In musical set theory

Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

, a Forte number is the pair of number

Number

A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s Allen Forte

Allen Forte

Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...

 assigned to the prime form of each pitch class

Pitch class

In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

 set

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects...

 of three or more members in The Structure of Atonal Music (1977, ISBN 0300021208). The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.
The prime form is the most compact (i.e., leftwards packed) of either the normal form of a set or its inversion

Inversion (music)

In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...

. The normal form of a set is that which is transposed

Transposition (music)

In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...

 so as to be most compact. For example, a (second inversion) major chord

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...

 contains the pitch classes 7, 0, and 4. The normal form would then be 0, 4 and 7. Its (transposed) inversion, which happens to be the minor chord

Minor chord

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

, contains the pitch classes 0, 3, and 7; and is the prime form.
The minor chord, is given Forte # 3-11, indicating that it is the eleventh (11) in Forte's ordering of pitch class sets with three pitches (3). In contrast, the Viennese trichord

Viennese trichord

In music theory, a Viennese trichord , named for the Second Viennese School, is prime form . It has Forte #3-5.As opposed to Hindemith and 037 , "Composers such as Webern .....

, pitch classes 0,1, and 6, is given Forte # 3-5, indicating that it is the fifth (5) in Forte's ordering of pitch class sets with three pitches (3). The normal form of 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...

, such as C major; 0, 2, 4, 5, 7, 9, and 11; is 11, 0, 2, 4, 5, 7, and 9; while its prime form is 0, 1, 3, 5, 6, 8, and 10; and its Forte # is 7-35, indicating that it is the thirty-fifth (35) of the seven member sets (7).

List of Forte Numbers

Forte Number	Pitch-class Set	Interval Vector	Forte Number	Pitch-class Set	Interval Vector
3-1	[0,1,2]	<2,1,0,0,0,0>	9-1	[0,1,2,3,4,5,6,7,8]	<8,7,6,6,6,3>
3-2	[0,1,3]	<1,1,1,0,0,0>	9-2	[0,1,2,3,4,5,6,7,9]	<7,7,7,6,6,3>
3-3	[0,1,4]	<1,0,1,1,0,0>	9-3	[0,1,2,3,4,5,6,8,9]	<7,6,7,7,6,3>
3-4	[0,1,5]	<1,0,0,1,1,0>	9-4	[0,1,2,3,4,5,7,8,9]	<7,6,6,7,7,3>
3-5	[0,1,6]	<1,0,0,0,1,1>	9-5	[0,1,2,3,4,6,7,8,9]	<7,6,6,6,7,4>
3-6	[0,2,4]	<0,2,0,1,0,0>	9-6	[0,1,2,3,4,5,6,8,10]	<6,8,6,7,6,3>
3-7	[0,2,5]	<0,1,1,0,1,0>	9-7	[0,1,2,3,4,5,7,8,10]	<6,7,7,6,7,3>
3-8	[0,2,6]	<0,1,0,1,0,1>	9-8	[0,1,2,3,4,6,7,8,10]	<6,7,6,7,6,4>
3-9	[0,2,7]	<0,1,0,0,2,0>	9-9	[0,1,2,3,5,6,7,8,10]	<6,7,6,6,8,3>
3-10	[0,3,6]	<0,0,2,0,0,1>	9-10	[0,1,2,3,4,6,7,9,10]	<6,6,8,6,6,4>
3-11	[0,3,7]	<0,0,1,1,1,0>	9-11	[0,1,2,3,5,6,7,9,10]	<6,6,7,7,7,3>
3-12	[0,4,8]	<0,0,0,3,0,0>	9-12	[0,1,2,4,5,6,8,9,10]	<6,6,6,9,6,3>
4-1	[0,1,2,3]	<3,2,1,0,0,0>	8-1	[0,1,2,3,4,5,6,7]	<7,6,5,4,4,2>
4-2	[0,1,2,4]	<2,2,1,1,0,0>	8-2	[0,1,2,3,4,5,6,8]	<6,6,5,5,4,2>
4-3	[0,1,3,4]	<2,1,2,1,0,0>	8-3	[0,1,2,3,4,5,6,9]	<6,5,6,5,4,2>
4-4	[0,1,2,5]	<2,1,1,1,1,0>	8-4	[0,1,2,3,4,5,7,8]	<6,5,5,5,5,2>
4-5	[0,1,2,6]	<2,1,0,1,1,1>	8-5	[0,1,2,3,4,6,7,8]	<6,5,4,5,5,3>
4-6	[0,1,2,7]	<2,1,0,0,2,1>	8-6	[0,1,2,3,5,6,7,8]	<6,5,4,4,6,3>
4-7	[0,1,4,5]	<2,0,1,2,1,0>	8-7	[0,1,2,3,4,5,8,9]	<6,4,5,6,5,2>
4-8	[0,1,5,6]	<2,0,0,1,2,1>	8-8	[0,1,2,3,4,7,8,9]	<6,4,4,5,6,3>
4-9	[0,1,6,7]	<2,0,0,0,2,2>	8-9	[0,1,2,3,6,7,8,9]	<6,4,4,4,6,4>
4-10	[0,2,3,5]	<1,2,2,0,1,0>	8-10	[0,2,3,4,5,6,7,9]	<5,6,6,4,5,2>
4-11	[0,1,3,5]	<1,2,1,1,1,0>	8-11	[0,1,2,3,4,5,7,9]	<5,6,5,5,5,2>
4-12	[0,2,3,6]	<1,1,2,1,0,1>	8-12	[0,1,3,4,5,6,7,9]	<5,5,6,5,4,3>
4-13	[0,1,3,6]	<1,1,2,0,1,1>	8-13	[0,1,2,3,4,6,7,9]	<5,5,6,4,5,3>
4-14	[0,2,3,7]	<1,1,1,1,2,0>	8-14	[0,1,2,4,5,6,7,9]	<5,5,5,5,6,2>
4-z15	[0,1,4,6]	<1,1,1,1,1,1>	8-z15	[0,1,2,3,4,6,8,9]	<5,5,5,5,5,3>
4-16	[0,1,5,7]	<1,1,0,1,2,1>	8-16	[0,1,2,3,5,7,8,9]	<5,5,4,5,6,3>
4-17	[0,3,4,7]	<1,0,2,2,1,0>	8-17	[0,1,3,4,5,6,8,9]	<5,4,6,6,5,2>
4-18	[0,1,4,7]	<1,0,2,1,1,1>	8-18	[0,1,2,3,5,6,8,9]	<5,4,6,5,5,3>
4-19	[0,1,4,8]	<1,0,1,3,1,0>	8-19	[0,1,2,4,5,6,8,9]	<5,4,5,7,5,2>
4-20	[0,1,5,8]	<1,0,1,2,2,0>	8-20	[0,1,2,4,5,7,8,9]	<5,4,5,6,6,2>
4-21	[0,2,4,6]	<0,3,0,2,0,1>	8-21	[0,1,2,3,4,6,8,10]	<4,7,4,6,4,3>
4-22	[0,2,4,7]	<0,2,1,1,2,0>	8-22	[0,1,2,3,5,6,8,10]	<4,6,5,5,6,2>
4-23	[0,2,5,7]	<0,2,1,0,3,0>	8-23	[0,1,2,3,5,7,8,10]	<4,6,5,4,7,2>
4-24	[0,2,4,8]	<0,2,0,3,0,1>	8-24	[0,1,2,4,5,6,8,10]	<4,6,4,7,4,3>
4-25	[0,2,6,8]	<0,2,0,2,0,2>	8-25	[0,1,2,4,6,7,8,10]	<4,6,4,6,4,4>
4-26	[0,3,5,8]	<0,1,2,1,2,0>	8-26	[0,1,3,4,5,7,8,10]	<4,5,6,5,6,2>
4-27	[0,2,5,8]	<0,1,2,1,1,1>	8-27	[0,1,2,4,5,7,8,10]	<4,5,6,5,5,3>
4-28	[0,3,6,9]	<0,0,4,0,0,2>	8-28	[0,1,3,4,6,7,9,10]	<4,4,8,4,4,4>
4-z29	[0,1,3,7]	<1,1,1,1,1,1>	8-z29	[0,1,2,3,5,6,7,9]	<5,5,5,5,5,3>
5-1	[0,1,2,3,4]	<4,3,2,1,0,0>	7-1	[0,1,2,3,4,5,6]	<6,5,4,3,2,1>
5-2	[0,1,2,3,5]	<3,3,2,1,1,0>	7-2	[0,1,2,3,4,5,7]	<5,5,4,3,3,1>
5-3	[0,1,2,4,5]	<3,2,2,2,1,0>	7-3	[0,1,2,3,4,5,8]	<5,4,4,4,3,1>
5-4	[0,1,2,3,6]	<3,2,2,1,1,1>	7-4	[0,1,2,3,4,6,7]	<5,4,4,3,3,2>
5-5	[0,1,2,3,7]	<3,2,1,1,2,1>	7-5	[0,1,2,3,5,6,7]	<5,4,3,3,4,2>
5-6	[0,1,2,5,6]	<3,1,1,2,2,1>	7-6	[0,1,2,3,4,7,8]	<5,3,3,4,4,2>
5-7	[0,1,2,6,7]	<3,1,0,1,3,2>	7-7	[0,1,2,3,6,7,8]	<5,3,2,3,5,3>
5-8	[0,2,3,4,6]	<2,3,2,2,0,1>	7-8	[0,2,3,4,5,6,8]	<4,5,4,4,2,2>
5-9	[0,1,2,4,6]	<2,3,1,2,1,1>	7-9	[0,1,2,3,4,6,8]	<4,5,3,4,3,2>
5-10	[0,1,3,4,6]	<2,2,3,1,1,1>	7-10	[0,1,2,3,4,6,9]	<4,4,5,3,3,2>
5-11	[0,2,3,4,7]	<2,2,2,2,2,0>	7-11	[0,1,3,4,5,6,8]	<4,4,4,4,4,1>
5-z12	[0,1,3,5,6]	<2,2,2,1,2,1>	7-z12	[0,1,2,3,4,7,9]	<4,4,4,3,4,2>
5-13	[0,1,2,4,8]	<2,2,1,3,1,1>	7-13	[0,1,2,4,5,6,8]	<4,4,3,5,3,2>
5-14	[0,1,2,5,7]	<2,2,1,1,3,1>	7-14	[0,1,2,3,5,7,8]	<4,4,3,3,5,2>
5-15	[0,1,2,6,8]	<2,2,0,2,2,2>	7-15	[0,1,2,4,6,7,8]	<4,4,2,4,4,3>
5-16	[0,1,3,4,7]	<2,1,3,2,1,1>	7-16	[0,1,2,3,5,6,9]	<4,3,5,4,3,2>
5-z17	[0,1,3,4,8]	<2,1,2,3,2,0>	7-z17	[0,1,2,4,5,6,9]	<4,3,4,5,4,1>
5-z18	[0,1,4,5,7]	<2,1,2,2,2,1>	7-z18	[0,1,4,5,6,7,9]	<4,3,4,4,4,2>
5-19	[0,1,3,6,7]	<2,1,2,1,2,2>	7-19	[0,1,2,3,6,7,9]	<4,3,4,3,4,3>
5-20	[0,1,5,6,8]	<2,1,1,2,3,1>	7-20	[0,1,2,5,6,7,9]	<4,3,3,4,5,2>
5-21	[0,1,4,5,8]	<2,0,2,4,2,0>	7-21	[0,1,2,4,5,8,9]	<4,2,4,6,4,1>
5-22	[0,1,4,7,8]	<2,0,2,3,2,1>	7-22	[0,1,2,5,6,8,9]	<4,2,4,5,4,2>
5-23	[0,2,3,5,7]	<1,3,2,1,3,0>	7-23	[0,2,3,4,5,7,9]	<3,5,4,3,5,1>
5-24	[0,1,3,5,7]	<1,3,1,2,2,1>	7-24	[0,1,2,3,5,7,9]	<3,5,3,4,4,2>
5-25	[0,2,3,5,8]	<1,2,3,1,2,1>	7-25	[0,2,3,4,6,7,9]	<3,4,5,3,4,2>
5-26	[0,2,4,5,8]	<1,2,2,3,1,1>	7-26	[0,1,3,4,5,7,9]	<3,4,4,5,3,2>
5-27	[0,1,3,5,8]	<1,2,2,2,3,0>	7-27	[0,1,2,4,5,7,9]	<3,4,4,4,5,1>
5-28	[0,2,3,6,8]	<1,2,2,2,1,2>	7-28	[0,1,3,5,6,7,9]	<3,4,4,4,3,3>
5-29	[0,1,3,6,8]	<1,2,2,1,3,1>	7-29	[0,1,2,4,6,7,9]	<3,4,4,3,5,2>
5-30	[0,1,4,6,8]	<1,2,1,3,2,1>	7-30	[0,1,2,4,6,8,9]	<3,4,3,5,4,2>
5-31	[0,1,3,6,9]	<1,1,4,1,1,2>	7-31	[0,1,3,4,6,7,9]	<3,3,6,3,3,3>
5-32	[0,1,4,6,9]	<1,1,3,2,2,1>	7-32	[0,1,3,4,6,8,9]	<3,3,5,4,4,2>
5-33	[0,2,4,6,8]	<0,4,0,4,0,2>	7-33	[0,1,2,4,6,8,10]	<2,6,2,6,2,3>
5-34	[0,2,4,6,9]	<0,3,2,2,2,1>	7-34	[0,1,3,4,6,8,10]	<2,5,4,4,4,2>
5-35	[0,2,4,7,9]	<0,3,2,1,4,0>	7-35	[0,1,3,5,6,8,10]	<2,5,4,3,6,1>
5-z36	[0,1,2,4,7]	<2,2,2,1,2,1>	7-z36	[0,1,2,3,5,6,8]	<4,4,4,3,4,2>
5-z37	[0,3,4,5,8]	<2,1,2,3,2,0>	7-z37	[0,1,3,4,5,7,8]	<4,3,4,5,4,1>
5-z38	[0,1,2,5,8]	<2,1,2,2,2,1>	7-z38	[0,1,2,4,5,7,8]	<4,3,4,4,4,2>
6-1	[0,1,2,3,4,5]	<5,4,3,2,1,0>
6-2	[0,1,2,3,4,6]	<4,4,3,2,1,1>
6-z3	[0,1,2,3,5,6]	<4,3,3,2,2,1>
6-z4	[0,1,2,4,5,6]	<4,3,2,3,2,1>
6-5	[0,1,2,3,6,7]	<4,2,2,2,3,2>
6-z6	[0,1,2,5,6,7]	<4,2,1,2,4,2>
6-7	[0,1,2,6,7,8]	<4,2,0,2,4,3>
6-8	[0,2,3,4,5,7]	<3,4,3,2,3,0>
6-9	[0,1,2,3,5,7]	<3,4,2,2,3,1>
6-z10	[0,1,3,4,5,7]	<3,3,3,3,2,1>
6-z11	[0,1,2,4,5,7]	<3,3,3,2,3,1>
6-z12	[0,1,2,4,6,7]	<3,3,2,2,3,2>
6-z13	[0,1,3,4,6,7]	<3,2,4,2,2,2>
6-14	[0,1,3,4,5,8]	<3,2,3,4,3,0>
6-15	[0,1,2,4,5,8]	<3,2,3,4,2,1>
6-16	[0,1,4,5,6,8]	<3,2,2,4,3,1>
6-z17	[0,1,2,4,7,8]	<3,2,2,3,3,2>
6-18	[0,1,2,5,7,8]	<3,2,2,2,4,2>
6-z19	[0,1,3,4,7,8]	<3,1,3,4,3,1>
6-20	[0,1,4,5,8,9]	<3,0,3,6,3,0>
6-21	[0,2,3,4,6,8]	<2,4,2,4,1,2>
6-22	[0,1,2,4,6,8]	<2,4,1,4,2,2>
6-z23	[0,2,3,5,6,8]	<2,3,4,2,2,2>
6-z24	[0,1,3,4,6,8]	<2,3,3,3,3,1>
6-z25	[0,1,3,5,6,8]	<2,3,3,2,4,1>
6-z26	[0,1,3,5,7,8]	<2,3,2,3,4,1>
6-27	[0,1,3,4,6,9]	<2,2,5,2,2,2>
6-z28	[0,1,3,5,6,9]	<2,2,4,3,2,2>
6-z29	[0,2,3,6,7,9]	<2,2,4,2,3,2>
6-30	[0,1,3,6,7,9]	<2,2,4,2,2,3>
6-31	[0,1,4,5,7,9]	<2,2,3,4,3,1>
6-32	[0,2,4,5,7,9]	<1,4,3,2,5,0>
6-33	[0,2,3,5,7,9]	<1,4,3,2,4,1>
6-34	[0,1,3,5,7,9]	<1,4,2,4,2,2>
6-35	[0,2,4,6,8,10]	<0,6,0,6,0,3>
6-z36	[0,1,2,3,4,7]	<4,3,3,2,2,1>
6-z37	[0,1,2,3,4,8]	<4,3,2,3,2,1>
6-z38	[0,1,2,3,7,8]	<4,2,1,2,4,2>
6-z39	[0,2,3,4,5,8]	<3,3,3,3,2,1>
6-z40	[0,1,2,3,5,8]	<3,3,3,2,3,1>
6-z41	[0,1,2,3,6,8]	<3,3,2,2,3,2>
6-z42	[0,1,2,3,6,9]	<3,2,4,2,2,2>
6-z43	[0,1,2,5,6,8]	<3,2,2,3,3,2>
6-z44	[0,1,2,5,6,9]	<3,1,3,4,3,1>
6-z45	[0,2,3,4,6,9]	<2,3,4,2,2,2>
6-z46	[0,1,2,4,6,9]	<2,3,3,3,3,1>
6-z47	[0,1,2,4,7,9]	<2,3,3,2,4,1>
6-z48	[0,1,2,5,7,9]	<2,3,2,3,4,1>
6-z49	[0,1,3,4,7,9]	<2,2,4,3,2,2>
6-z50	[0,1,4,6,7,9]	<2,2,4,2,3,2>

External links

• "All About Set Theory: What is a Forte Number?", JayTomlin.com.