The Math of Music

The waveform of a sound is a graph of the way the pressure changes between the wavefronts. This is usually a very convoluted pattern and the actual sound of the wave is not apparent from looking at the waveform. As a matter of fact, the waveform does not usually repeat exactly from one cycle to another.

Sine starts at the middle and smoothly curves up and down. Cosine starts at the top.

The waveform produced by simple harmonic motion is the SINE WAVE. We graph a sine wave by plotting the function:

F of T = A sine 2 pi F T

To do this we divide up our graph paper horizontally into equal chunks to represent a time scale, and for each time t we want to plot, we multiply t by 2[pi]f (f=frequency) and look up the sine of the result. That sine value is what gets used for the vertical part of the graph.

There is also a function called a cosine wave.

F ot T = A cosine 2 pi F T

and it looks just like the sine wave. The difference is that the cosine of an angle is equal to the sine of an angle 90 degrees bigger. When we have two waveforms which have the same shape and frequency but are offset in time, we say they are out of phase by the amount of angle you have to add to the 2[pi]ft term of the first to move them together. In other words the wave defined by sin(2[pi]ft) is out of phase with the wave defined as sin(2[pi]ft+p) by the angle p.

The second simplest waveform is probably the combination of two sine waves. Any combination of waves is interpreted by the ear as a single waveform, and that waveform is merely the sum of all of the waves passing that spot. Here are a few rules about the addition of two sine waves:

  • If both have the same frequency and phase, the result is a sine wave of amplitude equal to the sum of the two amplitudes.
  • If both have the same frequency and amplitude but are 180 degrees out of phase, the result is zero. Any other combinations of amplitude produce a result of amplitude equal to the difference in the two original amplitudes.
  • If both are the same frequency and amplitude but are out of phase a value other 180 degrees, you get a sine wave of amplitude less than the sum of the two and of intermediate phase.
  • If the two sine waves are not the same frequency, the result is complex. In fact, the waveform will not be the same for each cycle unless the frequency of one sine wave is an exact multiple of the frequency of the other.

If you explore combinations of more than two sine waves you find that the waveforms become very complex indeed, and depend on the amplitude, frequency and phase of each component. Every stable waveform you discover will be made up of sine waves with frequencies that are some whole number multiple of the frequency of the composite wave.

Fourier Analysis

The reverse process has been shown mathematically to be true: Any waveform can be analyzed as a combination of sine waves of various amplitude, frequency and phase. The method of analysis was developed by Fourier in 1807 and is called Fourier Analysis.

The actual procedure for Fourier analysis is too complex to get into here, but the result (with stable waveforms) is an expression of the form:

A sine omega T + B cosine omega T + C sine omega 2 T +D cosine omega 2 T ...

and so forth. The omega (looks like a w) represents the frequency in radians per second, also known as angular frequency. The inclusion of cosine waves as well as sine waves takes care of phase, and the letters represent the amplitude of each component. This result is easily translated into a bar graph with one bar per component. Since the ear is apparently not sensitive to phase, we often simplify the graph into a sine waves only form. Such a graph is called a spectral plot:

Vertical lines spaced horizontally by frequency.

The lowest component of the waveform is known as the FUNDAMENTAL, and the others are HARMONICS, with a number corresponding to the multiple of the fundamental frequency. The second harmonic is twice the fundamental frequency, the third harmonic is three times the fundamental frequency, and so forth. It is important to recognize that the harmonic number is not the same as the equivalent musical interval name, although the early harmonics do approximate some of the intervals. The most important relationship is that the harmonics numbered by powers of two are various octaves.

Non-repeating waveforms may be disassembled by Fourier means also, but the result is a complex integral that is not useful as a visual aid. However, if we disregard phase, these waveforms may also be represented on a spectral plot as long as we remember that the components are not necessarily whole number multiples of the fundamental frequency and therefore do not qualify as harmonics. We should not say that a non-harmonic waveform is not pitched, but it is true that the worse the spectral plot fits the harmonic model the more difficult it is to perceive pitch in a sound.

There are sounds whose waveforms are so complex that the Fourier process gives a statistical answer. (These waveforms are the sounds commonly called noise.) You can express the likelihood of finding a particular frequency as a component over a large enough time but you cannot assign any component a constant amplitude. To describe such sounds on a spectral plot, we plot the probability curve. A very narrow band of noise will sound like a pitched tone, but as the curve widens, we lose the impression of pitch, aware only of a vague highness or lowness of the sound.

Noise that spreads across the entire range of hearing is called WHITE NOISE if it has equal probability of all frequencies being represented. Such noise sounds high pitched because of the logarithmic response of the ear to frequency. (Our ears consider the octave 100 hz to 200 hz to be equal to the octave 1000 hz to 2000 hz, even though the higher one has a much wider frequency spread, and therefore more power.) Noise with emphasis added to the low end to compensate for this is called PINK NOISE.

Sonograms

A sound event is only partially described by its spectral plot. For a complete description, we need to graph the way the sound changes over time. There are two ways in which such graphs are presented. In the Sonogram, the horizontal axis is time, the vertical axis is frequency, and the amplitude is represented by the darkness of the mark. There is a machine that produces this kind of chart by mechanical means.

A Sonogram is a two dimensional image.

Horizontal blotches spaced above each other.

A Spectragraph is a three dimensional graph.

Several envelopes behind one another.

The three dimensional graph gives a clearer sense of how the amplitudes of various components of a sound change. This really shows amplitude envelopes for each partial of the sound. In this case, frequency is represented by the apparent depth into the screen. Most analysis programs allow you to either show high frequency behind low, as this one does, or low behind high. This and the ability to swap parameters among the three axes allows you to pick a view with the least information hidden.

The Mathematics of Electronic Music

One of the difficult aspects of the study of electronic music is the accurate description of the sounds used. With traditional music, there is a general understanding of what the instruments sound like, so a simple notation of 'violin', or 'steel guitar' will convey enough of an aural image for study or performance. In electronic music, the sounds are usually unfamiliar, and a composition may involve some very delicate variations in those sounds. In order to discuss and study such sounds with the required accuracy, we must use the tools of mathematics.

Hertz

In dealing with sound, we are constantly concered with frequency, the number of times some event occurs within a second. In old literature, you will find this parameter measured in c.p.s., standing for cycles per second. In modern usage, the unit of frequency is the Hertz, (abbr. hz) which is officially defined as the reciprocal of one second. This makes sense if you remember that the period of a cyclical process, which is a time measured in seconds, is equal to one over the frequency. (P=1/f) Since we often discuss frequencies in the thousands of Hertz, the unit kiloHertz (1000hz=1khz) is very useful.

Exponential functions

Many concepts in electronic music involve logarithmic or exponential relationships. A relationship between two parameters is linear if a constant ratio exists between the two. In other words, if one is increased, the other is increased a proportianal amount, or in math expression:

Y=kX (linear)

where k is a number that does not change (a constant).

A relationship between two parameters is exponential if the expression has this form:

Y=k^x (exponential)

In this situation, a small change in X will cause a small change in Y, but a moderate change in X will cause a large change in Y. The two kinds of relationship can be shown graphically like this:

Linear function increases gradually, while exponential function curves up out of sight.

One fact to keep in mind whenever you are confronted with exponential functions: X^0=1 no matter what X is.

Logarithms

A logarithm is a method of representing large numbers. It is the inverse of an exponential relationship. If Y=10^X, X is the logarithm (base 10) of Y. This system has several advantages; it keeps numbers compact (the log of1,000,000 is 6), and there are a variety of mathematical tricks that can be performed with logarithms. For instance, the sum of the logarithms of two numbers is the logarithm of the product of the two numbers-if you know your logs (or have a list of them handy), you can multiply large numbers with a mechanical adder. (This is what a slide rule does.) Two times the logarithm of a number is the log of the square of that number, and so forth.

We find logarithmic and exponential relationships within many places in music. 

For instance the octave's numerical relationship may be expressed as Freq= F*2^n...
where F is the frequency of the original pitch and n is the number of octaves you want to raise the pitch.

Decibels

The strength of sounds, and related electronic measurements are often expressed in decibels (abbr. dB). The dB is not an absolute measurement; it is based upon the relative strengths of two sounds. Furthermore, it is a logarithmic concept, so that very large ratios can be expressed with small numbers. 

The formula for computing the decibel relationship between two sounds of powers A and B is 10 log(A/B).

Can sound Kill?

The Spectral Plot

A spectral plot is a map of the energy of a sound. It shows the frequency and strength of each component.

Vertical bars of differing height spaced horizontally by frequency.

Each component of a complex sound is represented by a bar on the graph. The frequency of a component is indicated by its position to the right or left, and its amplitude is represented by the height of the bar. The frequencies are marked out in a manner that gives equal space to each octave of the audible spectrum. The amplitude scale is not usually marked, since we are usually only concerned with the relative strengths of each component. It is important to realize that whenever a spectral plot is presented, we are talking about the contents of sound. In the example, the sound has four noticable components, at 500 hz, 1000, just below 2000 hz, and just above 2000 hz.

Envelopes

Envelopes are a very familiar type of graph, showing how some parameter changes with time.

A line curves up quickly and back down to the bottom gradually.

This example shows how a sound starts from nothing, builds quickly to a peak, falls to an intermediate value and stays near that value a while, then falls back to zero. When we use these graphs, we are usually more concerned with the rate of the changes that take place than with any actual values.

A variation of this type of graph has the origin in the middle:

A curve swings over and under a horizontal line several times.

Even when the numbers are left off, we understand that values above the line are positive and values below the line are negative. The origin does not represent 'zero frequency', it represents no change from the expected frequency.

Spectral Envelopes

The most complex graph you will see combines spectral plots and envelopes in a sort of three dimensional display:

Several envelopes are shown lined up behind one another.

This graph shows how the amplitudes of all of the components of a sound change with time. The 'F' stands for frequency, which is displayed in this instance with the lower frequency components in the back. That perspective was chosen because the lowest partials of this sound have relaltively high amplitudes. A different sound may be best displayed with the low components in front.

Frequency Response

When we are discussing the effects of various devices on sounds, we often are concerned with the way such effects vary with frequency. The most common frequency dependent effect is a simple change of amplitude; in fact all electronic devices show some variation of output level with frequency. We call this overall change frequency response, and usually show it on a simple graph:

A straight horizontal dotted line and a not so straight solid line.

The dotted line represents 0 dB, which is defined as the 'flat' output, which would occur if the device responded the same way to all frequencies of input. This is not a spectral plot; rather, it shows how the spectrum of a sound would be changed by the device. In the example, if a sound with components of 1 kHz, 3kHz, and 8kHz were applied, at the device output the 1kHz partial would be reduced by 2dB, the 8kHz partial would be increased by 3dB, and the 3kHz partial would be unaffected. There would be nothing happening at 200Hz since there was no such component in the input signal.

When we analyze frequency response curves, we will often be interested in the rate of change, or slope of the curve. This is expressed in number of dB change per octave. In the example, the output above 16kHz seems to be dropping at about 6 dB/oct.

Waveforms

Once in a while, we will look at the details of the change in pressure (or the electrical equivalent, voltage) over a single cycle of the sound. A graph of the changing voltage is the waveform, as:

One cycle of a sine curve

Time is along the horizontal axis, but we usually do not indicate any units, as the waveform of a sound is more or less independent of its frequency. The graph is always one complete period. The dotted line is the average value of the signal. This value may be zero volts, or it may not. The amplitude of the signal is the maximum departure from this average.

Sine waves

The most common waveform we will see is the sine wave, a graph of the function v=AsinT. Understanding of some of the applications of sine functions in electronic music may come more easily if we review how sine values are derived.

A series of angles inscribed in circles. (Like a hand moving around a watch.)

You can mechanically construct sine values by moving a point around a circle as illustrated. Start at the left side of the circle and draw a radius. Move the point up the circle some distance, and draw another radius. The height of the point above the original radius is the sine of the angle formed by both radii. The sine is expressed as a fraction of the radius, and so must fall between 1 and -1.

Imagine that the circle is spinning at a constant rate. A graph of the height of the point vs. time would be a sine wave. Now imagine that there is a new circle drawn about the point that is also spinning. A point on this new circle would describe a very complex path, which would have an equally complex graph. It is this notion of circles upon circles upon circles which is the basis for the concept of breaking waveforms into collections of sine waves.

Cartoon of a machine with lots of circular parts drawing a complex
curve.

This fanciful machine shows how complex curves are made up of simple ones.

The Harmonic Series

A mathematical series is a list of numbers in which each new member is derived by performing some computation with previous members of the list. A famous one is the Fibonacci series, where each new number is the sum of the two previous numbers (1,1,2,3,5,8 etc.) In music, we often encounter the harmonic series, constructed by multiplying a base number by each integer in turn. The harmonic series built on 5 would be 5,10,15,20,25,30 and so forth. The number used as the base is called the fundamental, and is the first number in the series. Other members are named after their order in the series, so you would say that 15 is the third harmonic of 5. The series was called harmonic because early mathematicians considered it the foundation of musical harmony. (They were right, but it is only part of the story.)

Temperament

One of the aspects of music that is based on tradition is which frequencies of sound may be used for 'correct' notes. The concept of the octave, where one note is twice the frequency of another is almost universal, but the number of other notes that may be found between is highly variable from one culture to another, as is the tuning of those notes. In the western European tradition, there are twelve scale degrees, which are generally used in one or two assortments of seven. For the past hundred and fifty years or so, the tunings of these notes have been standardized as dividing the octave into twelve equal steps. The western equal tempered scale can then be defined as a series built by multiplying the last member by the twelfth root of two (1.05946). The distance between two notes is known by the musical term interval. (Frequency specifications are not very useful when we are talking about notes.) The smallest interval is the half step, which can be further broken down into one hundred units called cents.

Equal temperament has a variety of advantages over the alternatives, the most notable one being the ability of simple keyboard instruments to play in any key. The major disadvantage of the system is that none of the intervals beside the octave is in tune. To justify that last statement we have to define "in tune". When two musicians who have control of their instruments attempt to play the same pitch, they will adjust their pitch so the resulting sound is beat free. (Beating occurs when two tones of almost the same frequency are combined. The beat rate is the difference between the frequencies.) If the two attempt to play an interval expected to be consonant, they will also try for a beat free effect. This will occur when the frequencies of the notes fall at some simple whole number ratio, such as 3:2 or 5:4. If the instruments are restricted to equal tempered steps, that 5:4 ratio is unobtainable. The actual interval (supposed to be a third) is almost an eighth of a step too large.

It is possible to build scales in which all common intervals are simple ratios of frequency. It was such scales that were replaced by equaltemperament. We say scales-plural, because a different scale is required for each key; if you build a pure scale on C and one on D, you find that some notes which are supposed to occur in both scales come out with different frequencies. String instruments, and to some extent winds can deal with this, but keyboard instruments cannot. If you combine a musical style that requires modulation from key to key with the popularity keyboards have had for the last two centuries you have a situation where equal temperament is going to be the rule.

I wouldn't even bring this topic up if it weren't for two factors. One is that the different temperaments have a strong effect on the timbres achieved when harmony is part of a composition. The other is that the techniques of electronic music offer the best of both systems. It is possible to have the nice intonation of pure scales and the flexability for modulation offered by equal temperament. Composers are starting to explore the possibilities, and some commercial instrument makers are including multi-temperament capability on their products, so the near future may hold some interesting developments in the area.

The Science of Electronic Music

Digital vs Analogue

What is wrong with analogue?
Not much really. If you keep them serviced, manually controlled and test that everything works every day.

What is wrong with digital?
The more general question is "What is wrong with digital processing?" And I'm sorry to say that in some cases the answer is quite a lot. Basically it comes down to either not enough silicon or not enough knowledge, or both. There are certainly digital audio products which are engineered to superlative standards but there's also a lot of stuff, particularly inside PCs which truncates (not dithers) the audio signal to ridiculously small internal word lengths, or doesn't interpolate coefficients, or uses on-screen controls with far to little accuracy or other basically silly techniques.

Making Waves From Numbers

Wavetables

Nearly all digital music systems use some form of wavetable synthesis to generate signals. The wavetable is a section of memory that contains a list of values corresponding to the desired waveform. The computer reads the numbers from the list at a steady rate (the sampling rate), repeating the table when the end is reached. If the table contains a single cycle of the waveform, the frequency produced would simply be the sample rate divided by the number of values in the table:

F=SR/n

The output is a very high fidelity copy of the waveform:

Fig. 1 Using all values in the wavetable gives an exact copy of the stored waveform.

To produce higher pitches, the system skips some values each time. The number of values skipped is the sampling increment. A sampling increment of 4 (reading every fourth value) gives an output two octaves higher than the original.

Fig. 2 Effect of increases sampling increment.

The frequency produced is the original multiplied by the sampling increment.

F=SIxSR/n

It is possible to have fractional increments; the computer interpolates between listed values, or simply reads a number twice..

Point of fact, the number of the most recent value chosen is kept in a register known as the phase accumulator, which has more precison than neccessary to handle the size table used. The high part of the p.a. points into the table, and the low bits contain the fraction. The sampling increment is added to the phase accumulator during each iteration, which will produce the appropriate stepping action.

Digitally speaking, the value obtained from one wavetable lookup is added to the sampling increment for another wavetable lookup.

Fig. 3 A sampling increment of .5.

Amplitude control can be added with a variety of techniques. The straightforward way is to simply multiply the sample value by a number derived in a similar manner from an envelope table. A more efficient technique is available if the waveform is a sine. During each sample period two values are taken from the table: one found the usual way, and another at a location offset from the first according to the envelope. The two values are then added before moving to the output. The sum of two sine waves that are out of phase is a sine of amplitude determined by the phase difference. If the offset equals half the table size, the output will be zero.

Frequency Modulation

Frequency modulation is a very powerful algorithm for creating sounds. The heart of the technique is the way extra tones (sidebands) are created when one oscillator is used to modulate the frequency of another. The carrier is the oscillator we listen to; the modulator is an oscillator that changes the frequency of the carrier at an audio rate. These sidebands are symmetrically spaced about the frequency of the carrier (If all numbers are read twice, the pitch is one octave down), and the size of the spaces is equal to the frequency of the modulator. Increasing modulation increases the number of sidebands, but the amplitude of the sidebands varies in a rather complex way as the modulation changes.

Fig. 4 Spectrum of simple frequency modulation

There are three kinds of relationship between the frequencies of the carrier and modulator, and each produces a different family of sounds.

If the modulator and carrier are the same frequency, all of the sidebands will be harmonics of that frequency, and the sound will be strongly pitched. You may wonder how that can be if there are supposed to be sidebands at frequencies lower than the carrier. If the spacing of the sidebands is the same as the carrier frequency (as it will be if modulator equals carrier), the sideband just below the carrier will be zero in frequency. The sideband just below that will be the carrier frequency, but negative. When that concept is applied in reality, the result is the carrier frequency, but 180deg. out of phase. That sideband therefore weakens or strengthens the fundamental, depending on the modulation index. Further low sidebands interact with upper sidebands in the same way. The regularity of the sidebands produces the strongly harmonic sound usually associated with synthesizers, but if the modulation index is changed during the note (dynamic modulation) the intensity of the sidebands will change in some very voicelike effects.

Fig. 5 Harmonic spectrum generated with FM

If the frequencies of the carrier and modulator are different but rationally related, the result will again be strongly harmonic, and the pitch will be the root of the implied series. (For instance, frequencies of 400hz and 500hz imply a root of 100hz. ) If the carrier is the higher frequency, the resultant sound will be quite bright, sounding like a high pass effect at low modulation and becoming very brash as the modulation increases. The frequency of the carrier is always prominent. If the carrier is the lower frequency, the sound will have "missing" harmonics, and those that are present will appear in pairs (see figure 6). At low modulation index, you will hear two distinct pitches in the tone; as the index is increased, the timbre of the upper pitch seems to become brighter.

Fig. 6 FM with modulator frequency higher than carrier

If the frequencies of the carrier and modulator are not rationally related, the tone will have a less definite pitch, and will have a rich sound. Very often the effect is of two tones, a weak pure tone at the carrier frequency, plus a rough sound with a vague pitch. With careful adjustment of the operator level of the modulator, the carrier tone can be nearly eliminated. If the frequencies of the carrier and modulator are close to, but not quite harmonic, timbral beating will occur at a rate that equals the difference.

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Fig. 7 Nonharmonic FM spectrum

A particularly powerful aspect of frequency modulation as a music generating technique is that the timbres can be dynamically varied. By applying an envelope function to the amount of modulation or the frequencies of carrier and modulator, sounds can be produced that have a life and excitement far beyond that available with the older synthesis methods.

Midi, the Music Math in the Computer


Part of the lure of electronic music is the ability for one musician to perform highly complex compositions, or for the composer to hear his music without the need for performers at all. Splicing and digital editing allows this of course, but it is very tedious. As soon as analog synthesis became affordable, music engineers began looking for methods of automatic control for the systems.

Computer control was too expensive to contemplate in the early days (computer rental was over a million dollars), so a variety of techniques were tried: punched paper tape (Babbit's work on the RCA machine), recorded control signals (Subotnik's Butterflies) and elaborate digital sequencers (early Tangerine Dream). Some decent music was produced this way, but it was still hard work and the results were not really that complex. Electronic music that approaches orchestral music in scope had to wait for the appearance of cheap personal computers.

The first schemes (1974-84) for connecting synthesizers to computers were homemade or sold in small quantities by tiny companies. This led to a variety of systems that were mutually incompatible and so idiosyncratic that only their inventors could write software for them. The usual approach was to connect extra circuitry to the computer that either generated sounds directly or provided several channels of voltage control for modular synthesizers.

In 1983, several synthesizer manufacturers agreed on a communications protocol that would allow keyboard synthesizers to control each other (MIDI). This was very quickly picked up for computer applications, and today we have a mix and match situation, where any of several computers can be connected to one or more synthesizers, provided you have the proper software. MIDI is not perfect (the keyboard orientation and the rather slow data rate cause hassles), but it has provided an impetus for the development of software, has lowered the costs of computer assisted music, and has attracted many new musicians into the field.

The Musical Instrument Data Interface specification defines both the organization of the information transmitted and the circuitry used to connect systems together. The wiring is similar to that used for microphone cables, two wires within a shield. (The MIDI connector has five pins on it, but two of those are not connected. This is done for economy: five pin DIN plugs, widely used overseas for stereo gear, cost less than the three pin model.) Exactly one input may be attached to each output. Multiples are not allowed, but most devices have a "MIDI-THRU" output that simply passes data to the next device down the line. The basic configuration of equipment is a daisy-chain, with one master device controlling a series of slave synthesizers. An alternative arrangement is sometimes used where the data from the controller goes to a splitter box that feeds the data to several outputs, each connected to one synthesizer.

MIDI is a serial system. That means data is fed down a single wire one bit at a time. The bits are generated at the rate of 31,250 per second, but it takes ten bits to make a character and up to three characters to make a message, so it takes most of a millisecond to get anything said. As a rule, each action taken on the keyboard (such as releasing a key) generates a message. The typical message contains a channel number, a code for the key or other control affected, and descriptive data, such as key velocity. The channel number indicates which instruments are to respond to the data. There are sixteen channel numbers.

It is surprisingly easy to generate a lot of MIDI data. For instance, many keyboards have aftertouch; a feature that measures how hard you press on a key as you hold it down and feeds that information into the data stream. If you hit a chord and wiggle your wrists, you might generate several thousand bytes of data. This data may be vital, or it may be useless, depending on exactly how other instruments in the MIDI chain are voiced. When the data stream gets too full, bizarre things begin to happen. Instruments slow down, or messages can get lost. For this reason, many instruments and programs have a filter feature which removes selected types of data. You can even buy a special purpose box to do this.

Two streams of MIDI data cannot be mixed together in the simple manner two analog signals can. The group of bits that makes up a message must be kept intact or the meaning will be garbled. A device that combines MIDI signals, called a Merger, has a microprocessor in it that can recognize messages, assess a priority to them, knows how long the message should be, and prevents potential collisions by storing low priority messages until the output line is available. (This process is like switching freight trains onto a common track with out getting the cars mixed up.)

There are some other special tricks available in boxes. For instance there is a MIDI Delay which simply stores data a while before sending it along. If you connect an instrument's MIDI out to its own MIDI in through one of these, you get some complex echo effects. Another type of box is a Mapper which can change data to compensate for differences in synthesizers. For instance, instruments often vary in the number of presets the can store. If you are using a fancy machine to control several simple ones, the fancy machine may implement all 128 preset locations, and the cheapies may only have 32. When you select preset 33 on the main synthesizer, it will send program change 33, which may have a unpredictable result on the slave. The mapper can be set to change that program 33 to anything you desire. [These features are also available as a part of better computer programs. Any synthesizer with more than 128 presets must have some sort of mapping feature.]

A type of box that is very popular is the MIDI patcher. This device has a lot of inputs and outputs, say eight of each. Controls on the box electrically switch inputs to various outputs, so you don't have to fish around for the MIDI cables to change your system configuration. A particularly intriguing feature is that a configuration can be assigned a program number, so that the patch can be controlled over the MIDI line.

Problems

The MIDI protocol is often badmouthed because the original intentions of the designers are misunderstood. The system was created to allow a simple, cheap, and universal interconnection scheme for instrument controllers and synthesizers. The specification was developed by a committee made up of representatives from several companies, and contains many compromises between various needs and opinions. The specification was inadvertently modified in translation to Japanese, but since the company that made the mistake sells more synthesizers than all other companies combined, their implementation became the standard. The MIDI committee is still active, and adds features to the specification from time to time.
Speed

The complaint heard most often about MIDI is that it is too slow. It takes one millisecond (1/1000 sec) to send the command that starts a note. This is musically imperceptible ( in normal notation, MM=60,000) in simple pieces, but the delay across a twenty note chord can be noticed by a keen ear. The actual effect of this problem on the music is arguable (very few bands are together within twenty milliseconds). Probably the worst case for a performer is when the delay is unpredictably varied. The activities that generate the most frustration are elaborate computer controlled performances. The series connection MIDI system can clog up quickly when detailed control of a lot of instruments is attempted. The cure for this is to use a parallel connection scheme where the computer itself has several MIDI outputs.
Keyboardism

Another complaint is that MIDI sends the wrong information. It is clear that the standard was written with keyboard controllers in mind, and that is sensible, since the organ type keyboard is the most common controller for polyphonic single performer instruments. It is quite difficult but not impossible to design controllers with a continuous effect, such as a wind or bowed string instrument has, but the speed problem becomes extreme in such cases.

There is a proposal for a new standard, called "ZIPI" that addresses these two problems.
Stuck Notes

A perplexingly common occurrence is the stuck note. This happens because each note needs a separate message for note on and note off. If the note on is received, but the note off gets lost because of a loose cable, the note will sound forever. With many synthesizers the only way to get the note to shut up is to press many keys or turn the power off. (Most will quit if you change presets.)
Channels

The channelization scheme chosen causes a lot of confusion, but is not a problem. The channel numbers are really a tag on each command, and instruments have the option of ignoring commands that are not tagged a certain way. Difficulties arise when sending devices and receiving devices are not set to the same channel. The newer instruments can be set up to follow different channels with different voices, and this operation is often not clearly explained. The worst problem is that channel setting is usually hidden deep within an instrument's menus rather than on the front panel where it belongs.
Program Numbers

There is also some confusion about program numbers. The MIDI spec allows for 128 programs, numbered 0-127. Many manufacturers seem to feel that musicians are not ready to accept the concept of program zero, and number their buttons 1-128. Even worse are the systems that use funny numbering schemes, such as 88 meaning program 8 of bank 8.

The problems arise when one encounters a maverick corporation such as E-mu or Oberheim that calls a zero a zero; and when you need to enter program changes directly into a computer program. Of course the widespread belief that 128 programs are not enough has thrown another monkey wrench into the works as each company develops its own scheme for calling up to 1000 presets.
Modes

One of the most troublesome features is omni mode. A synthesizer set to omni will respond to any MIDI message, regardless of channel assignments. A typical problem this can cause is found when using Concertware: the player sends initial program changes for all eight voices at the beginning of a selection, even if there is nothing in some of the voices. A synthesizer in omni mode will respond to all of the program changes and wind up with the program number requested by voice eight. It is a good idea to check the mode of the synthesizer first off, since you don't know what the previous student was doing.(The only point to omni mode is to make synthesizers easy to demonstrate. I think it ought to be called "Salesman Mode".)

Overcoming these problems is a challenge, but is similar to challenges musicians are already familiar with. Here are a few guidelines to maintain sanity.

Use a simple configuration, and stay with it. The MIDI system is designed to have one master controller running a bunch of slaves. Mergers allow the use of two or more controllers, and switchers allow quick reconfiguration of the system, but there is usually little to be gained. The people who repatch the MIDI lines a lot are usually trying to use a black box sequencer and a keyboard as controllers at the same time.

Don't overload the system. Always filter out unnecessary information. Aftertouch, for instance should never be sent unless some device is responding to it. If you are playing with a sequenced track, the pedals are probably of interest only to the synthesizer you are playing.

Know the difference between OUT and THRU. OUT is information generated by the instrument. THRU is a copy of the input data. A few devices such as the Fadermaster provide a mix of the input and its own data at the OUT jack.

Take care of your cables. The MIDI connector is not noted for ruggedness and reliability. It is possible for a plug to look like it is in, but be loose enough to stop the data.

Read the manual. Read the Manual. READ THE MANUAL. Especially the part in the back that shows which MIDI features actually work. Pay particular attention to how to set the channel number and how to turn OMNI mode off.
Midi & the Machine

A MIDI message can consist of from one to several thousand bytes of data. The receiving instrument knows how many bytes to expect from the value of the first byte of the message. This byte is known as the status byte, the others are data bytes. Status bytes always have the most significant bit (msb) equal to one and data bytes have an msb of zero (If a status byte is received where a data byte is expected, the system assumes a transmission error has occurred). Because the msb of data bytes is always zero, actual values are limited to numbers less than 128. This restricts many things in the MIDI universe, such as the number of presets available.

Status bytes inform the receiver as to what to do with incoming data. Many of the commands include the channel number (0-15) as the four least significant bits of the status byte.

Commands are defined for about everything you would expect a synthesizer to do, to wit:

    * Note On
    * Note Off
    * Control Change
    * Program Change
    * Aftertouch (for the entire keyboard, set by the heaviest push)
    * Polyphonic aftertouch (values for each key down)
    * Pitch bend Note Messages

Note On and Note Off

The most common status is note on. [The actual bit values are: 1001nnnn, where nnnn gives the channel number.] Note on is followed by two data bytes, the first is the note number, the second is key velocity. If a keyboard is not equipped to sense velocity, it is supposed to send the value 64. Not too surprisingly, there is a status called note off, with the same data format. Note off is actually not used very much. Instead, MIDI allows for a shorthand, known as running status (Running Status applies to any status message, so this system can be used to send a series of program changes or aftertouch messages just as easily). Once a note on is received, an instrument interprets each pair of data bytes as instructions about a new note. If the velocity data is zero, the instrument performs a note off with velocity of 64.

This manner of thinking, requiring separate actions to start and stop a note, greatly simplifies the design of receiving instruments (the synthesizer does not have to keep time), but creates the potential for hung notes when a note off gets lost. The MIDI designers provided some features to compensate for this problem. There is a panic command, all notes off, which is generated by some keyboards and even some special effects boxes.

The note numbers start with 0 representing the lowest C. "Middle C" is supposed to be note 60. Middle C is usually known as "C4", but for some reason most manufactures call it C3.
Control Change

There is a group of commands called control changes, that relate to actions of things like foot pedals, modulation wheels, and sliders. Each command has two parts, defining which control to change and what to change it to. These are not very rigidly defined, so many systems allow assignment of controllers as part of preset definition. These are some of the official definitions: (numbers are actual data numbers)

    * 1 Mod wheel
    * 2 Breath controller
    * 4 Foot controller
    * 5 Portamento time
    * 6 Data entry knob
    * 7 Main Volume
    * 8 Balance
    * 10 Pan
    * 11 Expression

A controller usually has a single data byte, giving a range of 0-127 as the value. This is rather coarse , so the controllers from 32 to 63 are reserved to give extra precision to those assigned from 0 to 31.

The numbers from 64 to 69 are switches or pedals:

    * 64 Sustain 65
    * Portamento
    * 66 Sostenuto
    * 67 Soft

The numbers from 98 to 101 allow extened control changes called NRPN & RPN values.

NRPN (pronounced "nurpin") stands for Non Registered Parameter number. This allows companies to define their own extensions to the list of control changes. The approach is to first tell what to change, then what to change it to. There are two messages devoted to what to change- MSB and LSB, or Most Significant and Least Significant Bytes. Together, they indicate which parameter of the instrument to change. This limits a manufacturer to 16,000 or so controls per instrument. The value is transmitted in the "Data Entry", "Data increment" or "Data decrement" controller messages. Data Entry (which has an optional LSB like any of the frist 32 controls) sets the value. Increment and Decrement add or subtract from the current value.

There are also a couple of RPNs or Registered Parameter Numbers. RPNs allow the MMA to add defined controllers as the need for them becomes apparent. They are almost the same format as NRPN:

There are some specialized control messages:

    * 121 Reset all controllers
    * 122 Local Control
    * 123 All Notes Off
    * 124 Omni Mode Off
    * 125 Omni Mode On
    * 126 Mono Mode On
    * 127 Poly Mode On

Reset Controllers and All Notes Off have the obvious effects. What is not so obvious is that neither will work on a synthesizer set to Omni mode.

Local Control allows you to disconnect a keyboard from the synthesizer it is built into. The Keyboard still sends MIDI Data, and the synthesizer still responds to MIDI data, but pressing a key will not necessarily produce a sound. This is useful when you are using a computer based sequencer and want the computer to have total control of the sounds.

Controller 0 is the Bank change message. A bank change followed immediately by a program change should take you to a new sound on a different bank, but the actual use varies from instrument to instrument.

The bank change command is controller 0, optionally followed by its LSB partner, controller 32. The change should not take effect until a program change is received, so the messges sent should be:

    * cntrl 0 bank number MSB
    * cntrl 32 bank number LSB
    * program change

No synthesizers have enough banks to need both MSB and LSB, so a few expect the number in the MSB and don't need an LSB. Others demand an MSB of 0, with the bank number in the LSB. They are supposed to wait for the program change before changing sounds, but a few change instantly upon receipt of the bank change.

The actual bank numbers are not always what you would expect. Many machines use 0 1 2.. etc, but the Roland JD 990 numbers its two banks 80 and 81. There is also the same confusion about starting with 0 or 1 that we have on the program numbers. Some instruments also use bank changes to switch between performance and voice play mode.
Channel Modes

The modes take some explaining. When all this was set up, most synthesizer keyboards were monophonic, like the Moog. (Monophonic here means they would only play one note at a time.) A few instruments could play chords, these were Polyphonic. The original MIDI spec assumed you would use a MIDI channel to control each oscillator on an instrument or you would have instruments that would play chords from one channel. No one foresaw the current situation, where multitimbral synthesizers can play chords in response to several if not all of the MIDI channels.

There are four possible combinations of the mode messages:

    * Omni On, Poly On or Mode 1: The synthesizer plays everything it gets.
    * Omni On, Mono (Mode 2): The synthesizer plays only the most recent note.
    * Omni Off, Poly (Mode 3): The synthesizer plays chords on one channel.
    * Omni Off, Mono (Mode 4): The synthesizer plays the most recent note received on its base channel. It also plays the most recent note received on the next channel, and the one after that, until it's out of oscillators.

There is no message for Multi mode, so it has to be chosen from the synthesizer panel.
Program Change

The sound of a synthesizer is determined by the connections between the modules and settings of the module controls. Very few current models allow repatching of the digital subroutines that substitute for modules, but they have hundreds of controls to set. The settings are just numbers, and are stored in computer type memory. In a computer, a particular group of settings would be called a file. In synthesizers, it's a Patch, Preset, Voice, or Tone for different brands, but the official word is program. A MIDI message may call one of up to 128 of these by sending data of 0 to 127.

Most modern synthesizers have more than 128 presets. Different manufacturers and models implement a variety of ways to make these accessible by MIDI commands:

Maps On some instruments, 128 presets are called up by the Program Change commands, but you can choose ahead of time which presets are called by which command. You can assign preset 4 to Pgm Change 1, preset 205 to Pgm Change 2, and so forth. This kind of list is called a Map, and is occasionally used for other operations too.

Banks Many instruments organize the presets in groups of 64 or 128. Then you pick which group is in use at any time by pressing buttons on the instrument. At least one of the banks will be writeable, and you can copy presets into it if you want to combine some from different permanent banks. Bank switching may be possible via MIDI, but the method for doing this is not standardized.

Performances Many instruments let you define a multi channel (or complex keyboard) setup that combines various presets. These Performance setups (also called Multis, or Mixes) are stored in a bank of their own. The Program Change command then picks among these. Performance setups can also have settings for processors, volume, pan, and so on.

(When an instrument is in multi channel performance mode, program changes may change the performance setup, or may change the program on a particular channel. This depends on a setting hidden somewhere in the MIDI setup of the instrument.)

Program changes have data values of 0 to 127, but are supposed to be called Programs 1-128. Many Synthesizer and Software companies do not, so you basically have to experiment to find out what will happen when a particular application sends a program change to a particular instrument.
Pitch Bend

Most of the wheels and knobs on a synthesizer generate control change messages, but one gets a status message of its own. This is the Pitch Bender. A dedicated message makes it possible to efficiently send a bend value of 14 bits. If you try to do pitch bend with only seven bits of precision, you either have to restrict the range or you get audible steps. Unfortunately, no manufacturer takes advantage of this.
Aftertouch

On many keyboards, if you lean into the key as you hold it down, you generate controller messages. This is a very expressive feature. On normal aftertouch (also known as Channel Pressure) the values sent correspond to the key with the most pressure.
Polyphonic Aftertouch

Polyphonic Aftertouch sends separate pressure information for each key. This is a tremendous amount of information, and only a couple of synthesizers respond to it.
System Messages

The preceding messages are Channel Voice Messages which apply only to instruments set to the specified channel. System Messages apply to all machines:

    * Song Pointer
    * Song Select
    * Start
    * Stop
    * Continue
    * Clock
    * Midi Time Code
    * Active Sensing
    * System reset

With the first of these commands, several sequencers or computers can be cued to a preset point in a composition and run together. The clock command is a single byte that is "broadcast" by a master sequencer at the rate of 24 per quarter note. Sequencers can follow this clock and stay in tempo. This clock can be recorded on tape and played back with a suitable adapter. If this recording happens to be on a multi-track tape deck, complex sequences can be built up using many passes with a single synthesizer.

Song Select and Song Pointer cue up sequencers and drum machines, and Start, Stop and Continue control their operation.

An even more sophisticated synchronization system called MIDI Time Code is now available. In this system, time markers are recorded continuously on the tape. When the tape is played, sequencers will be automatically cued to match the tape. (This is a version of SMPTE time code, which does the same thing for video and audio editors.) Moreover, sequencers can be set to start doing their thing at arbitrary points in the composition, allowing such techniques as "slipping tracks" and eliminating the tedious process of composing long sequences of rests.

Active sensing warns an instrument if there is a serious malfunction. Once the active sensing command has been received, the instrument expects something on the MIDI line at least every 300 milliseconds (If the controller has nothing to say, it sends more active sensing messages.). If nothing is received the instrument shuts all notes off.

System Reset is supposed to return synthesizers to their power Up state. Hardly any recognize this.

The final group of commands are the SYstem EXclusive commands. These are commands that the manufacturer may define as they like. (Each manufacturer is assigned an ID code to prevent confusion.) The data stream may be arbitrarily long, terminating with a command known as End of Exclusive (EOX.) These messages are used for passing preset information, sequences, and even sound samples from one machine to another, and provide the foundation for the editor/librarian computer programs. Messages are not limited to program data; on the Yamaha instruments, system exclusive commands can be used to control everything, including the power switch.
Extensions To Midi

The Midi Manufactures Association has not stopped their work. Since the initial definitions they have produced the following:

MIDI Time Code Described above, MTC made it possible to link MIDI systems to video and other time based operations.

Sample Dump Standard This allows samples to be transferred from one brand of sampler to another.

Standard MIDI File This one allows MIDI tracks recorded on one sequencer program to be used by another, even if it runs on a different kind of computer.

MIDI Show Control This defines ways to automate theatrical productions, synchronizing lighting effects, sound, and even fireworks.

MIDI Machine Control This allows remote control of audio and video recorders. With this and Time Code, you can run an entire studio from the computer.

And then there's....
General MIDI

General MIDI is a response to a problem that arose with the popularity of the Standard MIDI file. As composers began exchanging compositions (and selling them) in SMF format, they discovered that pieces would change when played on different synthesizers. That's because the MIDI program commands simply provide a number for a preset. What sound you get on preset four is anybody's guess.

General MIDI defines a standard list of voices. (This list is a sort of snapshot of the synthesizers that were popular in 1991. The easiest way to get it is to buy a GM compliant synthesizer.) Not only the names are standardized-- envelope times are defined so the right sort of textures are maintained. Standard MIDI also defines channel 10 as the percussion channel, and gives a map of the drum sound to associate with each note. A GM instrument may create these sounds in any manner, so there's still a lot of variation, but you no longer get a tuba when you expect a bass drum.

Most synths that support General MIDI do so by providing a bank titled GM. This is mostly a rearrangement of sounds from other banks.

General MIDI is most important in the soundcards that plug into PCs. These allow game programmers to create MIDI based scores instead of including recorded sounds for the music cuts.

General MIDI is coming to Macintosh computers as part of the expanded QuickTime system. Midi scores will be playable with no synthesizers at all!
Sampling
In music, sampling is the act of taking a portion, or sample, of one sound recording and reusing it as an instrument or a different sound recording of a song. The wide spread use of sampling in popular music originated with the birth of hip hop music in New York in the 1970s, but in the groundbreaking music technology of granular synthesis, sampling will be used to breakdown a soundwave into a fundamental grain. Which can be applied to entirely new forms of composition or used in a stochastic musical form. Sampling is typically done with a sampler, which can be a piece of hardware or a computer program. Sampling is also possible with tape loops or with vinyl records on a phonograph. Breaking down a sample, soundwave or wave file into it's components opens an entirely new set of mathematical elements. Musical composers of the past sampled with tape, and by composers like Louis & BebeBarron (The Forbidden Planet Soundtrack) or Iannis Xenakis spent long hours cutting tape into samples to score their compositional works. Today this long process can be dramtically shortened by the use of sampling hardware or computer software. These products sample audio to a file that is either .WAV or .AIFF format.

WAVs and AIFFs

Both WAVs and AIFFs are compatible with Windows, Macintosh, and Linux operating systems. The format takes into account some differences of the Intel CPU such as little-endian byte order. The RIFF format acts as a “wrapper” for various audio compression codecs. Though a WAV file can hold compressed audio, the most common WAV format contains uncompressed audio in the linear pulse code modulation (LPCM) format. The standard audio file format for CDs, for example, is LPCM-encoded, containing two channels of 44,100 samples per second, 16 bits per sample. Since LPCM uses an uncompressed storage method which keeps all the samples of an audio track, professional users or audio experts may use the WAV format for maximum audio quality. WAV audio can also be edited and manipulated with relative ease using software. The WAV format supports compressed audio, using, on Windows, the Audio Compression Manager. Any ACM codec can be used to compress a WAV file. The user interface (UI) for Audio Compression Manager may be accessed through various programs that use it, including Sound Recorder in some versions of Windows. Beginning with Windows 2000, a WAVE_FORMAT_EXTENSIBLE header was defined which specifies multiple audio channel data along with speaker positions, eliminates ambiguity regarding sample types and container sizes in the standard WAV format and supports defining custom extensions to the format chunk.There are some inconsistencies in the WAV format: for example, 8-bit data is unsigned while 16-bit data is signed, and many chunks duplicate information found in other chunks.WAV files can also contain embedded IFF "lists", which can contain several "sub-chunks".

Mathematics in Music Composition

Analysis Music

Heinrich Schenker (June 19, 1868 - January 13, 1935) was a music theorist, best known for his approach to musical analysis now called Schenkerian analysis.

The Schenkerian analysis was founded on that most musical works have a fundamental tonal structure embracing the whole composition. The Schenkerian technique reduces a composition down into successive scores, each with fewer and fewer notes. The downward progression from one score to another involved grouping notes together and replacing each group by a single note. The final score, called the background, contained only one note that represented the work's fundamental tonal structure. This fundamental mathematical decomposition process certainly derives many new  musical works from the original.

For Schenker, the natural hierarchies of music were part of a naturally ordered universe, and tonal music inherently reflects this order no matter what choices the composer makes to detail the music. His analytical system yielded its most productive results when applied to music of the common practice. Schenker did not consider any musical compositions that failed to follow traditional principles of tonality. Among the Schenkerians musical practitioners that persists to this day, Oswald Jonas, a traditional disciple who is more strict about the theory than Schenker himself, promoted the viewpoint that the analysis belonged only in the realm of triadic tonal music. Hans Keller, who has worked on the Functional Analysis form of composition, sees music as a constant battle between repeated themes and new information. In 1967, Hans Keller had a famous encounter with the rock group Pink Floyd (then called 'The Pink Floyd') on the TV show The Look of the Week. During the interview, Keller is quoted saying that Pink Floyd's music was "...a little bit of a regression towards childhood".

Stochastic Music

Iannis Xenakis (May 29, 1922 – February 4, 2001) was an ethnic Greek, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers.  Xenakis pioneered the use of mathematical models such as applications of set theory, varied use of stochastic processes, game theory, etc., in music, and was also an important influence on the development of electronic music.

The use of the term stochastic is to mean based on the theory of probability. In mathematics, specifically in probability theory, the field of stochastic processes has been a major area of research. A stochastic matrix is a matrix that has non-negative real entries that sum to one in each row. Probability is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. 

Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more.

1.) Discrete probability theory deals with events that occur in countable sample spaces.

Examples: Throwing dice, experiments with decks of cards, and walking with no destination in mind.

2.) Continuous probability theory deals with events that occur in a continuous sample space.

3.) Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions therefore have gained special importance in probability theory. They are; Weak convergence, Convergence in probability & Strong convergence.

Stochastic music is the name given to a style of generation of musical ideas developed by Iannis Xenakis, described in his book "Formalized Music". This is not the same as random music, but rather describes a technique for developing a musical progress with a random walk-like method.

Stochastic music emerged in the years 1953-55, when Iannis Xenakis introduced the theory of probability in music composition. Xenakis decided to generalize the use of probabilities in music composition. The work Achorripsis was his first work towards this generalization. In Achorripsis, a small number of stochastic rules are applied to generate both the parameters of the notes and the global structure. The architecture of the piece can be read in a two-dimensional matrix that is defined in a space where seven rows representing seven groups of instruments evolve in time. During this time all the stochastic computations were made by hand or with the help of calculating machines that were rudimentary. In the 1960s, Xenakis started to use the computer to automate and accelerate the many stochastic operations that were needed, entrusting the computer with important compositional decisions that are usually left to the composer. In Xenakis' work ST10, the composition of the orchestra (expressed in percentages of groups of instruments) is computed by the machine, as well as the assignment of a given note to an instrument of the orchestra. At the end of the computation of the musical work, the numerical results were transcribed into traditional notation so that the music could be played by an orchestra. 

In the 1960s, Xenakis put forward the idea of extending the use of stochastic laws to all the levels of the composition, including sound production. Xenakis said, "Although this program gives a satisfactory solution to the minimal structure, it is, however, necessary to jump to the stage of pure composition by coupling a digital-to-analog converter to the computer". This proposition was renewed in 1971: Any theory or solution given on one level can be assigned to the solution of problems of another level. Thus the solutions in macro-composition (programmed stochastic mechanisms) can engender simpler and more powerful new perspectives in the shaping of microsounds than the usual trigonometric functions can ... All music is thus homogenized and unified. In the 1970s, at the University of Indiana, Xenakis experimented with new methods for synthesizing sounds based on random walks, the theoretical aspects of which are described in probability theory. In 1991 Xenakis returned to his dream of making music that would be entirely governed by stochastic laws and entirely computed. At CEMAMu, Xenakis wrote a program in Basic that runs on a PC. The program is called GENDY: GEN stands for Generation and DY for Dynamic; it generates both the musical structure and the actual sound.