Chapter 1 Basic Concepts and Definitions

l Octave equivalenceà same pitches separated by one or more octaves = octave-related pitches = functionally equivalent

l Pitchà a tone with a certain frequency

l Pitch classà a group of pitches with the same name

l Enharmonic equivalenceà same note on piano, but very different musical roles both melodically and harmonically. However, in post-tonal music, such distinctions are largely abandoned because where notes that are enharmonically equivalent are also functionally equivalent. Still, notation is functionally arbitrary, determined by simple convenience and legibility.

l Integer Notation: We will use a “fixed do” notationà the pitch class containing the Cs is arbitrarily assigned the integer 0 and the rest follows from there.

l Mod 12à Any number larger than 11 or smaller than 0 is equivalent to some integer from 0 to 11 inclusive. 12 is called *modulus*.

l Pitch spaceà pitches in an extended pitch space ranges in equal-tempered semitones from the lowest to the highest audible tone.

l (modular) Pitch-class spaceà which circles back on itself and contains only the 12 pitch classes.

l It is more accurate musically just to name intervals according to the # of semitones they contain.

Unison: 0

m2: 1

M2, dim3: 2

m3, aug2: 3

M3, dim4: 4

aug3, P4: 5

TT, aug4, dim5: 6

P5, dim6: 7

aug5, m6: 8

M6, dim7: 9

aug6, m7: 10

M7: 11

Octave: 12

m9: 13

M9: 14

m10: 15 etc…

l ip (pitch interval)à created when we move from pitch to pitch in pitch space.

l *Ordered pitch interval*à + – (direction upward or downward matters). Focus attention on the contour of the line, its balance of rising and falling motion.

l *Unordered pitch interval*à direction doesn’t matter.

l i (pitch-class interval)à created when we move from pitch class to pitch class in modular pitch-class space. It can never be larger than 11 semitones.

l *Ordered pitch-class intervals*à ordered interval from pitch class x to pitch class y is y-x (mod12). A to C# (4) is different from C# to A (8), becauseorder matters. They are each other’s *complement mod12*, because they add up to 12.

l *Unordered pitch-class Intervals* à it no longer matters whether you count upward or downward. Just count from one pitch class to the other by the shortest available route, x-y (mod12), or y-x (mod12). There are only 7 different unordered pitch-class intervals, because one never has to travel farther than 6 semitones.

l Interval Classà unordered pitch-class interval is also called interval class. Because of counpound intervals—intervals larger than an octave—are considered equivalent to their counterparts within the octave. Also, pitch-class intervals larger than six are considered equivalent to their complements mod 12.

l 4 ways of talking about intervals: 1. Ordered pitch interval, 2. Unordered pitch interval, 3. Ordered pitch-class interval, 4. Unordered pitch-class interval. C# à Ab(upward) +19, 19, 7, 5

l Interval class content: The different in the music sound is a reflection of the difference in their musical content.

l Interval-class vectorà Interval-class content is usually presented as a string of 6 numbers with no spaces intervening. It gives a convenient way of summarizing their basic sound.