3/2: Difference between revisions

'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[2/1|octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[EDO]]s.

For harmonic [[timbre|timbres]], the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it's easy to hear if it's mistuned.  

Variations of the perfect fifth (whether [[just]] or tempered) appear in most [[Approaches to musical tuning|music of the world]]. [[Historical temperaments|Historically]], European music treated the perfect fifth as consonant long before it treated the major third—specifically [[5/4]]—as consonant. In the present day, the dominant tuning [[12edo]] approximates 3/2 very accurately.

A [[Chain of fifths|chain of just perfect fifths]] generates [[Pythagorean tuning]]. The chain continues indefinitely and theoretically never returns to the starting note. A chain that ends at seven notes generates the historically important [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]]. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are Pythagorean, such as [[HEJI]], [[Sagittal notation|Sagittal]], [[Ups and downs notation|ups and downs]], [[FJS]] and [[color notation]].

[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.

* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4.

[[Schismatic|Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).

* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.

12edo approximates 3/2 to within only 2{{c}}. [[29edo]], [[41edo]], and [[53edo]] are even more accurate. In regards to [[telicity]], while 12edo is a 2-strong 3-2 [[telic]] system, 53edo is notably a 3-strong 3-2 telic system.

The following edos (up to 200) approximate 3/2 to within both 7{{c}} and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.

! Absolute <br>error ([[Cent|¢]])

|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 || &darr; || [[24edo|14\24]], [[36edo|21\36]]

|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || &uarr; ||  

|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 || &uarr; ||  

|  [[41edo|41]]  ||  24\41  || 0.484 || 1.654 || &uarr; || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]

|  [[53edo|53]]  ||  31\53  || 0.068 || 0.301 || &darr; || [[106edo|62\106]], [[159edo|93\159]]

|  [[65edo|65]]  ||  38\65  || 0.416 || 2.256 || &darr; || [[130edo|76\130]], [[195edo|114\195]]

|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 || &uarr; ||  

|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 || &darr; ||  

|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 || &darr; ||  

|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 || &uarr; || [[188edo|110\188]]

| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || &uarr; ||  

| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || &darr; ||  

| [[135edo|135]] ||  79\135 || 0.267 || 3.006 ||&uarr; ||  

| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || &darr; ||  

| [[147edo|147]] ||  86\147 || 0.086 || 1.051 || &uarr; ||  

| [[171edo|171]] || 100\171 || 0.200 || 2.859 || &darr; ||  

| [[176edo|176]] || 103\176 || 0.318 || 4.660 || &uarr; ||  

| [[183edo|183]] || 107\183 || 0.316 || 4.814 || &darr; ||  

| [[200edo|200]] || 117\200 || 0.045 || 0.750 || &uarr; ||  

* '''Superflat''' edos have fifths narrower than {{nowrap|4\7 {{=}} ~686{{c}}}}

* '''Diatonic''' edos have fifths between 4\7 and {{nowrap|3\5 {{=}} 720{{c}}}}

! Edo Category

| perfect edo

| supersharp edo

| superflat edo

| pentatonic edo

| superflat edo

| diatonic edo

| supersharp edo

| perfect edo

| pentatonic edo

| superflat edo

| diatonic edo

| supersharp edo

| diatonic edo

| pentatonic edo

| perfect edo

| diatonic edo

| superflat edo

| diatonic edo

| pentatonic edo

| diatonic edo

| diatonic edo

| perfect edo

| diatonic edo

| pentatonic edo

| diatonic edo

'''2:3''' is a 3-limit [[dyad]], known as the '''five chord''' (as in C5 not V), or as the '''power chord'''. This dyad is indispensable in certain musical genres such as [[African music#Equiheptatonic tunings|mbira music]]<nowiki/> and late medieval music. In the latter, when voiced as hi5add8, it's known as the '''trine''', a very common closing chord.

!Voices

![[EFR]]

![[Kite's thoughts on hi-lo notation|Hi-lo name]]

!Special properties

| rowspan="3" |2 voices

|1:3

|hi5

|AOV ([[Odd limit#Proposed extensions|all-odd voicing]])

|2:3

|basic

|CAOV (condensed AOV)

|3:4

|lo5

|1st inversion

|1:2:3

|hi5add8

|the trine

|2:3:4

|add8

|

|3:4:6

|addlo5

|2:3:4 melodically inverted

* {{OEIS| A060528 }} – sequence of edos with increasingly better approximations of 3/2 (and by extension 4/3)

* {{OEIS| A005664 }} – denominators of the convergents to log₂(3)

* {{OEIS| A206788 }} – denominators of the semiconvergents to log₂(3)