preprint no. 1265 (7_,-_)

FAST PITCH DETECTION

By

Russell O. HammGotham Audio CorporationNew York, New York

presented at the58th ConventionNovember 4-7, 1977NewYork

AN AUDIO ENGINEERING SOCIETY PREPRINT.This prep/tint has been reproduced from the author's advancemanuscript, without editing, corrections or consideration by theReview Board. For this reason there may be changes should fhispaper be published in the Audio Engineering Society JournalAdditiona/ preprints may be obtained by sending request andremittance to the Audio Engineering Society, Room 449, 60 East42nd Street, New York, N.Y. 10017.©Copyright 1977 by the Audio Engineering Society. ,N/ rightsreserved. Reproduction of this preprint, or any portion thereof, isnot permitted without direct permission from the pub/icationoffice of the Society.

FAST ?I?CI_ DLTIDT!ON

Ju_ell O. Har_ *

Gotham Audio Co:'pora_ion, Nel. Yon'k, N.Y.

I. Introduction

The definite trend _n audio equipment design engineering today is toward

digital systems. However, most of this effort is aimed at digital reali-

zations of well perfected analog systems in order to gain minor improvements

in their performance. The major breakthroughs which digital systems promise

in the areas of signal analysis, synthesis and information compression still

-seem a long way off. Perhaps.the major reason for this lag is the difficulty

of grasping the new digital concepts as opposed to the established ones.

Traditionally, analog signal systems are based on the concepts of integ_atlon

and differentiation where continuance of time and amplitude relationships are

involved. Such concepts are difficult to conceptualize in digital systems

where all operatlons must he structured as discrete, binary statements.

The pitch system this paper _ill describe is s small but important step

forward In digital signal analysis. It is a first step toward our goal of

making practical use of a power analytical tool, the Fast Fourier Transform.

This project was undertaken largely due to the individual interest and

continuous financial support of Mr. Walter E. Sear. Mr. Sear, a strong

proponent of the music synthesizer from its infancy, thought that musical

Instruments of the future must he structured as analysls/synthesizer systems.

His idea was to analyze the signal of a guitar, extract its basic pitch and

use this to control the Moog synthesizer thus producing sounds a _ibrating

string could not possibly generate. Of course, the test of _ny good idea

comes in its application. In this case, it required many years of dedicated

work by some very talented people.

II. Waveform Model

The first step in developing a complex wave analysis system is to develop a model

of the waveform that can be generated in the laboratory under controlled conditions.

This can prove to he the most important step of any design effort and sometimes the

most difficult. While there has been a _reat deal of study in mathematically modeling

the vibrating string, (i) (2) most of this deals with ideal conditions which are not

found in musical instruments. In the case of the Eultar, the vibration is inltiated

by plucklng the string with a pick. The string is supported at one end by a metal

bridge and at the other end by metal fretts along the neck. The fretts are spaced

to produce _ tone musical intervals over a scale of two octaves on each of the six

* Currently Application_ knz-ineer _-ithGoLham Audio

--1--

strings. While in overview the instrument's design seems straight forward_'_imost

ideal, the waveforms it generates are extremely complex end do not lend th_elves/

to simple mathematical modeling.

Perhaps a major _eason for this complexity is a result of changing the string length,

to produce changes in pitch, without changing the pluck point. For example, figure #I

shows the case of an open string note compared with that of a fretted note an octave

higher. Effectively the pluck point is moved relative to the wavelength produced by

the string vibration even though physically it remains at the same point. This one

principle of the guitar yields a vibrational system that dramatically changes para-

meters for each frett position---making an overall waveform model of the system very

difficult to construct.

Classically, we base mathematical models for string vibration on the Fourier series:

s(t) = _KL (ak sin kt + b k cos kt)k=l

Where the term k is usually assumed to be a series of whole number intergers 1,2,3 etc.

that are termed harmonics of the fundamental. Whether due to the lack of mathema-

tical idealness of the string, the varying pluck point relationship or other factors,

we found that models based on this formula did not conform to actual observation.

In fact we found, after extensive computor modeling, that the guitar harmonics are not

a whole number series. For example, the second harmonic term lies in the area of

1.9 to 2.1 times the fundamental and varys with frett position and string decay

time. Our observation is, furthermore, confirmed by Beauchamp (3) in his study of

cornet waveforms where he found the following expression to be closer to the behaviour

of actual musical instruments:

s (t)= _" Ck(t ) cos (k2w fa t + Ok(t))

k=I

where f = averages fundamental frequency

Ck = amplitude of k th harmonic

8k = absolute phase of k th harmonic

Plucking the guitar string immediately sounds the fundamental note followed a few

milliseconds later by the sounding of the second, third and higher harmonics.

-2-

This hehaviourholds true for almost all musical instruments (4) (5), however,

observations of the guitar string's harmonics revealed they actually change fre-

quency as they rise and decay forming clear beat note patterns with the fundamental.

Not only can these patterns be seen on an oscilloscope, they are also q,,_teaudible

on many guit@r frett posltions. Figure 2 shows the waveforms produced by playing

fretts A through C on the B string. Each photograph was taken 250milllseconds

after actually plucking the string to allow the second harmonic to reach full

amplitude. They clearly show the phase change of the second harmonic relative to

the fundamental for each note. Observing these same waveforms over longer periods

of time on an oscillograph reveals clear beat note patterns produced by the second

harmonic as it contlnuously changes its phase.

The waveform of Figure 3 while remarkahly resemhliug the previous oue was actually

produced by two laboratory function generators tuned to a frequency ratio of 1:2.1.

Comparing this closely wlth the guitar notes of Figure 2 shows a clear correlation

of the waveforms. And, pretuning the generators to frequency intervals such as

1:1.9 or 1:2.05 produces various heat patterns which very accurately resemble those

of the _ultar throughout most of its range. That this correlation of waveforms

exists and that the laboratory model should he so simple is to grossly underestimate

the untold hours of searching behind discovering this relationship. These para-

graphs only briefly cover a derivation that could easily be the basis for an entire

paper.

While our simple model does not totally describe the behaviour of the guitar waveform

under all conditions, by using it we were able to see problems in developing our

pitch tracking systems which would have taken much longer using the actual transient

waveform0f the guitar. With hindsight we can state that throughout this work we

found excellent correlation between the behaviour of our circuitry with this test

signal andwlth that of the actual guitar generated signal.

III. Analog Filtering

Using filters to separate the fundamental from the harmonics is a logical approach

to obtaining the pitch of a complex wave (6). Since each string of the guitar covers

a range of only two octaves, a low pass filter with a cut off frequency located at the

-j-

highest pitch for each string will eliminate all but the second and third harmonic

components from the waveform except for the open string where the fourth harmonic

will be _resent. However, in the previous section, analysls showed that the second

harmonic_Is the most troublesome component of the string waveform due to its

shifting phase as shown in Figure 2.

It is therefore necessary to eliminate all the second harmonic content which necessitates

using filters spaced less than one octave apart. Filtering each string separately

requires two filters per string or a total of t_elve for the whole instrument.

Since the frequency range of the guitar is only four octaves_ and the pitch system

will be monophonlc_ that is it will analyze only one pitch at a time, a better approach

is to use a bank of five filters spaced as shown in Figure 4. The level sensing

logic system of Figure 5 is then used to select the output of the fundamental filter

each time a new note is plucked. With proper logic switching only low pass, rather

than handpass, filters are required.

Results of our experiments show that filters must reject the second harmonic by more

than 40 dB for proper operation of the logic. We found this very critical sioce the

dynam/c range of the logic system must be able to track not only the string's full

dynamic range, but also discriminate low level in hand notes from those lying in the

frequency range which is only partially attenuated by a filter's skirt. To meet this

requirement we used ninth order eliptical filters with skirts approaching 500 dB/octave

(7). We constructed these using the state varlahle approach rather than other methods

since the circuitry is inherently easier to fine tune once the filter is assembled.

However, even using high precision components, we found the precise tuning of the

poles and zeros very tricky due to the rather high Q's involved. Figure 5 shows a

computor simulation of our final design for the 622 Hz filter while Figure 6 is a

spectrum analyzer sweep of the actual circuit. One thing our printout did not mention

is the fact that these filters are light sensitive due also to the high Q's and have

to be aligned in the dark.

With high Q filters, ringing is also a problem. Since the initial transient of a

plucked string presents a verysharp wavefront, it looks llke a pulse to the filter's

input. For a filter tuned to the fundamental frequency or above this transient has

little effect on the output. However, for the filters lower in frequency than the

fundamental this pulse will cause the high Q stages to ring. Therefore a slgnal

appears at the filter's outpu_ fo_ a period of time dependent on dampin_ eventhough

the wavetraln has no frequency in this range. This ringing triggers the logic clr-

cultry for that filter and this also has a release constant related to the frequencies

in its band. The end result is a time of uncertainty caused by the logic system's

inabillty to locate the filter with the true fundamental. Therefore, for a pitch

detection system using a bank of filters with logic switching, the uncertainty time

of one output will be directly related to the damping coefficients of the

lowest filter.

Oscillographs of actual pitch systems for the four octave

guitar range showed ringing could cause delays of 25 milliseconds in the

highest playing ranges of the guitar. In the lower ranges delay was minimal.

While 25milllseconds is not long in terms of the carts reaction time, this

problem combined with the difficulty of constructing such precision filters

required evaluation of an alternative approach.

IV. A Digital Approach

A detailed look at Figure 2 suggested to us that zero crossing detection in

conjunction with a software algorithm might be a method of extracting pitch

information from such complex waveforms. Using our test signal we generated

the wave pattern of Figure 7 which represents the behaviour of the guitar

signal over many cycles and clearly shows the beat pattern. Writing a com-

putor program that located negative zero crossing in the waveform and counted

the distance between them, we were able to construct Figure 8. For simplicity

we chose a fundamental frequency of I/2_ and a second harmonic of 2.125/_

producing a beat every eight cycles. Notice that the period 628 which

is the true pitch does not appear in the zero crossing pulse train. Most

of the pulses are spaced about one half this period but not exactly and never

quite the same Value. Since PLL (Phase Lock Loops) generally operate on this

principle of negative zero crossings, it is clear why such a device cannot

be used to track guitar forms. What actually is required is a type of control

algorlthmwhlch can selectively cancel out the extra zero crossings which

clearly are caused by the second harmonic beat note.

-5-

i ,I

I1 T

FIGURE _SA. BLOCK DIACRAM OF LOW PASS FILTER BANK WITH LEVEL DETECTOR

SWITCHING TO SELECT FUNDAMENTAL PITCH.

After a great deal of analysis of the zero crossing patterns produced by

various beat frequency present in guitar waveforms, we developed the following

algorithm:

÷ alternately if:

(A= B) or (A+ B = C) or (B + C = A) or (C = O)

A, B and C refer to the time periods between negative zero crossings shown in

Figure 8. A is the most recent period, B the period Just past, while C is

2 periods previous. The divide function which in reality drops pulses from

the train is performed alternately only if the statement is true. When it is

false the _ is reset, pulses pass through. In other words, at the first true

statement the system divides or eliminates the zero crossing from the pulse

train. If the algorithm of the next pulse period also yields a true state-

ment then the zero pulse is passed and the divide function is reset. If

the second statement had been false, the divide would again be reset and the

pulse wouid also pass. The bottom pulse train of Figure 8 illustrates its

application to the model waveform. All extraneous zero crossings are elimi-

nated and what remains is a series of almost evenly spaced pulses which are

close to the wave length of the fundamental. There is one exception, however.

For each complete cycle of the beat pattern, there is an odd period which

apparently is not related to the period of the pulse train or to the fundamental

frequency.

Scanning over the pulse train of Figure 8 clearly shows that the majority of

the periods fall within I% of each other. If the basic period is 603, as

shown here, then the long period, 804, is 201 units too long. If this error

of 201 units is nOw equally divided among the other periods, the pulse spacing

then becomes 628 or 2 _ which is exactly the period of the model's fundamental.

By adding a predictive clause to the basic algorithm it is possible to calculate

this error period in a real time waveform. Of course looking at a graph of

this function makes matters simple since it is possible to see the complete

picture hut in an actual system this is not possible. Therefore, it is ne-

cessary to recognize when the odd period is received before the end of its

duration so that the error can be corrected immediately. Since past history

can predict when the pulses should he received, if one is missing then it

must indicate an odd period.

In an actual system based on the model of Figure 8p the predicted period would

be 603. If a pulse is not received by 605 a pulse is automatically inserted

in pulse train and an error counter set to count the remainder of the odd

period. Immediately upon receiving the end pulse, the error counter stops,

divides the error by the number of past periods and begins adding this delay

in decreasing increments to each new pulse received. Thus the pulse train is

adjusted to correct pitch of the model waveformand the phase error is Just

cancelled as each new odd period is received. Figure 12.

V. PROGRAHSIHU[ATION

To check the accuracy of the algorithm over a range of conditions it was

programmed on a digital computer. Figure 9 is a simplified flow chart of t_e

program. Initially the computor's memory is filled with the zero crossing

dat_ generated by another program which simulates the model waveform. The

values for A, B and.C are then taken from the memory by the algorithm and"

processed resulting in the data tabulated in Figure I0. Column number one

shows the value of the A period for each calculation. The B and C periods

in each case are the two preceding values. Column number two shows the

resulting pitch values generated by the algorithm after dividing out the

unwanted zero crossings.

C = 0 is actually the algorithm's initializing condition which allows the

first two pulses of the attack transient to pass through. Depending upon the

algorithm's starting point on the pulse train the first two periods may be

at pitch or at the second harmonic, however, the C = 0 term insures that this

will only occur for two periods. The worst case for the guitar would be the

pitch E80 , the lowest note, where two cycles of the second harmonic would

cause incorrect pitch to sound for the initial 12.5 milliseconds. After

these two periods, the algorithm yields a eoeslstant pitch period slightly

shorter than the actual fundamental period except for the odd period. At

the first odd period, the program calculates the difference between where

-7-

the zero crossing should be and where it actually occurs. It then averages

this difference over the number of cycles which have passed since the last

odd period{ and adds this error to each new pitch period. Column three shows

the results after this averaging is applied to the pulse train.

The error term generated by the algorithm, in this case 2.956793005, averaged

over 8 periods results in a period of 628 which is exactly the fundamental.

This error may also be used to calculate the frequency of the harmonic.

In this case dividing the fundamental 2_ by the error term 2.95 yields 2.125

which is exactly the multlple of the second harmonic present in the waveform.

The computor program was used to evaluating the algorithm over a range of

frequency intervals. Sllghtmodlficatlon was required to differentiate the

beat note patterns produced by frequency intervals greater than an octave

from those less than an octave. Beat patterns (sharp & flat) of I in 8_ as

shown by the print out of Figure 10, gave excellent results. However, ratios

lower than I in 5 generated enough incorrect results to indicate further modl-

fication is needed for such cases. It is quite possible that further develop-

ment of the algorithm could lead to a true polyphonic pltch detection system.

However, since the ratio I to 5 is close to a musical interval of a m/nor 7th,

it is far lower than the range of the beat patterns produced by the gultar strings.

In this application then, the algorithm performs well.

VI. Problems of Digital Hardware

The preceding simulation of the algorithm operated at very slow speed on the

computor. Construction of circultry to speed this up to real time revealed

some major difficulties. Available micro-processors are much too slow to

execute such programs therefore a combination of high speed, wired logic

controlled by programming methods was necessary. One of these stumbling blocks

was noise, present in the real world but not in our models. To deal with

this the algorithm _as modified, changing the = to --'_, meaning + 5% which takes

into account Jitter in the zero crossing circuitry. Forming such a window

comparator digitally is difficult since speed was a factor and multiplication

is a long subroutine. Even wlth the fastest processors available this approxi-

_atlon took approximately 750us which left no time to execute the actual al-

gorithm when processing notes in the upper range of the guitar where time

between pulses is less than 1 millisecond. There were two possible solutions

-8-

to the problem. One was to use a lookup table progra_ed into a ROM (Read Only

Memory). Very high speeds are obtained this way, however, size of the ROM

becomes a problem for large tables. Since the prototype system employed 8 hit

digital works, the table would have to be 256 by 256 or 65k bits. While such

devices are available, theymust be specially programmed and are expensive in

small quantities, such a solution seemed superfluous in this case.

The second alternative was to use an analog comparison. While digital counting

systems are arithmetic in nature making binary comparisons very difficult,

analog systems are geometric and such comparisons are simply voltage ratios.

Therefore by translating the pulse spacings to Voltages the comparison cir-

cuit requires only resistive voltage dividers and two opamps, addltlonally

the output is digital. The only difficulty is converting the digital words

to analog voltages, but with the availability of monolithic D to A converters

this is really no problem. Figure II shows the eomparator used in the final

system. Its speed of operation is approximately 5us which is well within the

time available. Here then is a clear example where analog and digital cir-

cuitry are combined to deal with a problem neither could solve easily alone.

Another time consuming problem_..that of digital division, presents itself

when dealing with the odd per£od correction. Here the error must actually

be averaged over the pulse train in decreasing Increments to correct the pulse

spacing to the correct pitch period.

As shown previously, this rate is predicted from'the past history of the signai

and must just cancel itself as the next odd period is received. Again, this

is easy to calculate on a computor bu t in real times lt requires complex

circuitry operating at fairly high speed. After investigating different

electronic schemes, psychoacoustlcs gave the answer.

Listening tests with a guitar signal, but using only the original algorithm

without odd period correction, revealed that the pulse" train was so close

to actual pltch that the difference was not an audible problem. What was

audibl_ was the phase shift at each odd period. Therefore, attention was

shifted to correcting Just this phase shift without fully adjusting the pulse

spacing error.

The technique used to correct this error w e termed--"pulse stretching".

Essentially, Since we can predict the origin of the incoming pulse from past

history, if the pulse is not there then we need to add one. The system then

counts the error time until we actually receive the pulse; this error is

-9-

storedln a register until we receive the next long period. Again a pulse

is added, the error is measured and the result added to the error from the

previous periods. By varying the duty cycle of the resulting train of pulses

in direct proportion to the accumulated error we form the pulse train illus-

trated in Figure 13. Whenever our correction totals one full period, a pulse

is dropped from the train, the register is set to zero and the process begins

again. _n this manner the phase error is eliminated yet the circuitry consists

of only some slmple counters and uses no processing time. Figure 14 shows a

block diagram for one of the digital prototypes. The circuitry is _MOS type

logic operating at a clock rate of IMHz. The pitch unit described here

actually forms only one component of the overall system which consists of approx-

i.mtely 100 IC packages. A photograph of the pitch and envelope board is

shown in Figure 15,

Performance of this system wlth actual guitar waveforms was identical to thatprey

dieted By the waveform model, The worst pitch error was two cycles of the

second harmonic at the onset of the signal. The actual time delay varied

directly with the wavelength which measurements verified as about I milli-

second for IkHz and I0 milliseconds at 100Hz.

VII. Conclusions

We developed a model of the guitar waveformwhieh we used to test two

systems for extracting the fundamental pitch of guitar signals. Both these

systems were constructed, tested with our mode] waveform, and tested with

actual guitar signals. We found excellent correlation between these tests

indicating that our model, which differs significantly from those found in the

mathematleal literature, represents a significant new step in laboratory

simulation of musical waveforms.

A system for pitch extraction using filters was built and its performance

evaluated. Actual testing by guitar players showed the system operated with

excellent accuarcy. However, construction of the necessary filters was

difficult due to the sharp cut offs required by the system's logic switching.

These high Q filters also resulted in a certain amount of ringing on the

attack transient causing a delay of up to 25 milliseconds in the system's

response time.

-I0-

An algorithm suitable for programming on a digital logic system was deve-

f loped and debugged with the aid of eomputor modeling. Constructing this with

:real time hardware revealed additional problems which were solved by com-

bining digital logic with analog circuitry. Testing of the system by gui-

tarists showed that it too operated with excellent accuracy. While the

response of this digital system measures somewhat faster than that of the

filters, both systems operate well within the response time of the ear.

Perhaps the most unexpected feature of both designs was their ahillty to track

the decaying signal through incredible amounts of noise. The logic of both

systems automatically locked the pitch capture range as the signal decays

below a preset amplitude in order to avoid octave JumPing as the noise

becomes more prominent. This is especially a problem with decaying guitar

notes since the second harmonic can reach amplitudes two and three times

that of the fundamental. However, as a side benefit, this technique actually

allows tracking the fundamental through the high amount of noise generated

by the guitar's other strings.

While the emphasis in electronics technology today is on higher speed digi-

tal processors which, as we have seen, lend thenselves well to pitch extrac-

tion we do not underestimate the potential of filtering. _iltering is by far

the simplest, most efficient approach from a circuit standpoint, provided

that certain drawbacks can be overcome. New developements in the technology

of gyrators point toward easier to construct, more stable filters. Combining

this with some of the logic techniques we have presented should lead to the

developement of fast polyphonic p_tch detectors in the very near future. As

stated in the introduction, such digital analysis problems are not necessar-

ily solved by only digital methods; new concepts are involved.

Acknowledgement s

This paper cannot conclude without giving proper credit to guitarist

Mr. Robert Herne Jr. for his ability to get musical sounds from some of our

worst prototypes and supply us with the type of.musical feedback necessary

in such an engineering effort. Credit also to Josh Gindel, Joey Gindel, Bob

Gelman and Jean L. Hudson.

Much of the digital work for this paper was undertaken after leaving Mr. Scar's

employment. Many thanks to Stephen F. Temmer for encouraging us to continue

;he project.

References:

(i) HILLEK and KUIZ, "Synthesizing Musical Sounds by Solvln s the Wave

Equations for Vibrating Objects: Part I," JAES v.19, pp. 462-470

(1971).

(2) HILLER and RUIZ,

JAES 19, pp. 542-551 (1971).

(3) J.W. BEAUCHAMP, "Analysis and Synthesis of Cornet Tones UsingNoullnear Interharmonic Relationships," JAES v.23, pp. 778-795

(1975).

(4) J.A. MOORER, "The Synthesis of Complex Audio Spectra hy Means ofDiscrete Summation Formulas," JAES v.24, pp. 717-727 (1976).

(5) A.H. BENADE, "The Physics of Brasses," Sel American v. 229, pp. 24-35(1973).

(6) V.W. BOLIE et al, "Active-Network Realization of at 97-Channel

Audio Filter Bank", JAES v. 19, pp. 187-196 (1971).

(7) G. VOLPE, "Action Filter Design Handbook", General Instrument Corp.,

Hybrid Division (1976).

-;Z-

PLUCK_POINTNUT

_O_AVE FRETT BRIDGE

FIGURE _1. CROSS SECTIONAL VTEW OF A GUITAR ILLUSTRATING THE POSITIONOF THE PLUCK POINT FOR TWO DIFFERENT STRING WAVELENGTI_SONE

OCTAVE APART. THE OPEN STRING, DOTTEDLINE, IS PLUCKED ATAPPROXIMATELY ITS 1/8th WAVELENGTHWHILE THE OCTAVE, SOLID

LINE_ GETS PLUCKED CLOSER TO ITS I/4 WAVE POINT.

|

A A#

B C

FIGURE #2. WAVEFORMS PRODUCED BY GUITAR NOTES A, A#, B AND C PLAYED ON

THE B STRING. THESE PHOTOGRAPHS, EACH TAKEN 250 MILLISECONDS

AFTER PLU(_(ING, SHOW THE WIDE RANGE OF SECOND HARHONIC PHASESHIFT GENERATED BY ADJACENT FRETTS.

FIGURE #3. MODEL OF GUITAR WAVEFORM USING TWO FUNCTION GENERATORS.

E 2 • E 3 E 4

ESTRINGI ! I

I I

I I ID STRING _3 D4 D5I I

I c3 J JG4 G5G STRING J l I

J I B 3 I B4 I B5

B STRING I J |

J J E4 J IE s E6E STRING J l J

I I I I

FIGURE _4. FILTER CUT-OFF POINTS SHOWN RE'LATIVE TO PITCH RANGE OF THE

SIX GUITAR STRINGS,

.7, -___ --: _'._'_*,-.-'-?.__.---._:_,._ .... :,_- :

_ i. - _'_:_ -.:_._- - -._- -

: _ _ J :

r_: ",_. :. - ,,. _.....:

_ _- _: "_-_

2 _

_ __:: -.= _ = == _,, = _..._._r__...... :_.. = ._ =., z = == __a 5___= r.= . =, _, , , •

FIGURE #5. COMPUTOR SIMULATION OF 9TH ORDER ELIPTICAL FILTER WITH

IdB PASSBAND RIPPLE AND 60dB REJECTION AT 1.05 TIMES CUT-

OFF FREQUENCY.

FIGURE #6. KEALIZATION OF 622 Hz FILTER USING STATE VARIABLE CIRCUITRY.

FIGURE #7. SEVERAL CYCLES OF MODEL GUITAR WAVEFORM ILLUSTRATING PATTERN

PRODUCED BY T_E FUNDAMENTAL BEATING WITH ITS OUT OF TUNE 2ndHARMONI C.

AAAA A AA A AAAA _A AAIvV'UvUv /v vvwtvVvv

MODEL _AVEFORM

PULSE TRAIN "A"

I I II I II IIIPULSE TRAIN "B"

FIGURE #8. TOP WAVEFORM SHOWS BEAT PATTERN PRODUCED BY FREQUENCY RATIO OF i TO 2.125. PULSE TRAINA IS PRODUCED BY NEGATIVE ZERO CROSSINGS. PULSE TRAIN B SHOWS RESULTS OF APPLYING ALGORITHM

TO ZERO CROSSINGS.

FIGURE 39. DIAGRAM OF ALGORITHM FOR PROGRAMMING COMPUTOR MODEL OF PITCHSYSTEM.

COLUMN 2 _.._. "-,='4J'_3_/ 52'_5_5. zD_257.301 628@6_3._03683.30i 628_6_3.2_!6_3.25# 6280603.482,_J.;,#o e,2_8S83.28i603.603. 628660Z.482683.2_I 628_3._7"683.4@2 E,28_6_S.I£$

6@3.482 _28_6t_.2016@3.281 628_5.6a36@3.402 62806_3,345;_83."2#JI ERROR TERM 6289E _:_.}57

COLUMN 3 _ 2. 956 Z'93_:5'/'-'" 625_6g<_. 30;"_62806_3.6_] 628_6_3.38i

_28_68'3.T_5 $2866'3_.25=62806_.257 6288603.3486288EO3.ZOl 628B6_36636280E.83.30i 62E:_603,20i6280_3.254 628_S_3.4_2

FIGURE #I0. COMPUTOR PRINTOUT FOR ALGORITHMAPPLIED TO WAVEFORM OF FIGURE #8.COLUMN I LISTS THE ZERO CROSSING PERIODS, COLUMN2, THE RESULTSOF APPLYING THE ALGORITHM AND COLUMN 3, THE CORRECTEDVALUE FORTRUE FUNDAMENTALPITCH. 2# DIVIDED BY THE PHASE ERROR YIELDS THE2nd HARMONIC TERM, HERE 2.125 TIMES THE FUNDAMENTAL.

WORD A

l!!_ I!!l

I o"i I"I .,.IIII 'llll _WORD B

FIGURE #ii. DIGITAL/ANALOGWINDOW GOMPARATORBLOCK DIAGRAM.

16o16o3I 8o I o 16o3Io3IINPUT PULSES

_ n I F1 F1 F1 H.ERROR COUNTER

OUTPUT PULSES

FIGURE #12. PULSE SPACING TO CORRECT PITCH ERROR. TOP, ALGORITHM PULSE TRAIN,

BOTTOM, INCREMENTAL ERROR FACTOR.

160316°318o416o316o316o31INPUT PULSES

CUMULATIVE ERROR

OUTPUT PULSES

FIGURE #13. CORRECTING PHASE ERROR IN PULSE TRAIN BY "PULSE STRETCHING".NO CHANGE IS MADE IN PULSE SPACING. WHENACCUMULATEDPHASEERROK EQUALS ONE PERIOD IT IS CANCELLED AND PROCESS BEGINS _AGAI N.

__ ZERO [ I

'/..N DETECTOR COUNTER

pROGRAM I

[--OItjT COm'ARATOR D/A

FIGURE #14. PARTIAL BLOCK DIAGRAM OF A DIGITAL PITCH DETECTOR.

FIGURE #15. PHOTOGRAPH OF PITCH CIRCUIT BOARD. PART OF A DIGITAL GUITAR

SYNTHESIZER PROTOTYPE.

FIGURE _16. PHOTOGRAPH OF PROTOTYPE GUITAR SYNTHESIZER USING ANALOG FILTERING

COMBINED WITH DIGITAL LOGIC.