


[[Category:Fifth]]
[[Category:Fifth]]
[[Category:Augmented fifth]]
[[Category:Augmented fifth]]
[[Category:todo:expand]]
[[Category:todo:expand]]

Revision as of 20:05, 28 January 2023

Interval information
Ratio 25/16
Factorization 2-4 × 52
Monzo [-4 0 2⟩
Size in cents 772.6274¢
Names classic(al) augmented fifth,
diptolemaic augmented fifth
Color name yy5, yoyo 5th
FJS name [math]\displaystyle{ \text{A5}^{5,5} }[/math]
Special properties reduced,
reduced harmonic
Tenney norm (log2 nd) 8.64386
Weil norm (log2 max(n, d)) 9.28771
Wilson norm (sopfr(nd)) 18

[sound info]
Open this interval in xen-calc

25/16, the classic(al) augmented fifth or diptolemaic augmented fifth is the interval obtained by stacking two 5/4 major thirds, however, it gains additional isoharmonic identity from its position between 11/8 and 7/4, so it can frequently be used in conjunction with those, even in chords. It is 100/99 sharp of 99/64, and tempering out 100/99 makes 1-5/4-25/16 into the ptolemismic augmented triad.

See also

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