3/2: Difference between revisions

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{{Infobox Interval

'''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. Only the [[2/1|octave]] and the [[3/1|compound fifth]] have smaller numbers.

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 pythogorean, such as [[HEJI]], [[Sagittal notation|Sagittal]], [[Ups and downs notation|ups and downs]], [[FJS]] and [[color notation]].

Because 3/2 has very low [[harmonic entropy]], it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:

[[Meantone]] temperament flattens the fifth from just such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.

[[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.

[[Schismatic]] temperament flattens the fifth very slightly such that the ''diminished'' fourth generated by stacking eight fourths approximates 5/4. Thus a triad with 5/4 is written as C F♭ G (unless the notation has accidentals for [[81/80]], e.g. C vE G).

12edo approximates 3/2 to within only 2¢. [[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¢ and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.

! class="unsortable" | deg\edo

! Absolute<br>Error ([[Cent|¢]])

! Relative

Error (%)

! &#8597;

! class="unsortable" | Equally accurate

multiples

|  [[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 4\7 = ~686¢

*'''Perfect''' edos have fifths of exactly 4\7

*'''Diatonic''' edos have fifths between 4\7 and 3\5 = 720¢

*'''Pentatonic''' have fifths of exactly 3\5

*'''Supersharp''' edos have fifths wider than 3\5

|+Comparison of the fifths of edos 5 to 31

! Edo Category

|[[5edo]]

| pentatonic edo

|[[7edo]]

| perfect edo

| -16.241

|[[8edo]]

| supersharp edo

|[[9edo]]

| superflat edo

| -35.288

|[[10edo]]

| pentatonic edo

|[[11edo]]

| superflat edo

| -47.41

|[[12edo]]

| diatonic edo

| -1.955

|[[13edo]]

| supersharp edo

|[[14edo]]

| perfect edo

| -16.241

|[[15edo]]

| pentatonic edo

|[[16edo]]

| superflat edo

| -26.955

|[[17edo]]

| diatonic edo

|[[18edo]]

| supersharp edo

|[[19edo]]

| diatonic edo

| -7.218

|[[20edo]]

| pentatonic edo

|[[21edo]]

| perfect edo

| -16.241

|[[22edo]]

| diatonic edo

|[[23edo]]

| superflat edo

| -23.694

|[[24edo]]

| diatonic edo

| -1.955

|[[25edo]]

| pentatonic edo

|[[26edo]]

| diatonic edo

| -9.647

|[[27edo]]

| diatonic edo

|[[28edo]]

| perfect edo

| -16.241

|[[29edo]]

| diatonic edo

|[[30edo]]

| 17\30

| pentatonic edo

|[[31edo]]

| diatonic edo

| -5.181

* {{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)