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The Planets - Saturn, the Bringer of Old Age |
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Refer to Table 10.3 in the text. |
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Intervals are ratios of frequencies or lengths. |
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Octave (natural interval) |
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String Length Ratio = 2/1 = 2.00
or ½ =
0.50 |
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Frequency Ratio also = 2.00
or ½ = 0.50 |
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Perfect Fifth |
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Seven semitones |
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String Length Ratio = 3/2 =1.50
or 2/3
= 0.6667 |
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Frequency Ratio also = 1.50
or 2/3 =
0.6667 |
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Perfect Fourth |
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Five semitones |
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String Length Ratio = 4/3 = 1.33
or ¾ =
0.750 |
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String Length Ratio = 1.333
or ¾ =
0.750 |
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Pythagoras (582-507 BC) |
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Ratios for intervals: 1.000, 1.333, 1.500, 2.000
(unison, fourth, fifth, octave) |
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Used these ratios to construct a mathematical
scale. |
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Used string lengths since frequencies were not
known. |
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How do you divide an octave (1.00 to 2.00) into
8 equal parts? |
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Or in terms of frequencies an interval such as
220 –440 Hz? |
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Multiply or divide an existing length (ratio) by
3/2 (=1.500), factor of fifths. |
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If the result lies between 1 and 2, leave it as
it is. |
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If the answer is less than 1, double it (up an
octave) |
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If the answer is greater than 1, halve it (down
an octave) |
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Start with D4 = 1.000 (293.7 Hz) |
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Multiply D4= 1.00 by 1.50 to get 1.5
(the fifth) which is A4 (440 Hz). |
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Start with D4 = 1.000 (293.7 Hz) |
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Divide D4 = 1.00 by 1.50
to get 0.666 and double to get 1.333 (the fourth) which is G4
(392 Hz). |
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Start with A4 = 1.500 (440 Hz) |
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Multiply A4 = 1.50 by 1.50
to get 2.250 and halve to get 1.125 (the major second) which is E4
(229.6 Hz). |
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Start with G4 = 1.333 (292 Hz) |
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Divide G4 = 1.333 by 1.50
to get 0.88888 and double to get 1.777 (the minor seventh) which is C5
(523.3 Hz). |
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These first 5 notes D, E, G, A, C, and D again
constitute the 5-note Chinese scale called pentatonic (5 tones) |
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Greek scales had 7 notes called septatonic |
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Start with E4 = 1.125 (229.6 Hz) |
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Multiply E4 = 1.125 by
1.50 to get 1.6875 (the major sixth) which is B4 (493.9 Hz). |
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Start with C5 = 1.777 (523.3 Hz) |
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Divide C5 = 1.777 by 1.50
to get 1.1851 (the minor third) which is F4 (349.2 Hz). |
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Start with G4 = 1.333 (292 Hz) |
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Multiply G4 = 1.333 by
1.50 to get 2.00 (the octave) which is D5 (587.3 Hz). |
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Tone = T and Semitone = s |
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T s
T T T
s T |
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D E F
G A B
C D |
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Scales based on the white keys of the piano |
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Since there are seven different named keys A, B,
C, D, E, F, G, there are seven modes. |
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T T
s T T
T s |
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C D E
F G A
B C |
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T s
T T s
T T |
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A B C
D E F
G A |
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Major: J.S. Bach “Well Tempered Clavier Book II”
Prelude I in C major. (Track #1) |
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Minor: Prelude IV in C# minor. (Track #7) |
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C = Ionian (major scale) |
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D = Dorian |
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E = Phrygian (Spanish or Oriental) |
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F = Lydian (funny, comical) |
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G = Mixolydian |
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A = Aeolian (minor scale) |
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B = Locrian (not used) |
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(starts on E) |
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Vaughn Williams |
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Fantasia
on a Theme by Thomas Tallis |
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Multiples
of 1.500 generate the same 8 note scale that was found by musicians to be
the “right” ones for a musical scale. |
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The fifth is a multiple of 1.500. |
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The fifth is the 3rd harmonic. (3h1/2h1
= 1.50) |
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Third harmonic of A is E. |
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Third harmonic of E is B. |
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Third harmonic of B is F#. |
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of F# is
C#, of C# is G#, of G# is D#, of D# is A#, of A# is F, of F is C, of C is
G, of G is D, and of D is back to A. |
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This is the entire chromatic scale! |
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The 3rd harmonic is the lowest and
strongest harmonic that is not an octave. |
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Stringed instruments have the 3rd
harmonic. |
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A scale based on 3rd harmonics should
be the most “natural” or pleasing. |
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7 Greek modes or Church modes use all of the
white keys |
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Comprise 7 combinations of T=tone and s=semitone
sequences |
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Using a particular mode requires the scale to
start on one and only one note. |
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Need to place semitones anywhere. |
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Notes