Pitches

The traditional Western system of pitch naming uses two attributes: a letter and an optional accidental.

Traditional System of Pitch Naming	Format		Example
	letter	accidental	spoken	"A-flat"
	A B C D E F G		written	A

The Hunt system extends accidentals to include triples, and adds two optional attributes to indicate precise tuning: a JND inflection and a comma shift.

Hunt System of Pitch Naming	Format				Example
	JND inflection	comma shift	letter	accidental	spoken	"sharp sub A-flat"
		≈ ∼ + ‡	A B C D E F G		written	∼ A

NOTE: To see this page as intended, please use either Safari or IE. The characters ∼ and ≈ should be the same width, but the Firefox browser renders the HTML entity ∼ ( & s i m ; ) at 1½ the length of ≈, which is wrong.

Traditional System

In the traditional system, the letter is one of the first seven letters of the Latin alphabet, and the traditional accidental is one of five signs. Italics in the table headers below indicate that the accidental may or may not be present. The absence of an accidental designates a "natural". 12ET is the present day default tuning of the traditional system. In 12ET, the smallest scale step is a halfstep (h) of 100¢, so changes in pitch are shown below in h increments relative to the naturals.

Traditional System of Pitch Naming	7 letters
	A	B	C	D	E	F	G
	5 accidentals
	written
	spoken	"double flat"	"flat"	"natural"	"sharp"	"double sharp"
	± h, ¢	-2h, -200¢	-1h, -100¢	± 0h, ± 0¢	+1h, +100¢	+2h, +200¢
Examples	letter, accidental
	written	D	A	G or G	C	F
	spoken	"D-double flat"	"A-flat"	"G" or "G-natural"	"C-sharp"	"F-double sharp"
	± h, ¢	D-2h, -200¢	A-1h, -100¢	G ±0 h, ±0¢	C+1h, +100¢	F+2h, +200¢

The example pitches G and F sound exactly the same in 12ET. In modern terminology this is an enharmonic relationship. We can summarize all such relationships by looking at the diatonic naturals on a 12ET tone ruler. The C Major scale is shown below in 12ET.

C Major Scale in 12ET
natural	C		D		E	F		G		A		B	C
h	0	1	2	3	4	5	6	7	8	9	10	11	12

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

Chromatic Transposition of the C Major Scale in 12ET
double sharp, +2h					C		D		E	F		G		A		B	C
sharp, +1h				C		D		E	F		G		A		B	C
natural, ± 0h			C		D		E	F		G		A		B	C
flat, -1h		C		D		E	F		G		A		B	C
double flat, -2h	C		D		E	F		G		A		B	C

The ends of this parallelogram-shaped table can be wrapped around to form a rectangle, showing how the columns actually overlap at the ends. Each of the resulting columns shows a group of enharmonic names for the twelve pitches of 12ET. In all cases but one (A / G), there are three names per pitch. For example, the names B, C, and D all refer to the same pitch in 12ET.

Chromatic Transposition of the C Major Scale in 12ET
double sharp, +2h		B	C		D		E	F		G		A
sharp, +1h	B	C		D		E	F		G		A		B
natural, ± h	C		D		E	F		G		A		B	C
flat, -1h		D		E	F		G		A		B	C
double flat, -2h	D		E	F		G		A		B	C		D

Since the diatonic pattern deals directly with halfsteps in 12ET, it is a convenient way of illustrating these relationships, but the table below having pitches arranged in fifths order is more compact. There are 35 possible pitch names (7 letters × 5 accidentals) in the traditional system.

35 Traditional System Pitch Names
F	C	G	D	A	E	B	F	C	G	D	A
E	B	F	C	G	D	A	E	B	F	C	G
D	A	E	B	F	C	G	D	A	E	B

The shading of pitch names in the table above reflects the fact that not all of these names are used equally.

avoided	less desirable	less commonly used	common

For example, D is commonly used, while Eand C are less commonly used (although they are absolutely necessary for correct spellings of scales and harmonies).

These 35 pitch names correspond to only 12 actual pitches, as shown below in an enharmonic spiral.

35 Traditional System Pitch Names in Scale Order
B	B	C	D	D	E	E	A		G	A	A	B
C	C	D	E	E	F	F	G	G	A	B	B	C
D	D	E	F	F	G	G	A	A	B	C	C	D
0	1	2	3	4	5	6	7	8	9	10	11	12

Hunt System

In the Hunt system, the seven letters are maintained, and the five signs are expanded by one step to include triples. While the terms "sharp" and "flat" used in pitch names of the traditional system are called accidentals, musicians also use these terms to describe the intonation, or fine tuning, of pitches as "a bit higher" or "a bit lower" than one another, in which case they can be called inflections. Both usages are employed in the Hunt system. The accidental corresponds to the traditional concept of shifting by chromatic halfsteps (4Ç = 20J) while the inflection corresponds to small intonational changes of pitch of a single JND (1J). The default tuning of the Hunt system is 41ET (41Ç = 205J). Comma shifts (5J) designate changes in pitch of one step of 41ET. Note that inflections, shifts, and accidentals are all optional, indicated by italics in the table headers below.

Hunt System of Pitch Naming	7 JND inflections
	written
	spoken	"triple flat"	"double flat"	"flat"	"natural"	"sharp"	"double sharp"	"triple sharp"
	± J	-3J	-2J	-1J	±0J	+1J	+2J	+3J
	4 comma shifts
	written	≈		∼		+		‡
	spoken	"grave"		"sub"		"super"		"acute"
	± Ç, J	-2Ç, -10J		-1Ç, -5J		+1Ç, +5J		+2Ç, +10J
	7 letters
	A	B	C	D		E	F	G
	7 accidentals
	written
	spoken	"triple flat"	"double flat"	"flat"	"natural"	"sharp"	"double sharp"	"triple sharp"
	± Ç, J	-12Ç, -60J	-8Ç, -40J	-4Ç, -20J	±0Ç, ±0J	+4Ç, +20J	+8Ç, +40J	+12Ç, +60J
Examples	inflection, shift, letter, accidental
	written	∼ D		+ A		‡ G	≈ C	F
	spoken	"sharp sub D-double flat"		"double sharp super A-flat"		"double flat acute G"	"grave C-sharp"	"flat F-double sharp"
	± Ç, J (¢)	D-9Ç+1J D-44J (D-257.6¢)		A-3Ç+2J A-13J (A-76.1¢)		G+2Ç-2J G+6J (G+46.8¢)	C+2Ç C+8J (C+58.5¢)	F+8Ç-1J F+31J (F+228.3¢)

If we search for enharmonic relationships in the Hunt system via the major scale chromatic transposition approach used above for the traditional system, we find a more subtle situation. Shown below is a C major scale in 41ET.

C Major Scale in 41ET
natural	C							D							E			F							G							A							B			C
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 Transposition of the C Major Scale in 41ET
double sharp, +8Ç																	C							D							E			F							G							A							B			C
sharp, +4Ç													C							D							E			F							G							A							B			C
natural, ± 0Ç									C							D							E			F							G							A							B			C
flat, -4Ç					C							D							E			F							G							A							B			C
double flat, -8Ç	C							D							E			F							G							A							B			C

Wrapping around the ends of this parallelogram-shaped table to form a rectangle and removing the repetition of C, each of the resulting columns shows only one pitch name per column, with some columns left empty.

Chromatic Transposition of the C Major Scale in 41ET
double sharp, +8Ç						B			C							D							E			F							G							A
sharp, +4Ç		B			C							D							E			F							G							A
natural, ± 0Ç	C							D							E			F							G							A							B
flat, -4Ç				D							E			F							G							A							B			C
double flat, -8Ç							E			F							G							A							B			C							D

This illustrates that the 35 traditional pitch names define unique pitches in 41ET, with room to spare for more pitch names. This situation can also be illustrated by expanding the spiral of the traditional system into a larger circle.

Traditional		Hunt
	→

The empty columns in the non-enharmonic tables correspond to the gap shown above in the 35-tone circle. These spaces result from the absence of triple accidentals which are required in 41ET. Below, the staggered-row table is expanded with the shifted diatonic patterns of the triple accidentals. (The widths of some columns are collapsed in the table below to save space).

Chromatic Transposition of the C Major Scale in 41ET
triple sharp, +12Ç																									C							D							E			F							G							A							B			C
double sharp, +8Ç																					C							D							E			F							G							A							B			C
sharp, +4Ç																	C							D							E			F							G							A							B			C
natural, ± 0Ç													C							D							E			F							G							A							B			C
flat, -4Ç									C							D							E			F							G							A							B			C
double flat, -8Ç					C							D							E			F							G							A							B			C
triple flat, -12Ç	C							D							E			F							G							A							B			C

Wrapping the structure around shows how every column is filled, and the triple accidentals form enharmonic relationships.

Chromatic Transposition of the C Major Scale in 41ET
triple sharp, +12Ç			A							B			C							D							E			F							G
double sharp, +8Ç						B			C							D							E			F							G							A
sharp, +4Ç		B			C							D							E			F							G							A
natural, ± 0Ç	C							D							E			F							G							A							B			C
flat, -4Ç				D							E			F							G							A							B			C
double flat, -8Ç							E			F							G							A							B			C							D
triple flat, -12Ç			E			F							G							A							B			C							D

This table can be collapsed into a single column showing 49 unique fifths-based (Pythagorean) names for 41 pitches.

41 Pythagorean Names for Pitches of 41ET
C	B	A E	D	C	B F	E	D	C	B F	E	D	C G	F	E	D	G	F	E	D A	G	F	E	A	G	F	E B	A	G	F C	B	A	G	C	B	A	G D	C	B	A	D	C
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 gap in the 35-tone circle is filled in with new enharmonic triple accidentals, showing how 41ET tuning allows for 49 possible pitch names (7 letters × 7 accidentals), corresponding to 41 actual pitch categories.

This circle is the basis of pitch naming in the Hunt system, but it only represents the combination of letters and accidentals. Pitch names are further modified by comma shifts so that each pitch is part of a group of neighboring related pitches. Both single and double shifting is allowed, so that a single positive shift creates a super version (+) of a pitch, and a double positive shift creates an acute version (‡), while a negative shift creates a sub version (∼), and a double negative shift creates a grave version (≈), creating from each pitch a group of five pitches called a pitch family, as shown for the diatonic naturals below.

Chromatic Transposition of the C Major Scale in 41ET
acute, +2Ç					‡ C							‡ D							‡ E			‡ F							‡ G							‡ A							‡ B
super, +1Ç				+ C							+ D							+ E			+ F							+ G							+ A							+ B
(no shift) ± 0Ç			C							D							E			F							G							A							B
sub, -1Ç		∼ C							∼ D							∼ E			∼ F							∼ G							∼ A							∼ B
grave, -2Ç	≈ C							≈ D							≈ E			≈ F							≈ G							≈ A							≈ B

From this partial chart it can already be seen that comma shifted pitch families result in new enharmonic relationships; two such relationships shown above are + E / ≈ F, and ‡ E / ∼ F. Expanding this table further to include comma shifted sharps and flats will allow an important comparison to the traditional system. Because this table becomes quite large (open it in a separate window here), only a partial table is shown below.

Comma Shifted Chromatically Transposed C Major Scale in 41ET (partial)
acute, +2Ç													‡ C							‡ D							‡ E			‡ F							‡ G
super, +1Ç												+ C							+ D							+ E			+ F							+ G
(no shift) ± 0Ç											C							D							E			F							G
sub, -1Ç										∼ C							∼ D							∼ E			∼ F							∼ G
grave, -2Ç									≈ C							≈ D							≈ E			≈ F							≈ G
acute, +2Ç									‡ C							‡ D							‡ E			‡ F							‡ G
super, +1Ç								+ C							+ D							+ E			+ F							+ G
(no shift) ± 0Ç							C							D							E			F							G
sub, -1Ç						∼ C							∼ D							∼ E			∼ F							∼ G
grave, -2Ç					≈ C							≈ D							≈ E			≈ F							≈ G
acute, +2Ç					‡ C							‡ D							‡ E			‡ F							‡ G
super, +1Ç				+ C							+ D							+ E			+ F							+ G
(no shift) ± 0Ç			C							D							E			F							G
sub, -1Ç		∼ C							∼ D							∼ E			∼ F							∼ G
grave, -2Ç	≈ C							≈ D							≈ E			≈ F							≈ G

From this table it can be seen that pitch families create contiguous groups of overlapping pitch names. Pairs of comma shifted sharps and flats of adjacent letters become enharmonic, such as C / + D, and ∼ C / D, which are clarified versions of the enharmonics of the traditional system. The table below shows how the 35 traditional enharmonics are clarified by comma shifts in the Hunt system.

35 Traditional System Pitch Names F C G D A E B F C G D A E B F C G D A E B F C G D A E B F C G D A E B 35 Traditional Enharmonics in the Hunt System + F + C + G + D + A + E + B + F + C + G + D + A E B F C G D A E B F C G ∼ D ∼ A ∼ E ∼ B ∼ F ∼ C ∼ G ∼ D ∼ A ∼ E ∼ B avoided less desirable less commonly used common

Pitch naming preference conventions still hold in the Hunt system. For example, in both systems, the pitch name of F is an avoided name, although it can be seen that in the Hunt system a positive comma shift (+ F) is required for this pitch to be enharmonic with ∼ D and E. Below, the first column is taken from the tables above for further comparison of the two systems. As can be seen from the 41-tone circle above, an additional enharmonic appears for F, namely B, added in the middle version of the table. Next, doubly shifted extended enharmonics are added to the table.

Traditional Hunt F + F E E D ∼ D	→	Traditional Hunt F + B + F  E   E  D   ∼ D	→	Traditional Hunt ‡ C   F + B + F  E   E  D   ∼ D ≈ C ≈ G

The 49 pitch names are multiplied by 5 possible comma shifts per pitch family, so the number of categorical pitch names becomes 245 = (5 comma shifts × 7 letters × 7 accidentals), although as mentioned before not all possible pitch names in the Hunt system are used; for example, the extended triple accidentals become part of the avoided category. Each of the 41 unique categorical pitch names in the Hunt system may be spelled in five, six, or seven ways. While the pitches of a family can be called siblings, enharmonic relations can be called cousins. The following table shows all enharmonics more compactly than the diatonic tables by using fifths order. Each column in the table below shows all of the enharmonic cousins for each pitch.

245 Hunt System Pitch Names
‡ C	‡ G	‡ D	‡ A	‡ E	‡ B ‡ F	‡ F ‡ C	‡ C ‡ G	‡ G ‡ D	‡ D ‡ A	‡ A ‡ E	‡ E ‡ B	‡ B ‡ F	‡ C	‡ G	‡ D	‡ A	‡ E	‡ B	‡ F	‡ C
+ B + F	+ C	+ G	+ D	+ A	+ E	+ B	+ F	+ C	+ G	+ D	+ A	+ E	+ B	+ F	+ C	+ G	+ D	+ A	+ E	+ B
E	B	F	C	G	D	A	E	B	F	C	G	D	A	E	B	F	C	G	D	A
∼ D	∼ A	∼ E	∼ B	∼ F	∼ C	∼ G	∼ D	∼ A	∼ E	∼ B ∼ F	∼ F ∼ C	∼ C ∼ G	∼ G ∼ D	∼ D ∼ A	∼ A ∼ E	∼ E ∼ B	∼ B ∼ F	∼ C	∼ G	∼ D
≈ C ≈ G	≈ G ≈ D	≈ D ≈ A	≈ A ≈ E	≈ E ≈ B	≈ F ≈ B	≈ C	≈ G	≈ D	≈ A	≈ E	≈ B	≈ F	≈ C	≈ G	≈ D	≈ A	≈ E	≈ B	≈ F	≈ C
‡ G	‡ A	‡ A	‡ E	‡ B	‡ F	‡ C	‡ G	‡ D	‡ A	‡ E	‡ B	‡ F	‡ C	‡ G	‡ D	‡ A	‡ E	‡ B	‡ F
+ F	+ C	+ G	+ D	+ A	+ E	+ B	+ F	+ C	+ G	+ D	+ A	+ E	+ B + F	+ F + C	+ C + G	+ G + D	+ D + A	+ A + E	+ E + B
E	B F	F C	C G	G D	D A	A E	E B	B F	C	G	D	A	E	B	F	C	G	A	A
∼ A	∼ E	∼ B	∼ F	∼ C	∼ G	∼ D	∼ A	∼ E	∼ B	∼ F	∼ C	∼ G	∼ D	∼ A	∼ E	∼ B	∼ F	∼ C	∼ G
≈ G	≈ D	≈ A	≈ E	≈ B	≈ F	≈ C	≈ G	≈ D	≈ A	≈ E	≈ B	≈ F	≈ C	≈ G	≈ D	≈ A	≈ E	≈ B ≈ F	≈ F ≈ C

avoided	less desirable	less commonly used	common

These relationships can also be summarized in an enharmonic super-spiral.

Unshifted pitches are on the center ring, and its cousins are shown in the adjacent fields of each concentric ring in a ray. For example, the enharmonic cousins of D are + E, ∼ C, ‡ B and ‡ F, ≈ B and ≈ F. The D family consists of ≈ D, ∼ D, D, + D, and ‡ D, which may be found in adjacent rings on the spiral spaced every 12 fields clockwise and counterclockwise from D. The table below is organized in scale order, where enharmonic cousins are aligned in each column, showing between five and seven possible names for each pitch category. Comma shifted pitch families are found in diagonal lines from the bottom up; for example, the D family is found diagonally in columns 5 through 9.

Hunt System 245 Enharmonic Names for Pitch Categories of 41ET
≈ A	‡ D	‡ C	‡ B	‡ A ‡ E	‡ D	‡ C	‡ B ‡ F	‡ E	‡ D	‡ C	‡ B ‡ F	‡ E	‡ D	‡ C ‡ G	‡ F	‡ E	‡ D	‡ G	‡ F	‡ E
+ D	+ C	+ B	+ A + E	+ D	+ C	+ B + F	+ E	+ D	+ C	+ B + F	+ E	+ D	+ C + G	+ F	+ E	+ D	+ G	+ F	+ E	+ D + A
C	B	A E	D	C	B F	E	D	C	B F	E	D	C G	F	E	D	G	F	E	D A	G
∼ B	∼ A ∼ E	∼ D	∼ C	∼ B ∼ F	∼ E	∼ D	∼ C	∼ B ∼ F	∼ E	∼ D	∼ C ∼ G	∼ F	∼ E	∼ D	∼ G	∼ F	∼ E	∼ D ∼ A	∼ G	∼ F
≈ A ≈ E	≈ D	≈ C	≈ B ≈ F	≈ E	≈ D	≈ C	≈ B ≈ F	≈ E	≈ D	≈ C ≈ G	≈ F	≈ E	≈ D	≈ G	≈ F	≈ E	≈ D ≈ A	≈ G	≈ F	≈ E
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
‡ D ‡ A	‡ G	‡ F	‡ E	‡ A	‡ G	‡ F	‡ E ‡ B	‡ A	‡ G	‡ F ‡ C	‡ B	‡ A	‡ G	‡ C	‡ B	‡ A	‡ G ‡ D	‡ C	‡ B	‡ A
+ G	+ F	+ E	+ A	+ G	+ F	+ E + B	+ A	+ G	+ F + C	+ B	+ A	+ G	+ C	+ B	+ A	+ G + D	+ C	+ B	+ A	+ D
F	E	A	G	F	E B	A	G	F C	B	A	G	C	B	A	G D	C	B	A	D	C
∼ E	∼ A	∼ G	∼ F	∼ E ∼ B	∼ A	∼ G	∼ F ∼ C	∼ B	∼ A	∼ G	∼ C	∼ B	∼ A	∼ G ∼ D	∼ C	∼ B	∼ A	∼ D	∼ C	∼ B
≈ A	≈ G	≈ F	≈ E ≈ B	≈ A	≈ G	≈ F ≈ C	≈ B	≈ A	≈ G	≈ C	≈ B	≈ A	≈ G ≈ D	≈ C	≈ B	≈ A	≈ D	≈ C	≈ B	≈ A ≈ E
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 collapsed into a single row showing 52 preferred pitch names.

Hunt System 52 Preferred Pitch Names
Preferred Pitch Name	C	+ C	‡ C ≈ C	∼ C	C	+ C	∼ D ‡ C	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
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 names shown here correspond to a chromatic Key center of D, where the chain of fifths spans from A to G. Other chromatic Key centers alter pitch spellings accordingly; for example, a Key center of B shifts the fifths chain in preferrence of D over E. In every case there are 52 preferred names.

Each possible pitch name is fine tunable in JND inflections, allowing seven versions of each pitch category. For example, below are shown all possible inflections of A.

JND Inflections of A
A	A	A	A	A	A	A

The triple inflections are shaded because they are enharmonic. For example, if we consider any two pitches differing by a comma shift, we can show how inflections allow for seamless transitions between adjacent pitch categories, allowing overlapping enharmonic inflections to provide fine control of categorical flexibility when naming pitches. The smooth JND transtion from ∼ A to A is shown below.

Enharmonic JND Inflections
∼ A	∼ A	A ∼ A	A ∼ A	A	A

A scale centered on D using preferred names best shows the symmetry of the Hunt system, revealing clear relationships between pitch names, inflections, and harmonic limits.

Hunt System Preferred Names for Pitches of 41ET
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
	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

These relationships are further clarified in the section concerning intervals.

NEXT: Intervals