The intervals of 5-limit

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

 just intonation

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

 (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5

Regular number

Regular numbers are numbers that evenly divide powers of 60. As an example, 602 = 3600 = 48 × 75, so both 48 and 75 are divisors of a power of 60...

. The fundamental intervals are the superparticular ratios 2/1 (the octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

), 3/2 (the perfect fifth

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

) and 5/4 (the major third

Major third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

). That is, the notes of the major triad

Major chord

In music theory, a major chord is a chord having a root, a major third, and a perfect fifth. When a chord has these three notes alone, it is called a major triad...

 are in the ratio 1:5/4:3/2 or 4:5:6.

In all tunings, the major third

Major third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

 is equivalent to two major second

Major second

In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...

s. However, because just intonation does not allow the irrational ratio of √5/2, two different frequency ratios are used: the major tone (9/8) and the minor tone (10/9).

The intervals within the diatonic scale

Diatonic scale

In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...

 are shown in the table below.

Names	Ratio	Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...	ET Cents	Definition	53ET 53 equal temperament In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...  commas	Representation	Complement
unison	1/1	0.00	0		0		octave
syntonic comma	81/80	21.51	0	c or T − t	1
diesis diminished second	128/125	41.06	0	D or S − x	2		augmented seventh
lesser chromatic semitone minor semitone augmented unison	25/24	70.67	100	x or t − S or T − L	3		diminished octave
Pythagorean minor second Pythagorean limma	256/243	90.22	100	Λ	4		Pythagorean major seventh
greater chromatic semitone wide augmented unison	135/128	92.18	100	X or T − S	4		narrow diminished octave
major semitone limma minor second	16/15	111.73	100	S	5		major seventh
large limma acute minor second	27/25	133.24	100	L or T − x	6		grave major seventh
grave tone grave major second	800/729	160.90	200	τ or Λ + x or t − c	7		acute minor seventh
minor tone lesser major second	10/9	182.40	200	t	8		minor seventh
major tone Pythagorean major second greater major second	9/8	203.91	200	T or t + c	9		Pythagorean minor seventh
diminished third	256/225	223.46	200	S + S	10		augmented sixth
semi-augmented second	125/108	253.08	300	t + x	11
augmented second	75/64	274.58	300	T + x	12		diminished seventh
Pythagorean minor third	32/27	294.13	300	T + Λ	13		Pythagorean major sixth
minor third	6/5	315.64	300	T + S	14		major sixth
acute minor third	243/200	333.18	300	T + L	15		grave major sixth
grave major third	100/81	364.81	400	T + τ	16		acute minor sixth
major third	5/4	386.31	400	T + t	17		minor sixth
Pythagorean major third	81/64	407.82	400	T + T	18		Pythagorean minor sixth
classic diminished fourth	32/25	427.37	400	T + S + S	19		classic augmented fifth
classic augmented third	125/96	456.99	500	T + t + x	20		classic diminished sixth
wide augmented third	675/512	478.49	500	T + t + X	21		narrow diminished sixth
perfect fourth	4/3	498.04	500	T + t + S	22		perfect fifth
acute fourth	27/20	519.55	500	T + t + L	23		grave fifth
classic augmented fourth	25/18	568.72	600	T + t + t	25		classic diminished fifth
augmented fourth	45/32	590.22	600	T + t + T	26		diminished fifth
diminished fifth	64/45	609.78	600	T + t + S + S	27		augmented fourth
classic diminished fifth	36/25	631.29	600	T + t + S + L	28		classic augmented fourth
grave fifth	40/27	680.45	700	T + t + S + t	30		acute fourth
perfect fifth	3/2	701.96	700	T + t + S + T	31		perfect fourth
narrow diminished sixth	1024/675	721.51	700	T + t + S + S + S	32		wide augmented third
classic diminished sixth	192/125	743.01	700	T + t + S + L + S	33		classic augmented third
classic augmented fifth	25/16	772.63	800	T + t + S + T + x	34		classic diminished fourth
Pythagorean minor sixth	128/81	792.18	800	T + t + S + T + Λ	35		Pythagorean major third
minor sixth	8/5	813.69	800	(T + t + S + T) + S	36		major third
acute minor sixth	81/50	835.19	800	(T + t + S + T) + L	37		grave major third
major sixth	5/3	884.36	900	(T + t + S + T) + t	39		minor third
Pythagorean major sixth	27/16	905.87	900	(T + t + S + T) + T	40		Pythagorean minor third
diminished seventh	128/75	925.42	900	(T + t + S + T) + S + S	41		augmented second
augmented sixth	225/128	976.54	1000	(T + t + S + T) + T + x	43		diminished third
Pythagorean minor seventh Minor seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the...	16/9	996.09	1000	(T + t + S + T) + T + Λ	44		Pythagorean major second
minor seventh	9/5	1017.60	1000	(T + t + S + T) + T + S	45		lesser major second
acute minor seventh	729/400	1039.10	1000	(T + t + S + T) + T + L	46		grave major second
grave major seventh	50/27	1066.76	1100	(T + t + S + T) + T + τ	47		acute minor second
major seventh	15/8	1088.27	1100	(T + t + S + T) + T + t	48		minor second
narrow diminished octave	256/135	1107.82	1100	(T + t + S + T) + t + S + S	49		wide augmented unison
Pythagorean major seventh	243/128	1109.78	1100	(T + t + S + T) + T + T	49		Pythagorean minor second
diminished octave	48/25	1129.33	1100	(T + t + S + T) + T + S + S	50		augmented unison
augmented seventh	125/64	1158.94	1200	(T + t + S + T) + T + t + x	51		diminished second
octave	2/1	1200.00	1200	(T + t + S + T) + (T + t + S)	53		unison

(The Pythagorean minor second is found by adding 5 perfect fourths.)