Final Flashcards
1
Q
Unison
A
1 line space
0 half-steps is perfect
1 is augmented
2
Q
Second
A
2 line spaces 1 half-step is minor 2 is major 3 is augmented M, M, m, M, M, M, M
3
Q
Third
A
3 line spaces
3 half-steps is minor
4 is major
M, m, m, M, M, m, m
4
Q
Fourth
A
4 lines
5 half-steps is perfect
6 is augmented
P, P, P, A, P, P, P
5
Q
Fifth
A
5 lines
6 half-steps is diminished
7 is perfect
P, P, P, P, P, P, d
6
Q
Sixth
A
6 lines
8 half-steps is minor
9 is major
M, M, m, M, M, m, m
7
Q
Seventh
A
7 lines
10 half-steps is minor
11 is major
M, m, m, M, m, m, m
8
Q
Octave
A
8 lines
11 half-steps is diminished
12 is perfect
13 is augmented
9
Q
Equal tempered half-step
A
2^1/12 f
10
Q
Pythagorean white notes
A
f, 9/8f, 81/64f, 4/3f, 3/2f, 27/16f, 243/128f, 2f
It does not close the circle of fifths. Thirteen notes instead of twelve
11
Q
Just white notes
A
f, 9/8f, 5/4f, 4/3f, 3/2f, 5/3f, 15/8f, 2f
12
Q
Sequence of fifths
A
C, G, D, A, E, B, F#, D♭, A♭, E♭, B♭, F, C
13
Q
Counterpoint
A
A style of composition where a piece is composed of various independent voices and for which harmony is a result of these voices sounding together
14
Q
Fugue
A
A canonic composition that is less rigid with statements of theme
15
Q
Canon
A
It starts off with the theme, with other instruments or voices copying it
16
Q
Ricercar
A
A precursor to the fugue
17
Q
Three properties of tones
A
Pitch - the property of a sound that enables it to be ordered on a scale going from low to high
Loudness - intensity of a sound
Timbre - Quality/color of a sound
18
Q
Notes
A
Notes are tones with onsets and offsets
19
Q
Onset and offset
A
Onset is when the note begins and offset is when it ends
20
Q
Sound
A
The physical object perceived by tones
21
Q
Interval
A
Set of two pitches. Interval length is the distance in half-steps between them
22
Q
Shepard tones
A
They create the illusion of infinitely rising or falling tones. The average frequency of Shepard tones remains the same
23
Q
Chord
A
Three or more pitches sounded simultaneously or functioning as if they were sounded simultaneously.
24
Q
Triad
A
A 3-chord built out of stacked thirds. A root pitch is transposed up a major or minor third. That second pitch is then transposed up a major or minor third
25
Augmented triad
A M-M triad. C+
26
Major triad
A M-m triad. C
27
Minor triad
A m-M triad. c
28
Diminished triad
A m-m triad. c°
29
Octave equivalence
Two pitches are octave equivalent to each other if they are related by transposition by a number of octaves.
P1 = P2 + 12n
30
Seventh chords
A 4-chord built out of stacked chords. A root pitch is transposed up three times by thirds.
31
Major 7th
M-m-M. Cmaj7
32
Dominant 7th
M-m-m. C7
33
Minor/major 7th
m-M-M. cmaj7
34
Minor 7th
m-M-m. c7
35
Half-diminished 7th
m-m-M. cØ7
36
Diminished 7th
m-m-m. c°7
37
Augmented/major 7th
M-M-m. C+maj7
38
Chord-type
Two chords are of the same type if one is a transposition of the other
39
Whole-tone scales
They are generated by taking whole steps. They have rotational and inversional symmetry
40
WT0
{0, 2, 4, 6, 8, 10}
41
WT1
{1, 3, 5, 7, 9, 11}
42
Octatonic scales
They are generated by alternating between whole and half-steps. They have rotational and inversional symmetry
43
Oct0,1
{0, 1, 3, 4, 6, 7, 9, 10}
44
Oct0,2
{0, 2, 3, 5, 6, 8, 9, 11}
45
Oct1,2
{1, 2, 4, 5, 7, 8, 10, 11}
46
Petrushka chords
They are built out of a major triad and its transposition up by 6
47
FTA
Any integer n > 1 can be written as a prime factorization
n = p1^n1 · p2^n2 ···· pr^nr
48
Divisibility
m|n if there exists some q such that
| n = mq
49
Least common multiple
?
50
Greatest common denominator
?
51
Inverse elements
Any element g^-1 such that
g · g^-1 = e
It sends an element to the identity element.
52
Identity element
Any element e ∈ G such that
| e · g = g
53
Homomorphism
Two groups G and H are homomorphic if φ sends the identity of G to the identity of H; and if φ respects the group operation of the groups
54
Isomorphism
An isomorphism is a group homomorphism from G to H that is also invertible. It needs to be 1-1 and onto
55
1-1
φ sends distinct elements to distinct elements
56
Onto
φ "hits" all elements of Y
57
Order of elements
The smallest positive integer n such that g^n = e.
| It is how many times an element needs to be multiplied by itself to be the identity element
58
Equivalence relations
x = y
x ≡ y mod m
x is isomorphic to y
59
Quotient
Denoted X/~
| It is the set of all equivalence classes of X
60
Congruence modulo n
x is congruent to y if
n | (x - y).
Written x ≡ y mod n, which assets a relation. It is not formula
61
_mod n
This is a number. Compute it
62
Modular arithmetic
Integers x and y are congruent modulo m if
x = y + mq
for some q
63
ℝ_>0
The set of all positive real numbers. Its group operation is multiplication.
64
ℝ
The set of all real numbers. Its group operation is addition.
65
ℤ
The set of all integers
66
ℚ
The set of rational numbers
67
C_n
C_n = {e=x^0, x^1, x^,2,...,x^n-1}.
Its group operation is
x^i · x^j = x^(x+j) mod n
68
D_n
```
D_n = {1, x, x^2, ... x^n-1, xy, (x^2)y, ..., (x^n-1)y}
Its group operation is
x^n = 1 = y^2
x^i · x^j = x^(i + j) mod n
y · x^i = (x^n-i)y
```
69
ℤ/nℤ
```
For any integer r, [r]_n is the congruence class of all integers congruent to r modulo n.
Its group operation is
[r]_n + [s]_n = [r+s]_n
```
70
S_n
The set of all invertible functions from the set {1, 2, ..., n} to itself. Computation is done with cyclic notation
71
Cyclic notation
(12)(34)(134) = (142)(3)
72
Orbit
Denoted O_s. The orbit of a chord is its chord-type. It is the set of all transpositions by s
73
Stabilizer
Denoted Stab_s. The set of elements that map s to itself
74
Lagrange's theorem
```
#O_s = #G / #Stab_s
It answers how many chords of a given type there are.
```
75
X_freq
The set of all possible frequencies. It is equal to ℝ_>0
76
X_pitch
The set of all possible pitches. It is equal to ℝ
77
X_pc
The set of pitch-classes. It is equal to {[P1]_12, ..., [Pr]_12}
78
T_freq
The set of all transpositions. It acts on X_freq via multiplication
79
T_pitch
The set of all transpositions. It acts on X_pitch via addition
80
Isomorphism from T_freq to T_pitch
Φ(x) sends T_freq to T_pitch.
Φ(x) = 12log_2 (x)
Ψ(x) sends T_pitch to T_freq
Ψ(x) = 2^(x/12)
81
Pitch arithmetic
A pitch P is added to or subtracted from by n, where n is the number half-steps
82
Pitch-class arithemtic
A pitch [P]_12 is added to by [n]_12, where n is the number of half-steps
83
T_12
```
The group of equal-tempered pitch-class transpositions.
T_12 = {t_0, t_1, ... t_11}.
Its group operation is
t_i ◦ t_j = t_t+j
```
84
M_12
The group of equal-tempered transpositions and inversions.
M_12 = {e, t, t^2, ..., t^11, ti, (t^2)i, ..., (t^11)i}.
t^12 = i^2 = e
i(t^j) = (t^12-i)j
85
Rigid motions
The transformation of a space that preserves distance
86
Distance formula
In X_pitch, it is
d(P1, P2) = |P1 - P2|
In X_pc, it is the minimum length of the arcs between two points
87
Transpositional symmetry
If a chord has any rotation t_j such that t_j (X) = X, then that chord has transpositional symmetry
88
Inversional symmetry
If a chord has any inversion i_q0 such that i_q0 (X) = X, then that chord has inversional symmetry
89
Musical Offering. Canon I. a 2 cancrizans
By J.S. Bach. It is a crab canon. Instrument one plays through from left to right, then back. Instrument two plays from right to left, then back
90
Musica Ricercata No. 1
By György Ligeti. It is the first piece in Musica Ricercata. It only has two pitches. Each subsequent piece adds a pitch.
91
Musica Ricercata No. 11
By György Ligeti. It is the last of Musica Ricercata. It has all twelve pitches. It is an homage to Girolamo Frescobaldi's "Ricercata cromatico"
92
I Fiori Musicali: Ricercar cromatico post il Credo
By Girolamo Frescobaldi. It uses most pitches in its subject. Musica Ricercata No. 11 is an homage to it
93
Eleven Intrusions
By Harry Partch. He invented and composed with a just scale containing 43 distinct frequencies within the octave
94
Serenade for tenor, horn, and strings, Op. 31 I (Prologue) and II (Pastoral)
Written in the just scale. Some of its notes sound very different from our equal-tempered scale
95
Piano Sonata No. 12
By Mozart. He moves down the circle of fifths, using a pattern of rising fourths and falling fifths.
96
Fugue No. 2 in C minor
By Bach. He moves down the circle of fifths
97
Jordu
Written by Duke Jordan, performed by Clifford Brown. He moves down the circle of fifths, using rising fourths and falling fifths.
98
Bagatelle No. 2
By Beethoven. He moves up the circle of fifths
99
Quartet No. 14 in G major.
By Mozart. He moves up the circle of fifths
100
Frühlingstraum
By Franz Schubert. Melody is identical up to octave equivalence, but sounds very different depending on circumstance.
101
Die Krähe
By Franz Schubert. The melody is very uninteresting in pitch-class space
102
L'escalier du diable
By Ligeti. Etude 9. Meant to create an effect like that of the Shepard tones. Unlike actual Shepard tones, it rises
103
Vertige
By Ligeti. Etude 9. Meant to create an effect like that of the Shepard tones. Unlike actual Shepard tones, it rises.
104
Coloana infinita
By Ligeti. Etude 14. Meant to create an effect like that of the Shephard tones. Comes very close to doing so.
105
Contrapunctus No. 5
By Bach. The subject is reflected through A.
106
Free Variations
By Bartók. The right hand inverts the left hand.
107
From the Diary of a Fly
By Bartók. Melodic inversion through the G axis between G and A♭
108
Voiles
By Debussy. It is written in the symmetric whole-tone scale, but veers off into the pentatonic scale.
109
Regard du Père
By Messiaen. It is written in Oct0,1.
110
Chez Petroushka
By Stravinsky. It is the first use of the Petrushka chord.
111
Prime form of a chord
P1 = 0
Pr is as small as possible
The sequence is as small as possible with respect to lexicographic order
