Beyond keys: sets, rows and micromodes

Everything in the chapters before assumed a key: a home note, and chords that lean towards it. Much of the music of the 20th century has no key at all, and chord names don’t fit its harmonies. Composers and theorists found other ways to think about it, and Ferrous Lineup has three of them:

• pitch-class sets, a name for every group of notes, chord or not;
• twelve-tone rows, all twelve notes in an order that a whole piece is made of;
• micromodes, the twelve colours of three notes that the composer Francisco Kröpfl taught in Buenos Aires.

Each has a window in the piano roll’s toolbar (Row…, Sets…, Micromodes…), and each can be the key of a song.

Pitch-class sets

Pitch classes. Number the twelve notes from C: C is 0, C♯ is 1, up to B, 11. Octaves and spellings don’t count: every C is 0, and C♯ and D♭ are both 1. A set is a choice of some of the twelve.

Moving a set. Moving every note up n semitones is Tn: C E G moved up 3 is E♭ G B♭. Turning every interval upside down is inverting: C E G, up four semitones and then three, becomes C A♭ F, down four and then three. That is F minor: upside down, a major triad is a minor triad.

Set classes. A set with all its transpositions and inversions is a set class. All 24 major and minor triads are one class. The 4,096 possible sets fall into 224 classes, and each has a Forte number, from the list the theorist Allen Forte made in 1973: 3-11 is the class of major and minor triads (3 notes, the 11th class of 3 notes). A class is also written by its prime form, its most compact shape moved to start on 0: [037] for 3-11, C E♭ G.

(T stands for 10, so that each number is one character.)

The interval vector counts the intervals between every pair of notes in a set: how many minor seconds, major seconds, minor thirds, major thirds, fourths and tritones. A fifth counts as a fourth, being a fourth upside down. For the triad 3-11 it is <001110>: one minor third, one major third, one fourth. It is a fair summary of how a set sounds.

In the app

The piano roll names the set of the notes under the pointer, or of the notes you select, on the line under the chord buttons:

Set: 3-11 · normal form C E G · prime form [037] · intervals <001110> · major or minor triad · micromode Acorde menor-mayor

Sets… opens a list of every class of the size you choose, with its number, prime form, interval vector and, for the famous ones, a name. Search it by number or name (“4-Z15”, “octatonic”), or click Show it for the class of the selected notes. Pick one, then Move it up and Inverted (I) choose which of its forms: T3 is the set three semitones up, T3I its inversion moved up three. ▶ Hear it plays it, Draw with it puts the whole set down with each click in the piano roll, and Use as the key makes it the song’s key. On the chord sheet, a set is typed by its name or its notes: 3-11 T2, {D F A} or {0,1,4}.

One harmony, four names

A set is another way to name notes, not other notes. C major’s chord C is C, E and G however you name it. The chord sheet’s Show menu writes the same harmony four ways, without changing a note: As written, as Chords, as Pitch-class sets or as Micromodes.

How to read them:

• Tn is the prime form moved up n semitones: 3-11’s prime form is C E♭ G, a C minor triad, so 3-11 T2 is D F A, D minor.
• TnI is the prime form turned upside down first, then moved up: C E♭ G upside down is C A F, and moved up 7 it is G E C. So every major triad has an I in its name, and no minor triad has one.
• A micromode is always three notes, so G7, with four, keeps its set name in the Micromodes view.
• Chords names a set as a chord when its notes make one, and keeps the set’s name when they don’t.

For sets and micromodes, a second menu chooses which notes make each group: The sheet’s harmonies, Notes together (vertical), or Notes in a line (serial), with how many notes each group holds and which track to read. That reads the song’s own notes, so you can look at a tune as a chain of three-note groups.

The key works the same way. Pick pitch-class set in the Key menu of a song in C major, and nothing moves: the key is now named 7-35, and the harmony keys show the same seven chords with their set classes, six of them 3-11 and B D F 3-10. Notes move only when you pick a different set.

Twelve-tone rows

Around 1921, Arnold Schoenberg looked for a way to write music in which no note is home. His answer was to use all twelve notes in a fixed order, chosen for each piece: the row, or series. No note comes back until the other eleven have sounded, so none becomes the centre. This is the row of his Suite for Piano, Op. 25 (1925):

E F G C♯ F♯ D♯ G♯ D B C A A♯

A row comes in four forms:

Each can start on any of the twelve notes, which makes 48 forms. They are named by a letter and the number of the note they start on, with C as 0: Schoenberg’s row is P4, because it starts on E. An R form takes the number of the form it reverses, so R4 ends on E. (Some books number the forms from the row’s own first note instead, so that the row as written is always P0. The forms are the same; only their numbers move.)

The Row window

Row… in the piano roll opens it. Type a row in the box at the top, as note names (in letters or in solfège) or numbers, and click Use; or draw twelve notes in the piano roll and click From the notes; or take one from 🎲 Random, 🎲 All-interval or Famous and example rows: Schoenberg’s Suite, Berg’s Lyric Suite, Webern’s Symphony and Concerto, and four rows written for Ferrous Lineup.

The matrix holds all 48 forms in a square. Each line, read left to right, is a P form, and right to left an R form; each column, read down, is an I form, and up an RI form. The labels around the square say which: click one to choose that form. The diagonal from the top left is the same note all the way down, a quick check that the square is right.

Under the matrix, the window says what the row is made of: its steps in semitones, its groups of three notes, and which forms are combinatorial with it, whose first six notes are the six the row leaves out, so the two played together give all twelve notes in each half.

Then write a form into the pattern: Together says how many of its notes sound at once, as a chord; Then what comes after the form, Once, Repeat or P, R, I, RI; the register, Close (all in one octave), Smooth (each note near the one before) or Wide (leaping over three octaves, as in Webern). Serialism goes further, as Messiaen, Boulez and Babbitt did around 1950: it lets the row choose the Durations, Dynamics and Octaves too. Write P0 into current pattern replaces the channel’s notes, and Save to new pattern puts them in a pattern of their own.

A row as the key

Pick twelve-tone row in the Key menu, or click Use as the key in the Row window for the form you chose, and the row becomes the key. The piano roll numbers its notes 1 to 12 on the keyboard, in the row’s order, so you can draw the row by following the numbers. The harmony keys play the row’s groups instead of chords: 3 notes gives four keys, 1–3, 4–6, 7–9 and 10–12; 4 notes three, and 6 notes two.

With Voice leading ticked, the piano roll checks the rules of twelve-tone writing as Schoenberg’s students learnt them: the notes follow forms of the row, each in its order; notes next to each other in the row may sound together as a chord, and a note may be repeated straight away, but it comes back only after the other eleven; a form is played whole; no note is doubled in octaves; and no chord sounds like tonal harmony, such as a major or minor triad or a dominant seventh. A Row line under the chord names shows the forms it reads in the notes.

The Library’s Twelve-tone rows group has seven styles, and File > Demos > Twelve-tone plays each as a short piece.

Micromodes

Francisco Kröpfl (1931–2021) was an Argentine composer, a pioneer of electronic music who founded the first university studio for it in Latin America, in Buenos Aires in 1958, and a teacher of composition for decades. He taught his students to hear groups of three notes by their colour. Any three notes have three intervals between them, and those intervals give the group its sound, in any order, octave or spelling, moved up or turned upside down. By those rules, which are the rules of pitch-class sets, there are exactly twelve groups of three. Kröpfl called them micromodes (micromodos):

They come in three families. The menores all have a minor second, a semitone; the mayores have a major second and no semitone; and the rest are the three triads, named after them. The minor and the major triad are one micromode, Acorde menor-mayor, because each is the other upside down: every chord of a I, IV, V, I is the same colour, and tonal music works with which forms it uses.

Micromodes… in the piano roll shows the three families, each micromode drawn on a clock of the twelve notes, with its prime form, its intervals, its set and where it is heard:

Select some notes in the piano roll, and the window lists the micromodes inside them: the seven notes of C major hold 35 groups of three. Move it up and Inverted (I) choose a form; ▶ Hear it, Draw with it and Use as the key work as in the Sets window, and the set’s button opens the Sets window on the same class. On the chord sheet, type a micromode by name: Menor 3, or Menor 3 T4I for one of its forms.

The four groups of three in the row of Webern’s Concerto, Op. 24, in the picture of the Row window, are all Menor 3. The Library’s Micromodes (Kröpfl) group has seven styles, and File > Demos > Micromodes plays each of them.