Glossary
Set Class
Glossary
Set Theory
A set class is an unordered collection of notes without regard to which octave the notes are in, or what order they are played in, reduced to its prime form by transposition. Every possible collection of 3 to 9 distinct notes (whether considered a scale or chord or both) is included on the Set Classes page.
Set classes in Harmonious are given clock diagrams for icons. Black and white clock diagrams collect two or more transpositions (usually twelve, labeled “×12”) into a single diagram and a single page. (See Grouping Clocks for a detailed tutorial that explains these concepts, complete with animation and audio examples.)
There are 3,938 possible 3 to 9-note chords or scales, or 336 set classes in prime form, or 208 assigned a Forte number, and Harmonious has individual pages for all of them. Set classes are grouped into 115 rows by involution and complementarity (since complementary set classes have the same Lewin-Quinn FC-components), and labeled in vertical text on the sides with their Forte number.
The tables divide up the set classes by cardinality, the first component of the Forte number. Allen Forte’s system for numbering the set classes demonstrates an important pattern inherent in the structure of 12-TET. The number of 3-note set classes is 12, and 9-note set classes, 12. (3 + 9 = 12.) The number of 4-note set classes is 29, and 8-note set classes, 29. (4 + 8 = 12.) The number of 5-note set classes is 38, and 7-note set classes, 38. (5 + 7 = 12.) The number of 6-note set classes is 50. (6 + 6 = 12.)
Tap or click an icon of a clock to navigate to a page with details about the set class. For example the page 7-35, (013568t), Diatonic represents the set of all twelve transpositions of the diatonic (or major) scale. (Simply scroll down on that page to where the transpositions are listed.) The first part, “7-35” is the the Forte number of the set class: seven for the cardinality or number of distinct pitch classes in a given diatonic scale. The next number, 35, was assigned by Allen Forte (1977) and is basically arbitrary. The number in parentheses is the prime form, or a transposable, numerical representation of the clock diagram, reduced to its lowest (numerical) form. The label in italics, Diatonic, is for reference, but a given set class may have one or more labels or functions depending on the inversion.
The rows of a given table can be sorted by: evenness (labeled Distance, referring to the distance to the perfectly even C-note chord for cardinality C); FCi or Fourier Component, where i refers to the interval class, one through six (see Lewin-Quinn FC-components); Forte number, the labels in vertical text on the side, just tap or click on label Set Class or Complement. Since Allen Forte numbered complementary set classes with matching values for their second components, the outside columns always stay sorted together. Finally, tapping or clicking the same column heading more than once will toggle the sort order between ascending and descending.
See the Set Classes index.
Set Theory
A set class is an unordered collection of notes without regard to which octave the notes are in, or what order they are played in, reduced to its prime form by transposition. Every possible collection of 3 to 9 distinct notes (whether considered a scale or chord or both) is included on the Set Classes page.
Set classes in Harmonious are given clock diagrams for icons. Black and white clock diagrams collect two or more transpositions (usually twelve, labeled “×12”) into a single diagram and a single page. (See Grouping Clocks for a detailed tutorial that explains these concepts, complete with animation and audio examples.)
There are 3,938 possible 3 to 9-note chords or scales, or 336 set classes in prime form, or 208 assigned a Forte number, and Harmonious has individual pages for all of them. Set classes are grouped into 115 rows by involution and complementarity (since complementary set classes have the same Lewin-Quinn FC-components), and labeled in vertical text on the sides with their Forte number.
The tables divide up the set classes by cardinality, the first component of the Forte number. Allen Forte’s system for numbering the set classes demonstrates an important pattern inherent in the structure of 12-TET. The number of 3-note set classes is 12, and 9-note set classes, 12. (3 + 9 = 12.) The number of 4-note set classes is 29, and 8-note set classes, 29. (4 + 8 = 12.) The number of 5-note set classes is 38, and 7-note set classes, 38. (5 + 7 = 12.) The number of 6-note set classes is 50. (6 + 6 = 12.)
Tap or click an icon of a clock to navigate to a page with details about the set class. For example the page 7-35, (013568t), Diatonic represents the set of all twelve transpositions of the diatonic (or major) scale. (Simply scroll down on that page to where the transpositions are listed.) The first part, “7-35” is the the Forte number of the set class: seven for the cardinality or number of distinct pitch classes in a given diatonic scale. The next number, 35, was assigned by Allen Forte (1977) and is basically arbitrary. The number in parentheses is the prime form, or a transposable, numerical representation of the clock diagram, reduced to its lowest (numerical) form. The label in italics, Diatonic, is for reference, but a given set class may have one or more labels or functions depending on the inversion.
The rows of a given table can be sorted by: evenness (labeled Distance, referring to the distance to the perfectly even C-note chord for cardinality C); FCi or Fourier Component, where i refers to the interval class, one through six (see Lewin-Quinn FC-components); Forte number, the labels in vertical text on the side, just tap or click on label Set Class or Complement. Since Allen Forte numbered complementary set classes with matching values for their second components, the outside columns always stay sorted together. Finally, tapping or clicking the same column heading more than once will toggle the sort order between ascending and descending.
See the Set Classes index.