2. Chapter 15 Successive Tones: Reverberations, Melodic Relationships, and Musical Scales

3. Audibility of Decaying Sounds in a Room  The first of the tone we hear is the directly propagated wave. Because of the precedence effect, the direct wave will combine with the most direct reflections (within 30 to 50 milliseconds) and be perceived as one.

4. Picture of a Clearly Heard Tone Attack – heard as one because of Precedence Effect Decay – similar to the attack

5. Reverberation Time  The time required for the sound to decay to 1/1000 th of the initial SPL  Audibility Time Use a stopwatch to measure how long the sound is audible after the source is cut off Agrees well with reverberation time It is constant, independent of frequency, and unaffected by background noise

6. Why does Audibility Time Work?  Threshold of hearing temporarily shifted to 60 dB below a loud tone? 60 dB is 1000 times in SPL which then matches the definition of Reverberation Time Measurements show that this happens, but only for a few tenths of a second  Not long enough to make audibility time work

7. Why does Audibility Time Work?  The ear is responding to the rate of change of loudness? Look at example on next slide

9. Advantages of Audibility Time  Only simple equipment required  Many sound level meters can only measure a decay of 40-50 dB, not the 60 dB required by the definition  Instruments assume uniform decay of the sound, which may not be the case

10. Device to Study Successive Tones Tone Generator 2 Tone Generator 1 2 1 Switch Amplifier Speaker

11. Notes on Tone Switcher  Tone generators produce fundamental plus a few harmonics to simulate real instruments  Switching cannot be heard  Reverberation time at least ⅓ sec.

12. Experiment  Start with TG1 on C 4  Switch to TG2 and adjust  At certain frequencies the decaying TG1 will form beats with the partials or heterodyne components of TG2 The beats will be most audible when the amplitudes are equal.

13. Using Reverberation  These experiments show that we can use reverberation as an aid in performing It is easier to perform in a live room (shower) Noise can mask the decaying partials and make pitch recognition more difficult

14. Conclusions  We can set intervals easily for successive tones (even in dead rooms) so long as the tones are sounded close in time.  Setting intervals for pure sinusoids (no partials) is difficult if the loudness is small enough to avoid exciting room modes.  At high loudness levels there are enough harmonics generated in the room and ear to permit good interval setting.  Intervals set at low loudness with large gaps between the tones tend to be too wide in frequency.

15. The Beat-Free Chromatic (or Just) Scale  We will use the Tone Switcher to help find intervals that produce beat-free relationships to the fundamental. The fact that the frequency generators contain harmonics makes this possible Notice that the octave is a doubling of the frequency and the next octave would be four times the frequency of the fundamental

16. First Important Relationship  Three times the fundamental less an octave 3f/2 or an interval of 3/2 or a fifth Fundamental will have harmonics that contain the fifth Five such relationships can be found in the first octave

17. Just Intervals (with respect to C 4 ) ChromaticScales ListedInterval ComputedCent FrequencyNameRatioFrequencyDifference (equal-tempered)(beat-free) C261.63 E329.633 rd 5/4327.0414 F349.234 th 4/3348.842 G392.005 th 3/2392.45-2 A440.00Major 6 th 5/3436.0516 C523.25octave2/1523.260

18. Relationships Among Five Principles Note Frequency (equal-tempered) Interval Ratio Interval Name Resulting Frequency Note F349.233/25 th 523.85C E329.634/34 th 439.51A G392.004/34 th 522.67C F349.235/43 rd 436.54A E329.636/5Minor 3 rd 395.52G A440.006/5Minor 3 rd 528.00C

19. Finding the Missing Steps  Notice the B and D are not harmonically related to C  Finding B A fifth (3/2) above E gives 490.56 Hz A third (5/4) above G gives 490.00 Hz Difference is 2 cents – sensibly equal

20. The Trouble with D  A Fourth (4/3) below G gives 294.34 Hz  A Fifth (3/2) below A gives 290.70 Hz  Difference is 22 cents or 1¼%  Sounded together these “D’s” give clear beats

21. Intervals with B and D 5th CG D CE FAB 4th 5th 3rd

22. Filling in the Scale 3rd 4th Minor 6 G C D CE FAB Notice that C#, Eb, and Bb come into the scheme, but Ab/G# is another problem.

23. Putting numbers to the Ab/G# Problem FromatIntervalRatioGiving E327.04Third5/4408.80 C523.26Third5/4418.61

24. The Problem with F# 3rd min3 C D C EF AB G Other discrepancies exist but these highlight the problem.

25. Saving the Day  As the speed increases discrepancies in pitch are more difficult to detect.  The sound level is greater at the player’s ear than the audience. He can make small adjustments. He is always better tuned than the audience demands.

26. Working Toward Equal Temperament  The chromatic (Just) scale uses intervals which are whole number ratios of the frequency. Scales have unequal intervals E327.04F348.841.066616/15 B490.5C523.261.066716/15 but C#279.07D1D1 290.71.0417 F348.84F# 1 363.381.0417

27. Making the Interval Equal  An octave represents a doubling of the frequency and we recognize 12 intervals in the octave.  Make the interval  Using equal intervals makes the cents division more meaningful  The following table uses

28. Breaking Up One Interval Interval in CentsFrequency RatioFrequencyNote 01.00000261.63C4C4 101.00579263.15 201.01162264.67 301.01748266.20 401.02337267.75 501.02930269.30 601.03526270.86 701.04126272.43 801.04729274.00 901.05336275.59 1001.05946277.19D4D4

29. Comparison Frequency Ratio Musical IntervalCents (Just) Cents (Equal- Tempered) 1/1Unison000 2/1Octave1200 3/2Fifth702700 4/3Fourth498500 5/3Major sixth884900 5/4Major third386400 6/5Minor third316300 8/5Minor sixth814800

30. Pitch Discrepancy Groups  When pitch discrepancies exist in a scale, the cent difference from the equal-tempered interval cluster into three groups Low GroupMiddle GroupHigh Group 12 cents low Equal-tempered frequency 12 cents high  Each group has a range of about 7 cents If a player is asked to sharp/flat a tone, (s)he invariably goes up/down about 10 cents, moving from one group to another.

31. Complete Scale Comparison Interval Ratio to Tonic Just Scale Ratio to Tonic Equal Temperament Unison1.0000 Minor Second25/24 = 1.04171.05946 Major Second9/8 = 1.12501.12246 Minor Third6/5 = 1.20001.18921 Major Third5/4 = 1.25001.25992 Fourth4/3 = 1.33331.33483 Diminished Fifth45/32 = 1.40631.41421 Fifth3/2 = 1.50001.49831 Minor Sixth8/5 = 1.60001.58740 Major Sixth5/3 = 1.66671.68179 Minor Seventh9/5 = 1.80001.78180 Major Seventh15/8 = 1.87501.88775 Octave2.0000

32. Indian Music Comparisons Indiansaregamapadhanisa Westerndoremifasollatido LetterCDEFGABC Indian music uses a generalize seven note scale like the do re mi of Western music.

33. The Reference Raga  The rag is the most important concept of Indian music. The Hindi/Urdu word "rag" is derived from the Sanskrit "raga" which means "color, or passion". It is linked to the Sanskrit word "ranj" which means "to color".

34. The Alap  An Indian piece will usually open with an alap, notes going up and down the scale to establish position and relationship. They will play around a tone, the tone evasion becoming very elaborate. It becomes a game between the player and the listeners. Jazz has similar variations.

35. Indian Modes Play Bilawal Play Kafi

36. Pitch Variations  In Western music we have similar pitch wanderings (vibrato, for example) that the Indian musician would find strange.  We almost always make abrupt transitions from one note to the next without the slides of Indian music.