Intervals

In the traditional system, interval names reflect increasing tuning precision from right to left, as interval quality modifies diatonic ordinal size. Below, dark and light shadings indicate exclusive connections between these attributes.

Traditional System of Interval Naming	Format		Example
	quality	ordinal size	spoken	"minor third"
	dd d m P M a aa	1 2 3 4 5 6 7 8	written	m3

The Hunt system adds two optional indications for the precise tuning of each interval, modifying qualities by comma shift type, and further modifying by JND intonation quality.

Hunt System of Interval Naming	Format				Example
	JND intonation quality	comma shift type	quality	ordinal size	spoken	"Perfect Large minor third"
	dd d m P M a aa	N S L W	dd d m P M a aa	1 2 3 4 5 6 7 8	written	P.Lm3

Traditional System

According to modern default tuning, all intervals differ by increments of 1 halfstep (h) of 100¢, the chromatic scale step of 12ET. The table below shows relationships between adjacent qualities for both perfect and imperfect intervals.

Traditional System of Simple Interval Naming	ordinal size
	written	1		2	3		4	5	6	7		8
	spoken	"unison" or "prime"		"second"	"third"		"fourth"	"fifth"	"sixth"	"seventh"		"octave"
	default sizes h, ¢	0h 0¢		1-2h 100-200¢	3-4h 300-400¢		5-6h 500-600¢	6-7h 600-700¢	8-9h 800-900¢	10-11h 1000-1100¢		12h 1200¢
	quality - Perfect intervals: 1, 4, 5, 8
	written	dd			d		P		a		aa
	spoken	"doubly diminished"			"diminished"		"Perfect"		"augmented"		"doubly augmented"
	difference ±h, ¢		±1h, 100¢			±1h, 100¢		±1h, 100¢		±1h, 100¢
	quality - Imperfect intervals: 2, 3, 6, 7
	written	d			m			M			a
	spoken	"diminished"			"minor"			"Major"			"augmented"
	difference ±h, ¢		±1h, 100¢			±1h, 100¢				±1h, 100¢

We can summarize interval relationships in 12ET starting with a major scale on a 12ET tone ruler.

Diatonic Intervals of the Major Scale in 12ET
P1		M2		M3	P4		P5		M6		M7	P8
0h	1h	2h	3h	4h	5h	6h	7h	8h	9h	10h	11h	12h

To this table we can add staggered rows above and below showing how the diatonic pattern shifts to the right or to the left when interval qualities are made larger or smaller.

Chromatic Alteration of the Diatonic Intervals in 12ET
+2h					aa1		aa2		aa3	aa4		aa5		aa6		aa7	aa8
+1h				a1		a2		a3	a4		a5		a6		a7	a8
± 0h			P1		M2		M3	P4		P5		M6		M7	P8
-1h		d1		m2		m3	d4		d5		m6		m7	d8
-2h	dd1		d2		d3	dd4		dd5		d6		d7	dd8

The ends of this parallelogram-shaped table can be chopped off to form a rectangle, since the boundaries for simple intervals must begin with the unison and end with the octave. The table should also be extended with another row at the bottom to show alterations resulting in doubly diminished thirds, sixths, and sevenths, since these intervals are also possible. Each of the resulting columns shows a group of enharmonic names for the twelve intervals of 12ET. In all cases, save the octave and the unison, there are at least three names per interval. For example, the names a1, m2, and dd3 all refer to the same interval in 12ET.

Chromatic Alteration of the Diatonic Intervals in 12ET
+2h			aa1		aa2		aa3	aa4		aa5		aa6
+1h		a1		a2		a3	a4		a5		a6		a7
± h	P1		M2		M3	P4		P5		M6		M7	P8
-1h		m2		m3	d4		d5		m6		m7	d8
-2h	d2		d3	dd4		dd5		d6		d7	dd8
-3h		dd3					dd6		dd7

The table below is a more compact version of the table above with the empty spaces removed. This table shows 38 simple interval names from the traditional system.

38 Traditional System Simple Interval Names
	a1	aa1	a2	aa2	a3	aa3	aa4	a5	aa5	a6	aa6	a7
P1	m2	M2	m3	M3	P4	a4, d5	P5	m6	M6	m7	M7	P8
d2	dd3	d3	dd4	d4	dd5	dd6	d6	dd7	d7	dd8	d8

Interval names are shaded in the table above to show that not all of these names are used equally.

generally not used	rarely used	less commonly used	common

For example, the minor second (m2) is commonly used, while the augmented unison (a1) is less commonly used, and the doubly diminished third (dd3) is generally not used. More interval names are possible in the traditional system, but they are not used. For example, the interval F:B is a quintupally augmented fourth, (aaaaa4) — possible, but outrageous. Generally, the doubly augmented and doubly diminished qualities are not used, although there are exceptions, such as the doubly augmented fourth appearing in the English augmented sixth chord. Removing the doubly augmented and diminished intervals as exceptions to the rule, 26 interval names are left. In modern parlance, multiple names for a single tone are called enharmonic, so we arrive at 26 enharmonic simple interval names, as shown in the smaller table below.

26 Traditional Enharmonic Simple Interval Names
P1	m2	M2	m3	M3	P4	a4	P5	m6	M6	m7	M7	P8
d2	a1	d3	a2	d4	a3	d5	d6	a5	d7	a6	d8	a7
0	1	2	3	4	5	6	7	8	9	10	11	12

These intervals are arranged into four columns below, where the leftmost box in each row shows the number of halfsteps in each interval. Arrows show navigation of the four columns in a zig-zag pattern like a board game, in order to show inversional relationships between intervals.

START ↓ 0 P1, d2 1 m2, a1 2 M2, d3 3 m3, a2 →	→ 6 a4 5 P4, a3 4 M3, d4 ↑	↓ 6 d5 7 P5, d6 8 m6, a5 →	END ↑ 12 P8, a7 11 M7, d8 10 m7, a6 9 M6, d7 ↑

The two outer columns are inversional, and the two inner columns are inversional. The number of halfsteps in each pair of inversional intervals adds to 12. For example, from the inner columns, the perfect fourth (P4) is 5h and the perfect fifth (P5) is 7h: 5 + 7 = 12. Or looking at the outer columns, the Major second (M2) is 2h and the minor seventh (m7) is 10h: 2 + 10 = 12. Note that the augmented fourth (a4) and diminished fifth (d5) are both 6h in size, but are each given a separate row, so that the table is symmetrical.

Hunt System

In the Hunt system, ordinal sizes and qualities are called by the same names as in the traditional system, with adjacent qualities differing by 4 commas (4Ç). These qualities are then modified into different comma shift types by single comma shifts (1Ç = 5 JNDs). The final modification is an intonation quality, which is the precise indication of interval intonation in single JNDs (J). The table below shows these modifiers in order of increasing tuning precision from top to bottom.

Hunt System of Simple Interval Naming	ordinal size
	written		1					2		3			4		5		6		7				8
	spoken		"unison" or "prime"					"second"		"third"			"fourth"		"fifth"		"sixth"		"seventh"				"octave"
	quality - Perfect intervals: 1, 4, 5, 8
	written		dd						d				P				a				aa
	spoken		"doubly diminished"						"diminished"				"Perfect"				"augmented"				"doubly augmented"
	difference Ç, J				±4Ç, ±20J					±4Ç, ±20J					±4Ç, ±20J				±4Ç, ±20J
	quality - Imperfect intervals: 2, 3, 6, 7
	written		d						m						M						a
	spoken		"diminished"						"minor"						"Major"						"augmented"
	difference Ç, J				±4Ç, ±20J						±4Ç, ±20J								±4Ç, ±20J
	comma shift type
	written		N				S									L				W
	spoken		"Narrow"				"Small"				(unshifted)					"Large"				"Wide"
	difference Ç, J				±1Ç, ±5J				±1Ç, ±5J					±1Ç, ±5J					±1Ç, ±5J
	JND intonation quality
	written		dd				d		m		P						M		a				aa
	spoken		"doubly diminished"				"diminished"		"minor"		"Perfect"						"Major"		"augmented"				"doubly augmented"
	difference J				±1J			±1J		±1J						±1J		±1J		±1J

Shown below is a Pythagorean Major scale in 41ET.

Pythagorean Major Scale in 41ET
scale	P1							M2							M3			P4							P5							M6							M7			P8
Ç J	0 0	1 5	2 10	3 15	4 20	5 25	6 30	7 35	8 40	9 45	10 50	11 55	12 60	13 65	14 70	15 75	16 80	17 85	18 90	19 95	20 100	21 105	22 110	23 115	24 120	25 125	26 130	27 135	28 140	29 145	30 150	31 155	32 160	33 165	34 170	35 175	36 180	37 185	38 190	39 195	40 200	41 205

To this table we add staggered rows above and below the naturals showing how the diatonic pattern shifts according to the accidentals.

Chromatic Alteration of the Diatonic Intervals in 41ET
+4Ç													a1							a2							a3			a4							a5							a6							a7			a8
± 0Ç									P1							M2							M3			P4							P5							M6							M7			P8
-4Ç					d1							m2							m3			d4							d5							m6							m7			d8
-8Ç	dd1							d2							d3			dd4							dd5							d6							d7			dd8

This table can be chopped off at the sides, since simple intervals begin with the unison and end with the octave. The doubly diminished intervals can also be dispensed with, and the structure can be collapsed into a single row. A double Pythagorean chromatic scale is revealed, as shown below.

Double Pythagorean Chromatic Scale in 41ET
natural	P1			m2	a1		d3	M2			m3	a2		d4	M3			P4	a3		d5	a4		d6	P5			m6	a5		d7	M6			m7	a6		d8	M7			P8
Ç J	0 0	1 5	2 10	3 15	4 20	5 25	6 30	7 35	8 40	9 45	10 50	11 55	12 60	13 65	14 70	15 75	16 80	17 85	18 90	19 95	20 100	21 105	22 110	23 115	24 120	25 125	26 130	27 135	28 140	29 145	30 150	31 155	32 160	33 165	34 170	35 175	36 180	37 185	38 190	39 195	40 200	41 205

Note that the diminished second and augmented seventh do not appear on the table above, as they lie beyond the octave boundaries. There are many gaps in between the double tones of the double Pythagorean chromatic scale; whereas transposition of the diatonic intervals results in a chromatic superset in 12ET, this is clearly not the situation in 41ET. The gaps can be filled in by shifting the structure to the right (positive shift) and the left (negative shift) by comma steps of 41ET, keeping in mind that when this is done the diminished second and augmented seventh will shift within the octave boundaries. Both single and double shifting is allowed, so that a positive shift creates a large version (L) of an interval, and a double positive shift creates a wide version (W), while a negative shift creates a small version (S), and a double negative shift creates a narrow version (N), so that from each interval comes a group of five intervals called an interval family. Below, interval families of the double Pythagorean chromatic scale are shown.

Interval Families of the Double Pythagorean Chromatic Scale in 41ET
Wide, +2Ç		Wd2	W1			Wm2	Wa1		Wd3	WM2			Wm3	Wa2		Wd4	WM3			W4	Wa3		Wd5	Wa4		Wd6	W5			Wm6	Wa5		Wd7	WM6			Wm7	Wa6		Wd8	WM7
Large, +1Ç	Ld2	L1			Lm2	La1		Ld3	LM2			Lm3	La2		Ld4	LM3			L4	La3		Ld5	La4		Ld6	L5			Lm6	La5		Ld7	LM6			Lm7	La6		Ld8	LM7
(no shift) ± 0Ç	P1			m2	a1		d3	M2			m3	a2		d4	M3			P4	a3		d5	a4		d6	P5			m6	a5		d7	M6			m7	a6		d8	M7			P8
Small, -1Ç			Sm2	Sa1		Sd3	SM2			Sm3	Sa2		Sd4	SM3			S4	Sa3		Sd5	Sa4		Sd6	S5			Sm6	Sa5		Sd7	SM6			Sm7	Sa6		Sd8	SM7			S8	Sa7
Narrow, -2Ç		Nm2	Na1		Nd3	NM2			Nm3	Na2		Nd4	NM3			N4	Na3		Nd5	Na4		Nd6	N5			Nm6	Na5		Nd7	NM6			Nm7	Na6		Nd8	NM7			N8	Na7
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

This table can be clarified by grouping intervals together by ordinal size, showing where changes in quality overlap, revealing 120 simple interval names for the comma steps of 41ET.

120 Hunt System Enharmonic Simple Interval Names
P1	Nm2 L1	Sm2	m2	Lm2	Wm2 NM2	SM2	M2	LM2	Sm3	m3	Lm3	Wm3 NM3	SM3	M3	LM3	S4	P4	L4	Na4 Sd5	Sa4 d5	a4 Ld5	La4 Wd5	S5	P5	L5	Sm6	m6	Lm6	Wm6 NM6	SM6	M6	LM6	Sm7	m7	Lm7	Wm7 NM7	SM7	M7	LM7	S8 WM7	P8
				Nd3	Sd3	d3	Ld3	Wd3 Nm3	Na2 WM2	Sa2	a2	La2	Wa2					Nd5	W4			N5	Wa4					Nd7	Sd7	d7	Ld7	Wd7 Nm7	Na6 WM6	Sa6	a6	La6	Wa6
Ld2	Wd2	W1 Na1	Sa1	a1	La1	Wa1					Nd4	Sd4	d4	Ld4	Wd4 N4	Na3 NM3	Sa3	a3	La3	Wa3	Nd6	Sd6	d6	Ld6	Wd6 Wm6	W5 Na5	Sa5	a5	La5	Wa5					Nd8	Sd8	d8	Ld8	Wd8 N8	Na7	Sa7
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

We can compare this table of enharmonic interval names with the table of 26 traditional simple intervals, with empty fields added to show similarity of structure.

26 Traditional Enharmonic Simple Interval Names
P1	m2	M2	m3	M3	P4	a4	P5	m6	M6	m7	M7	P8
		d3	a2			d5			d7	a6
d2	a1			d4	a3		d6	a5			d8	a7
0	1	2	3	4	5	6	7	8	9	10	11	12

The larger chart is an expanded version of the smaller one, having comma shifted interval families for of each traditional interval. With two positively comma shifted names (Large and Wide) and two negatively comma shifted names (Small anb Narrow) in each of the 26 interval families, one would expect to find not 120 enharmonic names, but rather 26 × 5 = 130. The missing 10 names consist of two versions each of the unison and octave which are outside of the octave boundaries, and three versions each of the diminshed second, augmented seventh which also fall outside of the boundaries. Of these 120 names, the 52 shown in the top row of the large table are preferred names. These preferred interval names are shown with 52 preferred Hunt system pitch names below.

Hunt System 52 Preferred Interval and Pitch Names
Preferred Interval Name	P1	Nm2 L1	Sm2	m2	Lm2	Wm2 NM2	SM2	M2	LM2	Sm3	m3	Lm3	Wm3 NM3	SM3	M3	LM3	S4	P4	L4	Na4 Sd5	Sa4 d5	a4 Ld5	La4 Wd5	S5	P5	L5	Sm6	m6	Lm6	Wm6 NM6	SM6	M6	LM6	Sm7	m7	Lm7	Wm7 NM7	SM7	M7	LM7	S8 WM7	P8
Preferred Pitch Name	D	≈ E + D	∼ E	E	+ E	‡ E ≈ E	∼ E	E	+ E	∼ F	F	+ F	‡ F ≈ F	∼ F	F	+ F	∼ G	G	+ G	≈ G ∼ A	∼ G A	G + A	+ G ‡ A	∼ A	A	+ A	∼ B	B	+ B	‡ B ≈ B	∼ B	B	+ B	∼ C	C	+ C	‡ C ≈ C	∼ C	C	+ C	∼ D ‡ C	D
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

The scale centered on D clearly shows relationships between interval comma shift types and pitch comma shifts. The table below summarizes these relationships.

Notice that interval comma shift types refer directly to size while pitch comma shifts refer directly to position. This is correct usage of pitch and interval modifiers according to Western tradition. Other approaches to interval naming are in error when they employ the modifiers sub, super, acute, or grave to the names of intervals. As discussed in Chapter 1 of this text, the prefixes super and sub mean above and below respectively. In traditional Western music these prefixes have never been used to refer to the sizes of intervals; they have only been used to refer to positions of scale degrees, such as the subtonic (below the tonic) and the supertonic (above the tonic). A pitch can be above or below another pitch, but interval names have never been above or below anything. Intervals are distances between pitches, and modifiers for intervals must refer directly to the size of the interval. The traditional interval qualities Major, minor, diminished, and augmented refer directly to relative sizes, not positions. The Hunt comma shift types Narrow, Small, Large and Wide maintain the precedent of Western tradition by referring directly to size, not position. This is an important point, which ties the Hunt system in with Western tradition and sets it apart from other attempts to codify intervals which are seriously flawed.

The comma shift type is not the last interval modifier in the Hunt system. Every interval is further defined by a JND intonation quality, of which there are nine possible versions for each interval, from triply diminished to triply augmented, specifying distinctions between interval sizes in single JND units. This would appear to allow for 41 × 9 = 369 intervals per octave; however, four of these intonations result from enharmonic inflections, so that the total is actually 41 × 5 = 205 intervals per octave. Although an infinite number of tones may be mapped to this structure, a basic set of perfect intervals serves as a default expected standard for good intonation. For each of the categorical 41ET intervals, there is a basic prime limit harmonic interval which is taken as the perfect intonation for that interval. The top row from the large table above shows the most common interval names. Adding this row of names to tables of preferred pitch names and basic 3-Limit, 5-Limit, 7-Limit, 11-Limit, and 13-Limit intervals reveals clear and meaningful relationships between Hunt pitch names, interval names, and harmonic limits. Each of the following tables shows the basic intervals of a harmonic limit, spelled starting from D. All remarks below concerning JND intonation quality assume the unshifted, uninflected pitch D as the lower pitch of each interval.

The 3-Limit generates all the standard minor, Major, perfect, augmented and diminished intervals of the 13-tone Pythagorean gamut — the unshifted and uninflected intervals to which all shifted and inflected intervals are compared. For example, the interval 243:256 is a minor second m2, spelled D : E. The interval 8:9 is a Major second M2, spelled D : E, etc. The blue highlights appearing at the center of the large field of JND inflections are naturals () showing that these intervals do not require JND adjustments from the 41ET circle of fifths.

13 basic 3-Limit Intervals
Ratio		1 1			256 243				9 8			32 27				81 64			4 3			729 512	1024 729			3 2			128 81				27 16			16 9				243 128			2 1
Interval		P1			m2				M2			m3				M3			P4			d5	a4			P5			m6				M6			m7				M7			P8
Pitch		D			E				E			F				F			G			A	G			A			B				B			C				C			D
	aa a P d dd	2 1 0	7 6 5 4 3	12 11 10 9 8	17 16 15 14 13	22 21 20 19 18	27 26 25 24 23	32 31 30 29 28	37 36 35 34 33	42 41 40 39 38	47 46 45 44 43	52 51 50 49 48	57 56 55 54 53	62 61 60 59 58	67 66 65 64 63	72 71 70 69 68	77 76 75 74 73	82 81 80 79 78	87 86 85 84 83	92 91 90 89 88	97 96 95 94 93	102 101 100 99 98	107 106 105 104 103	112 111 110 109 108	117 116 115 114 113	122 121 120 119 118	127 126 125 124 123	132 131 130 129 128	137 136 135 134 133	142 141 140 139 138	147 146 145 144 143	152 151 150 149 148	157 156 155 154 153	162 161 160 159 158	167 166 165 164 163	172 171 170 169 168	177 176 175 174 173	182 181 180 179 178	187 186 185 184 183	192 191 190 189 188	197 196 195 194 193	202 201 200 199 198	205 204 203
Ç		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

Because perfect intonations of the 3-Limit intervals all lie directly on the 41ET circle of fifths, the intonation qualities can be shown directly in the table above next to the pitch inflections. Higher harmonic limits require the addition or subtraction of JNDs to fine tune intervals, such that intonation qualities cannot easily be shown directly with tables of the higher limits; therefore, an example of possible intonation qualities will be given for each of the higher harmonic limits. An example of possible intonation qualities for the perfect fourth P4 is shown in the table below.

Possible JND Intonation Qualities of the Perfect Fourth (P4)	dd.P4	d.P4	m.P4	P.P4 or P4	M.P4	a.P4	aa.P4
	D:G	D:G	D:G	D:G	D:G	D:G	D:G

Because it is possible for the Perfect intonation of an interval to require an inflection other than the natural, as well as the fact that the lower pitch may also be inflected, augmentations and diminishments are allowed to continue multiplying in either direction, as will be shown for intervals of higher harmonic limits.

Basic 5-Limit intervals introduce Large minor and Small Major intervals. The Large minors have perfect intonation when they are made smaller by one JND, and the Small Majors have perfect intonation when they are made larger by one JND. For example, the interval 15:16 is a Large minor second Lm2, spelled D : + E, and the interval 4:5 is a Small Major third SM3, spelled D : ∼ F. The 5-Limit also introduces comma shifted versions of the perfect, diminished and augmented intervals, as shown below.

14 Basic 5-Limit Intervals
Ratio	1 1	81 80			16 15		10 9					6 5		5 4					27 20		45 32	64 45		40 27					8 5		5 3					9 5		15 8			160 81	2 1
Interval	P1	L1			Lm2		SM2					Lm3		SM3					L4		Sa4	Ld5		S5					Lm6		SM6					Lm7		SM7			S8	P8
Pitch	D	+ D			+ E		∼ E					+ F		∼ F					+ G		∼ G	+ A		∼ A					+ B		∼ B					+ C		∼ C			∼ D	D
	2 1 0	7 6 5 4 3	12 11 10 9 8	17 16 15 14 13	22 21 20 19 18	27 26 25 24 23	32 31 30 29 28	37 36 35 34 33	42 41 40 39 38	47 46 45 44 43	52 51 50 49 48	57 56 55 54 53	62 61 60 59 58	67 66 65 64 63	72 71 70 69 68	77 76 75 74 73	82 81 80 79 78	87 86 85 84 83	92 91 90 89 88	97 96 95 94 93	102 101 100 99 98	107 106 105 104 103	112 111 110 109 108	117 116 115 114 113	122 121 120 119 118	127 126 125 124 123	132 131 130 129 128	137 136 135 134 133	142 141 140 139 138	147 146 145 144 143	152 151 150 149 148	157 156 155 154 153	162 161 160 159 158	167 166 165 164 163	172 171 170 169 168	177 176 175 174 173	182 181 180 179 178	187 186 185 184 183	192 191 190 189 188	197 196 195 194 193	202 201 200 199 198	205 204 203
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

Possible intonation qualities for the Small Major third SM3 are shown in the table below.

Possible JND Intonation Qualities of the Small Major third (SM3)	ddd.SM3	dd.SM3	d.SM3	m.SM3	P.SM3 or SM3	M.SM3	a.SM3
	D:∼ F	D:∼ F	D:∼ F	D:∼ F	D:∼ F	D:∼ F	D:∼ F

Basic 7-Limit intervals introduce Small minor and Large Major intervals. The Small minors have perfect intonation when they are made larger by one JND, and the Large Majors have perfect intonation when they are made smaller by one JND. For example, the interval 27:28 is a Small minor second Sm2, spelled D : ∼ E, and the interval 7:8 is a Large Major second LM2, spelled D : + E. The 7-Limit also introduces comma shifted versions of the perfect intervals, as well as narrow and wide intervals near the octave boundaries, as shown below.

12 Basic 7-Limit Intervals
Ratio	1 1	64 63	28 27						8 7	7 6						9 7	21 16									32 21	14 9						12 7	7 4						27 14	63 32	2 1
interval	P1	Nm2	Sm2						LM2	Sm3						LM3	S4									L5	Sm6						LM6	Sm7						LM7	WM7	P8
Pitch	D	≈ E	∼ E						+ E	∼ F						+ F	∼ G									+ A	∼ B						+ B	∼ C						+ C	‡ C	D
	2 1 0	7 6 5 4 3	12 11 10 9 8	17 16 15 14 13	22 21 20 19 18	27 26 25 24 23	32 31 30 29 28	37 36 35 34 33	42 41 40 39 38	47 46 45 44 43	52 51 50 49 48	57 56 55 54 53	62 61 60 59 58	67 66 65 64 63	72 71 70 69 68	77 76 75 74 73	82 81 80 79 78	87 86 85 84 83	92 91 90 89 88	97 96 95 94 93	102 101 100 99 98	107 106 105 104 103	112 111 110 109 108	117 116 115 114 113	122 121 120 119 118	127 126 125 124 123	132 131 130 129 128	137 136 135 134 133	142 141 140 139 138	147 146 145 144 143	152 151 150 149 148	157 156 155 154 153	162 161 160 159 158	167 166 165 164 163	172 171 170 169 168	177 176 175 174 173	182 181 180 179 178	187 186 185 184 183	192 191 190 189 188	197 196 195 194 193	202 201 200 199 198	205 204 203
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

Possible intonation qualities for the Large Major second LM2 are shown in the table below.

Possible JND Intonation Qualities of the Large Major second (LM2)	d.LM2	m.LM2	P.LM2 or LM2	M.LM2	a.LM2	aa.LM2	aaa.LM2
	D:+ E	D:+ E	D:+ E	D:+ E	D:+ E	D:+ E	D:+ E

Basic intervals of the 11-Limit and 13-Limit introduce Wide minor and Narrow Major intervals which lie in the same 41ET comma zone, two JNDs apart. The Wide minors have perfect intonation when they are made smaller by one JND, and the Narrow Majors have perfect intonation when they are made larger by one JND. For example, the interval 11:12 is a Narrow Major second NM2, spelled D : ≈ E, and the interval 12:13 is a Wide Minor second Wm2, spelled D : ‡ E. The 11-Limit and 13-Limit also introduce doubly comma shifted versions of the augmented and diminished intervals, as shown below.

6 Basic 11-Limit Intervals
Ratio	1 1					12 11							11 9							11 8			16 11							18 11							11 6					2 1
Interval	P1					NM2							Wm3							Sd5			La4							NM6							Wm7					P8
Pitch	D					≈ E							‡ F							∼ A			+ G							≈ B							‡ C					D
	2 1 0	7 6 5 4 3	12 11 10 9 8	17 16 15 14 13	22 21 20 19 18	27 26 25 24 23	32 31 30 29 28	37 36 35 34 33	42 41 40 39 38	47 46 45 44 43	52 51 50 49 48	57 56 55 54 53	62 61 60 59 58	67 66 65 64 63	72 71 70 69 68	77 76 75 74 73	82 81 80 79 78	87 86 85 84 83	92 91 90 89 88	97 96 95 94 93	102 101 100 99 98	107 106 105 104 103	112 111 110 109 108	117 116 115 114 113	122 121 120 119 118	127 126 125 124 123	132 131 130 129 128	137 136 135 134 133	142 141 140 139 138	147 146 145 144 143	152 151 150 149 148	157 156 155 154 153	162 161 160 159 158	167 166 165 164 163	172 171 170 169 168	177 176 175 174 173	182 181 180 179 178	187 186 185 184 183	192 191 190 189 188	197 196 195 194 193	202 201 200 199 198	205 204 203
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41
6 Basic 13-Limit Intervals
Ratio	1 1					13 12							16 13							18 13			13 9							13 8							24 13					2 1
Interval	P1					Wm2							NM3							Na4			Wd5							Wm6							NM7					P8
Pitch	D					‡ E							≈ F							≈ G			‡ A							‡ B							≈ C					D
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

The basic 3-Limit, 5-Limit, 7-Limit, 11-Limit, and 13-Limit intervals are summarized in the table below, showing relationships between interval names and intonations, pitch names and inflections, and harmonic ratios. NOTE: Ratios are given in the tables above; they are matched in this table by 205 degree and color.

Hunt System 41Ç × 5J = 205ET Scale
P I N	P1	Nm2 L1	Sm2	m2	Lm2	Wm2 NM2	SM2	M2	LM2	Sm3	m3	Lm3	Wm3 NM3	SM3	M3	LM3	S4	P4	L4	Na4 Sd5	Sa4 d5	a4 Ld5	La4 Wd5	S5	P5	L5	Sm6	m6	Lm6	Wm6 NM6	SM6	M6	LM6	Sm7	m7	Lm7	Wm7 NM7	SM7	M7	LM7	S8 WM7	P8
P P N	D	≈ E + D	∼ E	E	+ E	‡ E ≈ E	∼ E	E	+ E	∼ F	F	+ F	‡ F ≈ F	∼ F	F	+ F	∼ G	G	+ G	≈ G ∼ A	∼ G A	G + A	+ G ‡ A	∼ A	A	+ A	∼ B	B	+ B	‡ B ≈ B	∼ B	B	+ B	∼ C	C	+ C	‡ C ≈ C	∼ C	C	+ C	∼ D ‡ C	D
	2 1 0	7 6 5 4 3	12 11 10 9 8	17 16 15 14 13	22 21 20 19 18	27 26 25 24 23	32 31 30 29 28	37 36 35 34 33	42 41 40 39 38	47 46 45 44 43	52 51 50 49 48	57 56 55 54 53	62 61 60 59 58	67 66 65 64 63	72 71 70 69 68	77 76 75 74 73	82 81 80 79 78	87 86 85 84 83	92 91 90 89 88	97 96 95 94 93	102 101 100 99 98	107 106 105 104 103	112 111 110 109 108	117 116 115 114 113	122 121 120 119 118	127 126 125 124 123	132 131 130 129 128	137 136 135 134 133	142 141 140 139 138	147 146 145 144 143	152 151 150 149 148	157 156 155 154 153	162 161 160 159 158	167 166 165 164 163	172 171 170 169 168	177 176 175 174 173	182 181 180 179 178	187 186 185 184 183	192 191 190 189 188	197 196 195 194 193	202 201 200 199 198	205 204 203
Ç	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41

The top row is labeled PIN to mean Preferrred Interval Name, and the second row PPN, meaning Preferrred Pitch Name. The third row gives a field of JNDs, showing the inlection of the upper pitch of each interval (when the lower pitch has a natural inflection) which is required to acheive perfect intonation, meaning with the "best tuning" of that interval, which in many cases means a tuning in which beating is nearly eliminated. The upper pitch of any interval fine tuned to a perfect JND intonation quality requires either a natural (), sharp (), or flat () on the upper pitch. In other words, perfect JND qualities are defined by intervals which are either uninflected, or inflected one JND larger or one JND smaller.

These basic intervals are arranged into four columns below showing harmonic ratios and interval names, where the leftmost box in each row shows the number of comma steps in each interval, and the rightmost box in each column shows the inflection required to achieve a Perfect JND intonation quality. The four columns are navigated in a zig-zag pattern like a board game, in order to show inversional relationships between intervals. The intervals sound below with C as the lower pitch

START ↓ 0 P1 1:1   1 L1 Nm2 80:81 63:64   2 Sm2 27:28 3 m2 243:256   4 Lm2 15:16 5 Wm2 NM2 12:13 11:12 6 SM2 9:10 7 M2 8:9   8 LM2 7:8 9 Sm3 6:7 10 m3 27:32   11 Lm3 5:6 →	→ 20 Sa4 d5 32:45 729:1024   19 Na4 Sd5 13:18 8:11 18 L4 20:27 17 P4 3:4   16 S4 16:21 15 LM3 7:9 14 M3 64:81   13 SM3 4:5 12 NM3 Wm3 13:16 9:11 ↑	↓ 21 Ld5 a4 45:64 512:729   22 Wd5 La4 9:13 11:16 23 S5 27:40 24 P5 2:3   25 L5 21:32 26 Sm6 9:14 27 m6 81:128   28 Lm6 5:8 29 Wm6 NM6 8:13 11:18 →	END ↑ 41 P8 1:2   40 S8 WM7 81:160 32:63   39 LM7 14:27 38 M7 128:243   37 SM7 8:15 36 NM7 Wm7 13:24 6:11 35 Lm7 5:9 34 m7 9:16   33 Sm7 4:7 32 LM6 7:12 31 M6 16:27   30 SM6 3:5 ↑

The two outer columns are inversional, and the two inner columns are inversional. The number of comma steps in each pair of inversional intervals adds to 41, and the inflections invert from sharp to flat and vice versa. For example, from the inner columns, the Small fourth (S4) is 16Ç and requires a sharp inflection, while the Large fifth (L5) is 25Ç and requires a flat inflection. The comma step sizes add to 41 (16 + 25 = 41). Or looking at the outer columns, the Large Major second (LM2) is 8Ç and requires a flat inflection, while the Small minor seventh (Sm7) is 33Ç and requires a sharp inflection, and the comma step sizes add to 41 (7 + 34 = 41).

To learn which inflections are required for perfect intonation of the basic harmonic intervals, the following tables are helpful. The first table shows all intervals which lie directly on the central natural 41ET chain with no JND fine tuning needed. This includes the 3-Limit 13-tone Pythagorean chromatic gamut and the 7-Limit comma 63:64 (comma of Archytas or average comma).

Unshifted Basic Intervals*
Size	3-Limit				7-Limit		Inflection
1							±0J to
2	m2	243:256	M2	8:9	Nm2*	63:64
3	m3	27:32	M3	64:81
4	P4	3:4	a4	729:1024
5	d5	512:729	P5	2:3
6	m6	81:128	M6	16:27
7	m7	9:16	M7	128:243
8

* The Nm2 is an exception to the set of unshifted basic intervals.

The next table shows all intervals which span adjacent chains, requiring 1 JND widening of the interval to acheive perfect intonation. This includes all the Small and Narrow intervals, with two exceptions. The 5-Limit Small Major, 7-Limit Small minor, 11-Limit and 13-Limit Narrow Major intervals are included, with the exclusion of the Sd5 and Nm2, and the inclusion of the La4.

Small and Narrow Basic Intervals*
Size	5-Limit		7-Limit		11-Limit		13-Limit		Inflection
1									+1J to
2	SM2	9:10	Sm2	27:28	NM2	11:12
3	SM3	4:5	Sm3	6:7			NM3	13:16
4	Sa4	32:45	S4	16:21	La4*	11:16	Na4	13:18
5	S5	27:40
6	SM6	3:5	Sm6	9:14	NM6	11:18
7	SM7	8:15	Sm7	4:7			NM7	13:24
8	S8	81:160

* The La4 is an exception to the set of Small and Narrow basic intervals.

The last table shows all intervals which span adjacent chains, requiring 1 JND narrowing of the interval to acheive perfect intonation. This includes all the Large and Wide intervals, with one exception. The 5-Limit Large Minor, 7-Limit Large Major, 11-Limit and 13-Limit Wide minor intervals are included, with the exclusion of the La4, and the inclusion of the Sd5.

Large and Wide Basic Intervals*
Size	5-Limit		7-Limit		11-Limit		13-Limit		Inflection
1	L1	80:81							-1J to
2	Lm2	15:16	LM2	7:8			Wm2	12:13
3	Lm3	5:6	LM3	7:9	Wm3	9:11
4	L4	20:27
5	Ld5	45:64	L5	21:32	Sd5*	8:11	Wd5	9:13
6	Lm6	5:8	LM6	7:12			Wm6	8:13
7	Lm7	5:9	LM7	14:27	Wm7	6:11
8

* The Sd5 is an exception to the set of Large and Wide basic intervals.

The table below summarizes the relationships between comma shift types, limits, and JND intonation qualities for the basic harmonic intervals of the Hunt system.

Comma Shift Types	Limits and Qualities	Included Exception	Perfect Intonation Inflection
Unshifted	M,m,P,a,d Nm2	Nm2	±0J, to
Small and Narrow	SM Sm NM NM	La4	+1J, to
Large and Wide	Lm LM Wm Wm	Sd5	-1J, to

Correct Interval Spelling & Naming

In the Hunt system, basic rational intervals derived in different ways can be spelled correctly, named correctly and tuned correctly. For example, consider the interval 16:25. How should this interval be spelled, and what should it be called? The Hunt system gives the following answers.

To find the derivation of any rational interval, the first task is to find its prime factorization. 25 breaks down to 5², or 5 × 5. So, we consider the harmonic interval 4:5. The interval 4:5 spelled from C spans up to sub E (C : E) which is a Major third, comma shifted once smaller, making it a small Major third (SM3). This is one of the basic categorical intervals of the system, as shown in the table.

13	SM3	4:5

Since 25 is 5 × 5, what we know about 4:5 gives us everything we need to correctly spell and name 16:25, because it is two 4:5 SM3 intervals stacked one on top of the other. Correct interval spelling of the second stacked 4:5 gives us sub E up to grave G-sharp ( E : G), so the interval 16:25 spans from the C up to the grave G-sharp (C : G). This is an augmented fifth which is comma shifted twice smaller, giving us a Narrow augmented fifth (Na5). Unlike the SM3, the Na5 is not one of the basic categorical intervals of the system. It is enharmonic with a Small minor sixth and a Wide fifth, as shown in the table below.

Partial 41ET Interval Table
P5	L5 Nm6	Sm6	m6	Lm6	Wm6 NM6	SM6
				Nd7	Sd7	d7
Ld6	Wd6	W5 Na5	Sa5	a5	La5	Wa5

So the Na5 will fall near the categorical interval Sm6.

26	Sm6	9:14

SM3 and Na5 are partial names for the intervals we have looked at. Naming the intervals is not complete until the JND intonation quality is also determined. Because the SM3 is a basic categorical interval in the system, we can go to the table.

13	SM3	4:5

The sharp in the table tells us that for the intonation quality to be perfect, the interval requires a 1 JND positive shift, so the 4:5 interval becomes C spans up to sharp sub E (C : E), a Perfect Small Major third, (P.SM3). The Na5 will be two such intervals, so that there must be 2 JND positive shifts, making 16:25 span from C up to doublesharp grave G-sharp (C : G), distinguishing it from the Sm6. The Na5 would be placed in the categorical table as shown below.

26	Na5 Sm6	16:25 9:14

With the correct JND quality, the interval 16:25 would be called a Perfect Narrow augmented fifth, (P.Na5).

Mapping Intervals by Calculation

If an interval spelling is not known, or the interval is irrational or very complex, a few equations can be useful for determining a possible spelling and name for the interval. What is needed is a way to first find the interval category, and then the JND intonation quality. A cent value is a good place to start when determining what an interval is going to sound like, since it directly relates to 12ET. The equation below determines the cent value, where x is an interval expressed as a decimal value or a ratio a:b or b/a where a < b and 1 ≤ a:b ≥ 2. The three equations below will give identical results.

Cents Mapping
1200*Log2(x)	1200*Log(x)/log(2)	3986.314*Log(x)
Type in a decimal value or a ratio. Interval:    ¢

In the Hunt system, the interval category is a step of 41ET, where each step is treated as a comma (Ç). The equation used to map to 41ET is similar to that used to find the cent value, only the results should be rounded.

41ET Step (Ç) Mapping
41*Log2(x)	41*Log(x)/log(2)	136.2*Log(x)
Type in a decimal value or a ratio. Interval:    Ç

Next we find the JND intonation quality. There are many ways to do this. One relatively simple way is to map directly to 205ET, rounding the results. The JND quality is given by the last digit in the answer, as shown in the table.

JND (J) Mapping: 205ET Method
205*Log2(x)	205*Log(x)/log(2)			681*Log(x)
Last digit in result	3 or 8	4 or 9	0 or 5	1 or 6	2 or 7
JND	-2 JND	-1 JND	±0 JND	+1 JND	+2 JND
Type in a decimal value or a ratio. Interval:    205 Step:    J

All of these calculations can be put together with a lookup table giving the name of the 41ET scale step. Note that this name will be a default basic interval name from the table above, which does not take into account the spelling of the interval in letters.

Hunt System Interval Calculator
Type in a decimal value or a ratio. Interval:    ¢ = Ç   J

NEXT: Online Hunt System Calculators