Lab 7: Musical Scales The Just Scale The Tempered Scale Transposition Tuning Triads Sensitivity to Tuning The Black Keys of the Keyboard The Missing Black Keys Problems With the Just Scale The Tempered Scale Transposition Major Scale Minor Scale

Notation for specific tones: C4 G4 E5 … Standard frequency of middle A: f (A4) = 440 Hz (or 442 Hz) Note: the standard frequency to be used in this lab is f (C) = 240 Hz.

Musical interval – characterized by the ratio of two frequencies Name Just Interval Octave 2 Fifth 3/2 Fourth 4/3 Major Third 5/4 Minor Third 6/5 Name Musical Interval Second 9/8 (Dissonant) The simple-number frequency ratios are consonant (“in harmony”) because “for these special ratios the various partials of the two tones being played together either coincide exactly or are so different as to avoid beats or roughness.” (textbook) Note: each key on our keyboard generates a pure tone – no higher harmonics.

Just Scale – constructed from three major triads G A B Just Scale – constructed from three major triads C-E-G, G-B-D' and F-A-C' Major Triad – a combination of three tones (chord) with frequency ratio of 4:5:6. For example, f (C): f (E): f (G) = 4:5:6 If f (C) = 240 Hz,

Tempered Scale Divides each octave into 12 identical semitone intervals. The ratio of frequencies of adjacent keys, x, is the same — equal intervals. For example, if f (C) = 240 Hz, The ratio becomes 2 when x is multiplied by itself 12 times or x12 = 2

Transposition Large interval = Whole-tone (1) interval C D E F G A B Transposition Large interval = Whole-tone (1) interval Small interval = Half-tone (½) (or semitone) interval Major Scale (e.g. C-Major) C D E F G A B C' 1 1 ½ 1 1 1 ½ G-Major G A B C' D' E' F' G' F'# Minor Scale (e.g. A-Minor) A B C' D' E' F' G' A' 1 ½ 1 1 ½ 1 1 D-Minor D E F G A B C' D' Bb In music, the same letter is used only once as a rule.