Chapter 16 Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano The Just Scale All intervals are integer ratios in frequency Major Scale Minor Scale A Note of Caution Major Scale Minor Scale Notes on the Just Scale
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Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano
Minor ScaleNotes on the Just Scale
The D corresponds to the upper D in the pair found in Chapter 15. Also, the tones here (except D and B) were the same found in the beat-free Chromatic scale in Chapter 15.
Here we use the lower D from chapter 15 and the upper Ab. In music theory two other minor 7th are recognized, the grave 7th (16/9) and the harmonic minor 7th (7/4).
Column jumps indicate octave changes
as L , f (longer strings lower tones)
as r , f (larger strings lower tones)
as T , f (more tension higher tones)
double the length and the frequency is up two octaves
the opposite behavior of the string, as r , f
And is small (about 0.00016)
We use a model where the string is firmly anchored at one end and can move freely on a vertical rod at the other end between springs
FS is the string natural frequency
FM is the natural frequency of the block and spring to which the string is connected.
Note: The values used here for the C4 partials are the same as were used previously to compare piano to organ tuning and introduced the inharmonic factor J. Also notice that the C5 is not 3 cents sharp of the second harmonic of C4.
The one cent difference suggested in the text would give 392.67 Hz. Two times this is 785.34 and three times the middle C is 784.89 (a beat frequency of 0.45).