COMBINATION TONES

The Science of Sound

Chapter 8

MUSICAL ACOUSTICS

LINEAR SUPERPOSITION OF TWO SIMPLE HARMONIC MOTIONS AT THE SAME FREQUENCY

SAME PHASE OPPOSITE PHASE

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

TWO POINTS P AND Q MOVE WITH THE SAME PERIOD T AND AMPLITUDE A AND MAINTAIN A

PHASE DIFFERENCE ΦP – ΦQ

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

ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45O 90O 135O

Practice problems:

ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45O 90O 135O

√ [16 + 9 + (1.4)(4)(3)] = 6.47

ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45O 90O 135O

√ [16 + 9 + (1.4)(4)(3)] = 6.47

√ [16 + 9] = 5

ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45O 90O 135O

√ [16 + 9 + (1.4)(4)(3)] = 6.47

√ [16 + 9] = 5

√ [16 + 9 - (1.4)(4)(3)] = 2.86

TWO TONES WITH FREQUENCIES ƒ1 AND ƒ2 : BEATS

32 Primary beats. Track 62

If f1 is fixed and f2 is variable, beats result for small differences between f1 and f2.

THE MUSICAL STAFF:

MUSICIAN’S GRAPH PAPER

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

COMBINATION TONES

COMBINATION TONES ON A MUSICAL STAFF

34 Aural combination Tones, Tr 68-69

DIFFERENCE TONES

MOST CLEARLY HEARD:

DIFFERENCE TONE (f2 – f1) [TARTINI TONES]

CUBIC DIFFERENCE TONE (2f1 – f2)

OTHER DIFFERENCE TONES: (3f1– 2f2), 4f1—3f2), etc,

DIFFERENCE TONE MELODY

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

Amplitude modulation results in “sidebands” as shown

AMPLITUDE MODULATION (AM) AND FREQUENCY MODULATION (FM)

BOTH AM AND FM ARE USED IN RADIO BROADCASTING

FREQUENCY MODULATION PRODUCES MANY SIDEBANDS AND THESE ARE USEFUL IN FM SYNTHESIS OF MUSIC

ORIGIN OF DIFFERENCE TONES

The simple difference tone with frequency f2-f1 (properly called the quadratic difference tone) can be easily heard up to about 1500 Hz.

The cubic difference tone with frequency 2f1-f2can be easily heard as well.

The quartic difference tone (3f1-2f2) and higher order difference tones are generally more difficult to hear.

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

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

INTERACTIONS BETWEEN HARMONICS OF TWO TONES SEPARATED BY DIFFERENT INTERVALS

CONSONANCEEXPECTED BY COMBINING A

TONE OF 250 Hz WITH ANOTHER

TONE

STATISTICAL ANALYSIS OF CHORDS

Bach “Trio Sonata for Organ” Dvorak “String Quartet op. 51”

(Plomp and Levelt, 2965)

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

33 Distortion, Tracks 64-67

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

32 Secondary beats, Track 63 Tone pairs having intervals slightly greater than an octave, fifth, and fourth

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?

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)

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 1326-1320 = 6 Hz

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

BINARUAL EFFECTS

AUDITORY DEMONSTRATIONS 36-39 ILLLUSTRATE BINAURAL EFFECTS THAT CAN BE HEARD THROUGH STEREO HEADPHONES

A BINAURAL AUDITORY ILLUSION

TONES OF 400 AND 800 Hz ALTERNATE IN BOTH EARS IN OPPOSITE PHASE (i.e., when right ears hears 400 Hz, left ear hears 800 Hz). MOST RIGHT-HANDED LISTENERS HEAR THE HIGH TONE IN THEIR RIGHT EAR AND THE LOW TONE IN THEIR LEFT EAR, REGARDLESS OF HOW THE HEADPHONES ARE ORIENTED (even when they are exchanged). LEFT-HANDED LISTENERS, ON THE OTHER HAND, ARE JUST AS APT TO HEAR THE HIGH TONE IN THE RIGHT EAR AS IN THE LEFT. THAT IS BECAUSE IN RIGHT-HANDED PEOPLE, THE LEFT HEMISPHERE IS DOMINANT (AND ITS PRIMARY AUDITORY INPUT IS FROM THE RIGHT EAR), WHEREAS IN LEFT-HANDED PEOPLE EITHER HEMISPHERE MAY BE DOMINANT. HIGH TONES ARE PERCEIVED AS BEING HEARD AT THE EAR THAT FEEDS THE DOMINANT HEMISPHERE (Deutsch, “Musical Illusions,” Scientific American, 1975).

36

ASSIGNMENT FOR MONDAY

MIDTERM EXAM ON CHAPTERS 1-8 OPTIONAL REVIEW ON FRIDAY AT 1000

EXERCISES 2,4,5,7,8,9 (p.172) Hand them in Thursday if you want them back to study for the exam

ASSIGNMENT FOR WEDNESDAY: READ CHAPTER 10

OPTIONAL SESSION ON FRIDAY, FEB. 14 TO DISCUSS THE EXAM

NO CLASS ON MONDAY, FEB. 17 (PRESIDENTS’ DAY)