461-3 Math Basics

Send your corrections, additions, comments, etc. to Bret Battey (bbattey@u.washington.edu).

Arithmetic vs. Geometric Progressions | Exponents and Exponential Curves | Logarithms and Logarithmic Curves | Logarithms and Pitch | Logarithms and Amplitude

A "progression" is a series of numbers created by some rule. Two common kinds of progressions are "arithmetic" and "geometric".

An arithmetic progression is built using addition. For example, we could use the rule "add 2" and create the following progression starting with the number 1:
```
1, 3, 5, 7, 9, 11... etc.
```
An arithmetic progression creates a straight line, like so:

A geometric progression is built using multiplication. For example, we could use the rule "multiply by 2" and create the following progression starting with the number 1:
```
1, 2, 4, 8, 16... etc.
```
Notice that a geometric progression will get big very quickly compared to an arithmetic progression. In fact, it keeps getting bigger faster.

A geometric progression creates a curved line, like so:

Recall that exponents refer to multiplying a number by itself a certain number of times.

The term "power" is sometimes used for this, as in "let's take 3 to the 5th power," which means "let's multiply 3 by itself 5 times".

The number we are going to multiply is called the "base", the number of times we are going to multiply it by itself is called the "exponent."

Consider the geometric progression shown above. It can also be thought of as a series of exponential calculations, like so:

In fact, the kind of curve created by a geometrical progression is called an "exponential curve". It is an essential tool in computer music.

When you use "linseg" or "line" or use the Gen 7 table creator in Csound, you are creating straight lines -- arithmetic progressions.

When you use "expseg" or the Gen 5 table creator in Csound, you are creating exponential curves -- geometric progressions.

Exponential Curve Examples

	An even integer exponent makes a parabola.
	Use a higher exponent, and the parabola is more "sticky" -- it hangs around the X axis for longer.
	An odd integer exponent makes a mirror around the x and y axis.
	Fractional exponents do the opposite of the examples above: they change quickly at the beginning, then slower and slower the larger your X.
	The smaller the fractional exponent, the "stickier" the y axis.

Cute Exponent Tricks

In computer music, we often use control values in a range from 0 to 1. For example, we might design a Csound instrument such that we can send a "brightness" value to it from the score, and in which 0 would be no brightness and 1 would be full brightness, with a full range of fractional in-between values available.

If you want to change a 0 to 1 linear curve into an exponential curve, all you have to do is multiply each point on the curve by itself, i.e. take it to the power of 2. 0 will still be the smallest output value (0*0=0), and 1 will still be the highest (1*1=1), but you'll get a nice exponential curve in-between. If you want a steeper curve, use a higher power.

For example, if you have a linear envelope from 0 to 1 which is used to change the cutoff frequency of filter, you can change the envelope into an exponential one simply applying an exponent to the output.

Here are the visuals. Note that in the examples below, the carat (^) indicates exponentiation.

	Integer exponentiation of a scalar between 0 and 1. A higher exponent provides a steeper curve.
	Fractional exponentiation of a scalar between 0 and 1. The mirror image of integer exponentiation!

You can think of logarithms as being the opposite of exponents.

When we compute a logarithm, we are asking "to what power do we need to take X number to get Y result." In other words, we know the base for an exponential calculation, and we know the desired result. Now what exponent do we need to make it work?

Three ways to ask the same question:

In the above example, 2 is called the "base" of the logarithm.

If you see a "log" button on your calculator, by default it is probably a base-10 logarithm, also called a "common logarithm". That is, for a given number X, it will answer the question "10 to what power gives us X?"

Note, however, that LISP's default is to use a base-e logarithm, called a "natural logarithm". So you have to explicitly provide a base if you don't want to use e as the base. That is, (log 256 2) would be used to implement the above example; (log 256 10) would use a base of 10 rather than 2.

Pitch is not organized in a linear scale.

If pitch were organized in a linear scale, then life would be simple. Then the difference in frequency between C4 and D4 would be exactly the same as the difference in frequency between, say, G5 and A5. Then the difference in frequency between C4 and C5 would be exactly the same as the difference in frequency between C5 and C6.

Instead, C4 is 261.9 Hz.

C5 is 523.2 Hz., and

C6 is 1046 Hz.

That is, C5 is twice the frequency of C4. C6 is twice the frequency of C5.

Here's a chart showing the equal-tempered notes between 440 Hz and 880 Hz. Notice that the distance between each successive note gets larger and larger.

What is constant between each interval is the ratio between the two notes. That is, if multiply the frequency of A4 by X number, you get A#4. If you multiply A#4 by the same number, you get B4. In other words, it is a geometric progression.

The following chart should help make this clear. The difference between the frequencies gets greater as we go up the scale. But the ratio remains the same.

Equal Tempered Notes	Frequency	Freq. Difference	Ratio (Higher/Lower)
A4	440.0
		26.2	1.05946
A#4	466.2
		27.7	1.05946
B4	493.9
		29.4	1.05946
C5	523.2
		31.1	1.05946
C#5	554.4
		33.0	1.05946
D5	587.3
		34.9	1.05946
D#5	622.2
		37.0	1.05946
E5	659.2
		39.2	1.05946
F5	698.4
		41.5	1.05946
F#5	740.0
		44.0	1.05946
G5	784.0
		46.6	1.05946
G#5	830.6
		49.4	1.05946
A5	880.0

What is a "Straight" Glissando?

Let's say we want to make a straight glissando from 400 Hz to 800 Hz. A straight glissando would mean that all parts of the pitch range between 400 Hz and 800 Hz would get "equal treatment" -- that is, we wouldn't hear any part of the pitch range more than any other.

At first it sounds easy! Just make a straight line between 400 Hz and 800 Hz over the duration of a note, right!?

No. If we make a mathematically straight line between 400 Hz and 800 Hz, we don't actually get a straight glissando.

You might think of it this way: there is more distance mathematically between notes near 800 Hz than at 400 Hz. Therefore, with our straight line, pitch will be perceived to move more slowly as it approaches 800 Hz.

It's kind of like driving from the city out into the country on the highway. You are going at a constant 70 miles per hour, but the towns keep occurring further and further apart.

Here's the visual. On the left is our straight line from 400 Hz to 800 Hz. On the right, we see the logarithm of the line, showing us the actual resulting pitch curve as we perceive it in "pitch space". Notice is takes off fast from 400 Hz and then gets shallower and slower as it approaches 800 Hz.


Mathematical Straight Line Controlling Frequency	The Perceptual Result is a Curve in Pitch Space

How do we get around this?

Here's one's way to think about it: if a mathematically straight line gives us an upward-bowed curve in pitch space, maybe a mathematically downward-bowed curve will result in a straight line in pitch space.

This would be like driving from the city out into the country on the highway, but instead of going a constant 70 miles per hour, you start slow and keep going faster and faster such that the towns come by at the same rate even though they are further and further apart. (Don't try this at home.)

Here's the visual. On the left is an exponential curve from 400 Hz to 800 Hz. By taking a logarithm of the line, we see the perceptual result: a straight line in pitch space!


Exponential Line Controlling Frequency	The Perceptual Result is a Straight Line in Pitch Space

Calculating Equal Tempered Intervals

The most common tuning system use in Western music is 12-tone equal tempered tuning. That means that one octave is divided into 12 equally spaced intervals in pitch space.

Recall that an octave involves a doubling of frequency: A4=440, A5=880. One might be tempted to guess that a point half-way in the octave (D#5) would be half way between these two values. But given the exponential nature of the pitch space, this would be terribly wrong!

To account for the exponential space, we use this formula:

Given m, the number of scale steps we want to move, this formula will return the multiplier that corresponds to that interval. We multiply that value by the frequency of our starting note to find the frequency of our new note.

Here's an example for jumping from A4 to D#5 as posed above. D# is 6 semitones (scale steps) above A, so m = 6:

A4 is 440 Hz. So D#5 will be 440 * 1.4142135623730961 = 622.25396744

This also works downward. Let's look at it in LISP this time:

? (* 440 (expt 2 (/ -6 12)))
311.1269837220809

It is easy to alter the formula to create divisions of the octave other than twelve: just replace the 12 with the desired number.

We've just demonstrated that pitch perception is logarithmically related to mathematical pitch.

Similarly, the subjective sense of "loudness" has a logarithmic relationship to amplitude. But it is a step more complex than our pitch example.

Amplitude is the amount of change in the air pressure for the sound.

Intensity (also called level or power) describes the energy delivered to the eardrum by the sound. Intensity is "proportional to the square of the amplitude". That is, intensity = amplitude squared.

Loudness is our subjective response to the intensity being delivered to our eardrum. Our sense of loudness has a nearly-logarithmic relationship to intensity (not amplitude).

Thus, skipping the intensity step, you could just say that loudness has a nearly-logarithmic relationship to the square of the amplitude.

This chain of relationships has significant implications for work in a computer music programming, where the "native" tongue of a system like Csound is amplitude values.

Here is the chain of relationships shown graphically.

On the bottom of the chart is a range of amplitude values we might use in Csound. The graph shows these values squared: the equivalent intensity.

The mind perceives the intensity logarithmically, resulting in the following relationship between input amplitudes and perceived result. Notice: the change from 1000 amps to 2000 amps is a perceptually more significant than the change from 7000 to 8000.

For this reason, the decibel scale (dB) is often used in audio and music applications rather than raw amplitude values.

Decibels can be calculated from intensities or from amplitudes. (Obviously true, since intensities can be derived from the amplitudes.) For our purposes in Csound, we use the decibel calculated based on amplitudes, which is:

Decibels are a relative measurement, not an absolute one. That is, they provide the relative measurement between pairs of values. So if one is going to make comparative measurements, one has to establish a standard reference value to use in all of the measurements. This reference values is used as A1 in the formula, and the comparison value becomes A2.

In the case of digital audio, the common reference amplitude A1 is the maximum amplitude of the computer's digital audio system. In 16-bit audio, that value is 2^15, or 32768. By using this maximum system amplitude as the reference amplitude, the resulting scale will be negative, with the maximum value being 0.

Why use 0 as the maximum? In part, this reflects the fact that in a digital audio system there is an absolute ceiling to amplitude. If 0 wasn't the top value, what would you designate to be the top value? 100? 1000? 2430.5? So let's just make it 0.

Here is our example 0-8000 amps range expressed on the dB curve. Looks the same as the log-of-intensity curve above except that the y-axis scale is different.

Why are decibel values useful? Think of the pitch example. The musical scale is a logarithmic scale: the distance between F and G is the same perceptual distance as that between G and A, even though the frequency numbers are not the same distance.

Similarly, the perceptual distance between -12 and -6 dB is the same as the distance between -6 and 0 dB, even though the raw amplitude differences are not the same.

The following chart is another way of looking at the decibel scale. At the bottom of the graph decibel values are spaced linearly. Corresponding amplitude values appear on the left. There is a perceptually-equivalent distance between -36 dB and -18 dB as between -18 dB and 0 dB. But the former involves far, far less change in raw amplitude values than the later.

This all said, some people still prefer to work with raw amplitude values. One can develop an intuition for working with amplitude values. The trick is to use multipliers to scale amplitudes rather than adding or subtracting dBs. Sometimes that is a more convenient or clear way of thinking.

Here a comparison of how one would do loudness scaling in dBs versus amps:

In order to: Add to the dBs OR Multiply the amps by

Double loudness 6 dB 2

Halve loudness -6 dB .5

Triple loudness 12 dB 3

One should become comfortable working with both approaches. And in music, after all, the ear is the ultimate arbiter, not the numbers.