3/2: Difference between revisions

'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Not only that, but there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third – specifically [[5/4]] – as consonant. 3/2 is the simplest [[just intonation]] interval to be very well approximated by [[12edo]], after the [[octave]].

Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely.  Nevertheless, even in Xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the [[moment of symmetry]] scales created by this tuning – specifically in the [[TAMNAMS]] system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators.  An example of such a scale is the familiar [[Wikipedia: Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]].

Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4 – or, in the case of [[quarter-comma meantone]] (see also [[31edo]]), identical. In such systems, and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In [[12edo]], and hence in most discussions these days, this is simplified to seven semitones, which is fitting seeing as 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. On the other hand, in just intonation, the perfect fifth consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.

Then there is the possibility of [[schismatic]] temperaments, which flatten the perfect fifth such that an approximated 5/4 is generated by stacking eight fifths downwards; however, without a notation system that properly accounts for the syntonic comma (such as [[Syntonic-Rastmic Subchroma notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]], and this in turn results in common chords such as conventional [[Wikipedia: Major chord|Major]] and [[Wikipedia: Minor chord|Minor]] triads being awkward to notate.

== Approximations by edos ==

The following [[edo]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&uarr;) or flat (&darr;).

! [[Edo]]

! Absolute<br>Error ([[Cent|¢]])

! Relative<br>Error ([[Relative cent|r¢]])

! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>

|+Comparison of edo approximations of 3/2 and "fifth classes" (from 5edo to 31edo)

! Edo

! Degree

! Cents

! Fifth Category

! Error (¢)

| [[5edo]]

| 3\5

| 720.000

| pentatonic edo

| +18.045

| [[7edo]]

| 4\7

| 685.714

| perfect edo

| -16.241

| [[8edo]]

| 5\8

| 750.000

| supersharp edo

| +48.045

| [[9edo]]

| 5\9

| 666.667

| superflat edo

| -35.288

| [[10edo]]

| 6\10

| 720.000

| pentatonic edo

| +18.045

| [[11edo]]

| 6\11

| 654.545

| superflat edo

| -47.41

| [[12edo]]

| 7\12

| 700.000

| diatonic edo

| -1.955

| [[13edo]]

| 8\13

| 738.462

| supersharp edo

| +36.507

| [[14edo]]

| 8\14

| 685.714

| perfect edo

| -16.241

| [[15edo]]

| 9\15

| 720.000

| pentatonic edo

| +18.045

| [[16edo]]

| 9\16

| 675.000

| superflat edo

| -26.955

| [[17edo]]

| 10\17

| 705.882

| diatonic edo

| +3.927

| [[18edo]]

| 11\18

| 733.333

| supersharp edo

| +31.378

| [[19edo]]

| 11\19

| 694.737

| diatonic edo

| -7.218

| [[20edo]]

| 12\20

| 720.000

| pentatonic edo

| +18.045

| [[21edo]]

| 12\21

| 685.714

| perfect edo

| -16.241

| [[22edo]]

| 13\22

| 709.091

| diatonic edo

| +7.136

| [[23edo]]

| 13\23

| 678.261

| superflat edo

| -23.694

| [[24edo]]

| 14\24

| 700.000

| diatonic edo

| -1.955

| [[25edo]]

| 15\25

| 720.000

| pentatonic edo

| +18.045

| [[26edo]]

| 15\26

| 692.308

| diatonic edo

| -9.647

| [[27edo]]

| 16\27

| 711.111

| diatonic edo

| +9.156

| [[28edo]]

| 16\28

| 685.714

| perfect edo

| -16.241

| [[29edo]]

| 17\29

| 703.448

| diatonic edo

| +1.493

| [[30edo]]

| 17\30

| 720.000

| pentatonic edo

| +18.045

| [[31edo]]

| 18\31

| 696.774

| diatonic edo

| -5.181

* The many and various 3/2 approximations in different edos can be classified as (after [[Kite Giedraitis]]):

** '''superflat''' edo – fifth is narrower than 686 cents.

** '''perfect''' edo – fifth is 686 cents wide (and 4/7 steps).

** '''diatonic''' edo – fifth is between 686.1 - 719.9 cents wide.

** '''pentatonic''' edo – fifth is exactly 720 cents wide.

** '''supersharp''' edo – fifth is wider than 720 cents.