1. The Science of Sound Chapter 8
MUSICAL ACOUSTICS
COMBINATION TONES
The Science of Sound
Chapter 8

2. LINEAR SUPERPOSITION OF TWO SIMPLE HARMONIC MOTIONS AT THE SAME FREQUENCY
SAME PHASE
OPPOSITE PHASE

3. SIMPLE HARMONIC MOTION AS THE PROJECTION OF POINT P (MOVING IN A CIRCLE)

4. TWO POINTS P AND Q MOVE WITH THE SAME PERIOD T AND AMPLITUDE A AND MAINTAIN A PHASE DIFFERENCE ΦP – ΦQ

5. FREQUENCY BUT WITH A PHASE DIFFERENCE ΦB – ΦA = 90º
LINEAR SUPERPOSITION OF TWO SHMs WITH THE SAME
FREQUENCY BUT WITH A PHASE DIFFERENCE ΦB – ΦA = 90º

6. TWO TONES WITH FREQUENCIES ƒ1 AND ƒ2 : BEATS

7. THE MUSICAL STAFF: MUSICIAN’S GRAPH PAPER

8. PROPOSAL FOR THREE NEW CLEFFS (See Appendix A.6)

9. DIFFERENCE TONE MELODY

10. STATISTICAL ANALYSIS OF CHORDS

11. COMBINATION TONES
COMBINATION TONES ON A MUSICAL STAFF

12. DIFFERENCE TONES MOST CLEARLY HEARD:
DIFFERENCE TONE (f2 – f1) [TARTINI TONES]
CUBIC DIFFERENCE TONE (2f1 – f2)
OTHER DIFFERENCE TONES: (3f1– 2f2), 4f1—3f2), etc,

13. OTHER NONLINEAR EFFECTS
AURAL HARMONICS
FLETCHER SUGGESTED A SIMPLE POWER LAW FOR EAR RESPONSE
x = a0 + a1 p + a2 p2 + a3 p3 + , , ,
THIS PREDICTS THAT FOR A 1 dB INCREASE IN SIGNAL LEVEL, THE SECOND HARMONIC WILL INCREASE BY 2 dB, THE THIRD BY 3 dB, etc.
SUMMATION TONES
IF THE EAR HAS A NONLINEAR RESPONSE, ONE MIGHT EXPECT TO HEAR SUMMATION TONES (f1 + f2) AS WELL AS DIFFERENCE TONES.
NO ONE HAS PRESENTED CONVINCING EVIDENCE FOR THEIR EXISTENCE

14. MODULATION OF ONE TONE (ƒ2) BY ANOTHER (ƒ1)

15. CONSONANCE AND DISSONANCE: MUSICAL INTERVALS
PLOMP AND LEVELT: IF THE FREQUENCY DIFFERENCE IS GREATER THAN A CRITICAL BAND, THEY SOUND CONSONANT
MAXIMUM DISSONANCE OCCURS WHEN THE DIFFERENCE IS ABOUT ¼ OF A CRITICAL BAND

16. INTERACTIONS BETWEEN HARMONICS OF TWO TONES SEPARATED BY DIFFERENT INTERVALS

17. CONSONANCE AND DISSONANCE

18. EFFECT OF PHASE ON TIMBRE
BUILDING UP COMPLEX TONES WITH THE SAME SPECTRUM OF PARTIALS BUT WITH DIFFERENT PHASES RESULTS IN TOTALLY DIFFERENT WAVEFORMS. DO THEY SOUND DIFFERENT?
THE ANSWER IS “SOMETIMES”
PLOMP (1970):
THE MAXIMUM EFFECT OF PHASE ON TIMBRE IS BETWEEN A COMPLEX TONE IN WHICH THE HARMONICS ARE IN PHASE AND ONE IN WHICH ALTERNATE HARMONICS DIFFER IN PHASE BY 90O

19. BEATS OF MISTUNED CONSONANCES
A SENSATION OF BEATS OCCUR WHEN THE FREQUENCIES OF TWO TONES f1 AND f2 ARE NEARLY, BUT NOT QUITE, IN A SIMPLE RATIO
IF f2 = 2f1 + δ, BEATS ARE HEARD AT A FREQUENCY δ.
WHEN f2 = n/m f1 + δ, mδ BEATS OCCUR. THESE ARE CALLED SECOND ORDER BEATS OR BEATS OF MISTUNED CONSONANCE, AND THEY RESULT IN PERIODIC CHANGES IN PHASE
THE EAR, WHICH IS A POOR DETECTOR OF STATIONARY PHASE, APPEARS TO BE SENSITIVE TO CYCLICAL VARIATIONS IN PHASE
BEATS OF MISTUNED CONSONANCES HAVE LONG BEEN USED BY PIANO TUNERS TO TUNE FIFTHS, FOURTHS, AND OCTAVES. VIOLINISTS USE THEM TO TUNE THEIR STRINGS

20. PRACTICE PROBLEMS:
IF TONES WITH FREQUENCIES OF 440 Hz AND 443 Hz ARE SOUNDED TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?
IF TONES WITH FREQUENCIES OF 442 Hz AND 330 Hz ARE HEARD TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?

21. PRACTICE PROBLEMS:
IF TONES WITH FREQUENCIES OF 440 Hz AND 443 Hz ARE SOUNDED TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?
SOLUTION: 443 – 440 = 3 Hz
IF TONES WITH FREQUENCIES OF 442 Hz AND 330 Hz ARE HEARD TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?
SOLUTION: 442 = 4/3(330) + 2, SO mδ = (3)(2) = 6 Hz
(THESE ARE BEATS OF A MISTUNED FOURTH)

22. PRACTICE PROBLEMS:
IF TONES WITH FREQUENCIES OF 440 Hz AND 443 Hz ARE SOUNDED TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?
SOLUTION: 443 – 440 = 3 Hz
IF TONES WITH FREQUENCIES OF 442 Hz AND 330 Hz ARE HEARD TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?
SOLUTION: 442 = 4/3(330) + 2, SO mδ = (3)(2) = 6 Hz
(THESE ARE BEATS OF A MISTUNED FOURTH)
Alternate solution:
3rd harmonic of 442 is 1326 Hz; 4th harmonic of 330 is 1320; if sounded together, beats would be heard at = 6 Hz

23. CENTRAL NERVOUS SYSTEM
AUTOCORRELATION AND CROSS-CORRELATION
AUTOCORRELATION IS A COMPARISON OF A PULSE TRAIN WITH PREVIOUS PULSE TRAINS IN ORDER TO PICK OUT REPETITIVE FEATURES (e.g., repetition pitch)
CROSS-CORRELATION IS A COMPARISON BETWEEN SIGNALS ON TWO DIFFERENT NERVE FIBERS (e.g., localization of sound)
CEREBRAL DOMINANCE
THE LEFT SIDE OF THE BRAIN (IN 97% OF THE POPULATION) IS SPECIALIZED FOR SPEECH PROCESSING, AND THE RIGHT SIDE FOR NON-LINGUISTIC FUNCTIONS SUCH AS MUSIC
SPEECH PROCESSING REQUIRES ANALYTIC AND SERIAL PROCESSING, MUSICAL PERCEPTION REQUIRES HOLISTIC PROCESSING