Mauro Graziani

A Max/MSP primer

Lesson 02
Numbers & Amplitude

This materials are part of the lecture notes from my lessons about Max/Msp
at the E.F. Dall'Abaco Conservatorium, Verona (Italy)

email me: mg AT maurograziani DOT org

Chapter Codes

Chapter topics are coded using title colors:

Black Title: normal chapter you must read.
Blue Title: tips or tricks maybe useful but not essential.
Maroon Title: advanced topic for advanced users. More information but not essential for current lesson. Try to read; skip if you don't understand.

English is not my first language. I apologize for errors. This material is in english to be worthwhile to more people and to give my students a little thrust (in local dialect is called "descanta-baucchi")

NumbersNumbers as Sliders

Now let's see another way to control the cycle~ frequency. Get two number box, one integer and the other floating from the palette and put on the patcher window. Connect with a cycle~ and connect the cycle~ with a dac~.
Note that I don't write a frequency value on the cycle~. This means that the cycle~ will start using its default frequency that is 0, i.e. don't play until a valid frequency value is received from the controls.
Add the usual start and stop messages to the dac~ as in figure and go to play mode.
In play mode, click on the integer number box and drag the mouse on high. You'll see the value in the box rapidly changing. When the number will reach an audible frequency (e.g. 20-30, but could be more, depending on the quality of your speakers; remember that the little computer speakers often don't work at all under 100 Hz), you will begin to hear a glissato moving from low to high frequencies.
Now, you see, the number box don't act as button, like messages. It works as a slider that you can control dragging the mouse up and down.
Now try the floating point number. This one works as the former, but a little difference: if you click on the left part of the box (near the little triangle) only the integer part of the number change; if you click on the right, the fractional part also change. So clicking on the left you have a coarse control and clicking on the left you get a more fine control.
You can also change the box dimension to be sure to see all the number: in edit mode, move the mouse near the lower right corner of the box until the cursor become a cross, then click and drag.
it is very important that you are always aware of the patch data flow.
You must be able to figure in your mind how and when the data flow in a Max/MSP patch.

It could be a little difficult for beginners, but only by this way you can understand the more complex patches.
So let's stop just a moment to understand exactly what's happening in this simple patch.
  1. When the number changes (and only when the number changes) a message start from the number box, runs through the wire and reach the cycle~. This number is not a frequency, it's a pure number. Its only meaning are its value and its format (integer or float).
  2. The cycle~ receive this number in its left inlet and uses it as a frequency value only because on the left inlet a frequency value is expected. If you connect the number box to the cycle~'s right inlet, you'll hear no frequency change because on the right inlet a phase value (not frequency) is expected.
  3. So it's the cycle~ that gives the number a meaning, depending on the inlet the number reach. Note that the cycle~ don't mind about the number format because a frequency can be written both way (with or without fractional part), but, with other objects, the right format also is requested by some inlet. On the computer, the number 50 is different from the number 50.0 so the right number with the wrong format could be seen as an error.

Floating InspectorThe Floating Inspector

As you see, while dragging on a number box, you can go up and down with no limits. You can also output negative numbers. Of course, setting minimum and maximum to some meaningful value should be better.
To do so, go to edit mode, select the number object and click the menu Windows → Show Floating Inspector.
This dialog allows you to set many options for the current selected object. Note that the floating inspector options are not the same for all objects, because any object has its own distinctive features. Some objects have no floating inspector at all.
For number box, integer or floating, the most important feature is the range. Here you can set minimum and maximum values.
Now we are using this number to control a frequency, so set the range to reasonable values in Hertz e.g. min 20-30, max 4000-5000 (the lower note on the piano is 27.5 Hz, the higher is 4186 Hz). Be sure the “No Min” and “No Max” check boxes are unchecked. Now you can freely drag the mouse.
By the floating inspector you can also set the colors of the number object, the display format and a bunch of options:

Amplitude Control

The cycle~ has no amplitude control in itself. It always output the wave at the maximum amplitude value.
To control the sound amplitude we must insert an amplifier between the signal generator (the cycle~ in our example) and the dac~.
To understand how an amplifier works on a digital signal, remember that a digital sound wave is a stream of number running at SR speed (e.g. 44100 numbers per second). Each number is an amplitude sample of  the sound wave. So the wave in this figure
Digital sound wave
is in fact a stream of numbers like this one

0 72 144 215 285 352 417 479 538 593 644 690 732 769 801 828 849 865 875 880 879 873 862 846 825 799 769 736 699 659 616 571 524 476 426 377 327 278 229 182 136 92 51 12 -23 -56 -85 -111 -133 -151 -165 -175 -182 -184 -184 -179 -171 -161 -147 -131 -113 -92 -71 -48 -24 0 24 48 71 92 113 131 147 161 171 179 184 184 182 175 165 151 133 111 85 56 23 -12 -51 -92 -136 -182 -229 -278 -327 -377 -426 -476 -524 -571 -616 -659 -699 -736 -769 -799 -825 -846 -862 -873 -879 -880 -875 -865 -849 -828 -801 -769 -732 -690 -644 -593 -538 -479 -417 -352 -285 -215 -144 -72 0 72 144 215 285 352 417 479 538 593 644 690 732 769 801 828 849 865 875 880 879 873 862 846 825 799 769 736 699 659 616 571 524 476 426 377 327 278 229 182 136 92 51 12 -23 -56 -85 -111 -133 -151 -165 -175 -182 -184 -184 -179 -171 -161 -147 -131 -113 -92 -71 -48 -24 0 24 48 71 92 113 131 147 161 171 179 184 184 182 175 165 151 133 111 85 56 23 -12 -51 -92 -136 -182 -229 -278 -327 -377 -426 -476 -524 -571 -616 -659 -699 -736 -769 -799 -825 -846 -862 -873 -879 -880 -875 -865 -849 -828 -801 -769 -732 -690 -644 -593 -538 -479 -417 -352 -285 -215 -144 -72 0 72 144 215 285 352 417 479 538 593 644 690 732 769 801 828 849 865 875 880 879 873 862 846 825 799 769 736 699 659 616 571 524 476 426 377 327 278 229 ...

where each number represent the amplitude of the wave at a given instant like a graph with the time on x-axis and the amplitude on y-axis.
So, to change the wave's amplitude we can simply multiply each number for a given value. If that value is greater that 1, the amplitude increase. If it is lower that 1, the amplitude decrease. In the following image, a plotted digital wave (red) is multiplied by 2.0 (blue) and by 0.5 (yellow). Here is the result. You can clearly see that only amplitude is affected.

Patch 02Applying amplitude control to a signal

To apply amplitude control to a signal we only need a multiplier between the signal generator(s) and the dac~ (see patch on the left). Be sure to pick the MSP multiplier (with tilde: *~) from the MSP Operators group, not the Max multiplier (with no tilde).
To set the amplitude value we can use a floating point number. Now we must set the range of the amplitude control.
In MSP, the maximum amplitude a dac~ can handle is 1, that is, when reaching the dac~, a wave must run between 1 to avoid clipping and distortion. Of course a zero amplitude is no signal.
The cycle~ generator already output a wave at maximum amplitude, so our amplifier is in fact a damper and will have the range 0 - 1 (here the reason we must use a floating point number).
If the amplitude control would reach a value greater that 1, a wave with range > 1 would reach the dac~ and be clipped: you would hear no amplitude increment, but harmonic distortion (try a value > 1).

Amplitude evaluation

Measuring the amplitude of an incoming signal could be very important when you work with external signals. Now we'll see different ways to do it using standard Max/MSP objects.
To make it simple, we'll apply the measurement to the cycle~ sine wave for now. Look at the the patch 03.
Patch 03
Here you can see and try different ways to evaluate a signal's amplitude.

                  control value
This is not a real measument.
Here we convert the given amplitude value to dB for control purposes only and see if the real evaluations match.
By the way, here you can find the atodb object that convert a number to deciBels. It is taken from the Max "Math" section on the object list.
The atodb output is connected to a number object to show the resulting dB. The number object is made read-only (check the option in the floating inspector).
                  signal amp
Here is the first real signal amplitude evaluation.
You can find the peakamp~ object on the "MSP Analysis" section. This object accept a signal in its left inlet and output the maximum absolute amplitude every k milliseconds, where k is the argument (100 in figure). You can also control the k value sending an integer number to the right inlet.
The peakamp~ object is very accurate and it's a good and simple way to evaluate the peak amplitude of a signal.
Level meters
In the palette you can find two graphic objects to evaluate and show a signal amplitude.
Peak level
The peak level meter accept a signal in its left inlet and display the amplitude by the usual led line. It also output the evaluated absolute amplitude.
The object appearance can be fully edited using the floating inspector.
Running the patch you'll see that when the input signal has a low frequency (set the frequency control to 50 Hz or lower) the result is rapidly changing. This is not a bug. It does not happens with natural sounds. Here it happens because:
  1. The peak level meter has a little buffer. When the wave frequency is low it see only a section of the wave cycle, but, more important:
  2. Here we supply a single sine wave without harmonics. In nature the signals have harmonics whose frequencies are higher than the fundamental, so the wave is more complex and the peak amplitude can be evaluated also with a little buffer.
This object's MSP name is meter~. If you to know more about it search for meter~ on the MSP Reference manual.
Level meter
This level meter seems like the previous one, with needle in place of leds, but it has some important differences.
It is not a peak level meter. It computes the RMS (root-mean-square) of the signal's amplitude. This is very important because the peak value is meaningful with a simple harmonic wave, but it makes little sense to talk about a peak amplitude when describing a complex sound (e.g. a drum stroke) or a noise. This is because there are many peak amplitudes and any one is not representative of the overall amplitude.
So, how should we measure the amplitude of a complex signal? We can't take the average of the instantaneous amplitudes since negative and positive values always cancel yielding an average value of zero. To prevent this we square each amplitude making all values positive. We then take the square root of the average of the squared values to get back to an amplitude measure. The resultant measure is referred to as the root-mean-square or RMS amplitude. The RMS amplitude is a more commonly accepted measure of amplitude because it applies to periodic sounds like sinusoids as well as to noise.
This object output the RMS amplitude value in dB, so there is no need of the atodb object.

This object's MSP name is levelmeter~. If you to know more about it search for levelmeter~ on the MSP Reference manual.

Now that you know more about the amplitude, you can set 0 dB on your mixer.
If you connect Max/MSP to a professional mixer with measurement leds on input and output, you can easily set the 0 dB. On the computer, play a test frequency (about 1000 Hz). Then set the board output to the maximum (or the level you normally use, but a good board should work well at the maximum with no distortion). Then set the amplitude control of the patch to the maximum (i.e. 1.0). This is the MSP 0 dB (you should see 0 dB on all the meters).
On the mixer, set the input slider to 0 dB and turn the input sensitivity control (if any) until the input signal level show 0 dB. Do the same for the mixer output.

Why the dac~ maximum amplitude is 1?

It's a good question. Why 1 and not 100 or 1000 or a power of 2, like 1024 or the real amplitude of the sound board? Is there some technical reason to set the range to 0 - 1? The answer lead us to the realm of computer arithmetic, so it could be a little complex business. This notes are for more advanced (or curious) users.
  1. Max/MSP must work with many different sound boards and the sound boards currently on the market can have different dac types, typically 16 or 24 bit. Maximum values are 32767 for 16 bit (2(16-1) - 1) and 8388607 for 24 bit (2(24-1) - 1).
    But the software can't work with absolute amplitude values simply because the patches you make, must work with all the sound boards, even if you change machine or the board itself. So a conventional maximum amplitude value has to be set. When sending sound data to the sound board all values will be scaled according to the maximum value the board support.
  2. MSP works with 32 bit floating point numbers. This is a very accurate way to make calculations on a computer because 32 bit floating point numbers can hold until 18 digits after the decimal point.
    Floating point is more accurate than 24 or 32 bit integer arithmetic. People often think that every operations on digital signals are noiseless, but with integer arithmetic it's not really true. A very simple example: suppose you are working with 24 bit integer arithmetic on a digital wave with amplitude 8388607 (the maximum). Now suppose you are mixing, and this signal's amplitude is lowered multiplying by, say, a 0.731 factor. The resulting amplitude is 6132071.717. OK: where do you think this 0.717 goes? It get lost because this is integer arithmetic and by this way every multiply you make creates a very little sampling error.
    An experienced digital user can argue that with integer arithmetic you never multiply by floating point numbers. You multiply by an integer number creating a double precision result that will be truncated to simple precision, but it's the same because here are the lower bits that get lost.
    The meaning of the story is: always try to work with 32 bit floating point when making computations on digital signals. You can't avoid the final rounding error, when data are sent to the board's dac or are written on audio CD in the usual format (44100, 16 bit, stereo), but at least you save the accumulation of little sampling errors. The current debate is about 48 bit integer versus 32 bit floating arithmetic (see this James A. Moorer's paper), but under the 48 bit integer arithmetic, 32 bit float is surely better.
  3. But why 0 - 1? Even with floating point arithmetic, a conventional amplitude of, say, 1000 could be easiest to understand.
    Because of the problem of floating point arithmetic: while number increase, you lose precision after the decimal point. This is normal business on real world. If someone owes you $10.95 all the digits are significant, but if someone owes you $100000.95 you could apply a little discount and save the $0.95.
    So, to hold the maximum accuracy, MSP choose to work with little numbers.

A useful patch: Hear the Fletcher/Munson Effect

Important note: to hear the Fletcher/Munson effect you must work with hi-fi speakers. This is because this effect is evident at low frequencies (under 100 Hz) and the little computer speakers are not effective at this frequencies. You can use haedphones but be careful. Never turn on a patch without controlling its loudness or the volume of your equipment. Set amplitude/loudness to 0, turn on the patch and then raise the volume.

Let's do this trial.
Now begins to lower the frequency (click on the left part of the control box to change the integer part only, going down faster). Never change the amplitude. Move down the frequency only.
Under 500 Hz, the loudness will be a little lower. About and under 100 Hz the loudness will be very low. Very low frequencies (60 Hz and lower) will be barely audible even if the amplitude is the same!.
Why? It's your speakers? Maybe, but the real answer is that the human ear frequency response (or better, the ear\brain combination response) is not the same for whole audible area. The ear works better approximately from 500 to 5000 Hz. Outside this area it needs more amplitude to hear equal loudness.
Fletcher/Munson curvesThis effect has been first studied in 1937 by Fletcher & Munson who traced the human ear response curves. In 1956 Robinson & Dadson refined the Fletcher/Munson curves and today the Robinson/Dadson curves are generally preferred over the Fletcher/Munson curves owing to their more reliable measurement results. The International Standards Organization agreed and adapted the Robinson/Dadson curves as the basis for ISO 226:1987; Normal Equal-Loudness Level Contours (ELLC) (see image on the left; the MAF line is the Minimum Audible Field).
This chart say that humans don't hear all frequencies at the same level. That is, our ears are more sensitive to some frequencies and less sensitive to other frequencies. Not only that, but the sensitivity changes with the sound pressure level (SPL), too. Take a look at the chart. You'll notice it's marked horizontally with a scale denoting the frequency of sound. Vertically it's marked in SPL. On the chart are a number of curved lines, each with a number (loudness level) marked.
Let's begin by looking at the lowest solid line marked with a loudness level of 10 phons. (The loudness level in phons is a subjective sensation--this is the level we perceive the sound to be at.) From about 500Hz to roughly 1,500Hz the line is flat on the 10dB scale. This means that for us to perceive the sound being a loudness level (LL) of 10 phons, (the overall curved line), frequencies from 500Hz to 1,500 Hz must be 10dB. Make sense so far? OK, now look further into the higher frequencies, say 5,000Hz. Notice the line dips here--this says we perceive 5,000Hz to be 10 phons when the source is actually only 6dB. To perceive 10,000Hz at the same level (10 phons), it would need to be about 20dB. From this we can clearly see the ear is more sensitive in the 2,000Hz to 5,000Hz range, yet not nearly as sensitive in the 6,000Hz and up range.
Lets take a look down at the lower frequencies now, say 100Hz. For us to perceive 100Hz as loud as we do 1,000Hz (when the source is at 10dB), the 100Hz source must be at 30dB–that's 20dB higher than the 1,000Hz signal! Looking even farther down, a 20Hz signal must be nearly 75dB (65dB higher than the 1,000Hz signal)! We can clearly see our ears are not very sensitive to the lower frequencies, even more so at lower SPL levels.
Why is this? A simply physical explanation is that resonance in the ear-canal amplifies frequencies between 1500 Hz and 5000 Hz with a peak around 3000 Hz. In addition, the response in the range between 500 and 1500 is flat. All other frequencies are damped. Note that the main communication area lies in the range 500 - 5000. In this range we find human and animal voices and in the upper part of this range we find the children voices.
Now you know why the double-bass sound box is so big. Looking at the musical instruments construction, we can see many ways to compensate the Fletcher/Munson effect. For instance, the mechanical device of piano lower keys is adjusted to struck the string a little harder with respect to medium and high registers. So the player can achieve about the same loudness using the same force.

OK. We heard the Fletcher/Munson effect and we also know what is and why it happens. But now we can also measure it. Load the fletcher patch.
Fletcher patch
Here you can see two versions of the Fletcher patch. Now look at the simplest and not hidden (left). There is a reference frequency fixed at 1000 Hz on right channel and a second cycle~ that can be set to various frequencies (8ve distance) on the left channel.
To measure the Fletcher/Munson effect you must choose a frequency on the left, then move the amplitude control of the 1000 Hz (up on the right), until you hear the two frequencies have equal loudness. Then you read the amplitude difference in dB in the read-only number box down on the right.
The atodb object convert the amplitude from absolute value to dB value. This value is negative because it say how much the 1000 Hz must be damped to hear equal loudness.
The second patch version is the same with a more pleasant user interface, but two main differences:
  1. it uses a slider as amplitude control: note that the slider range is 0-100 (set by the floating inspector); then is multiplied by 0.01 to achieve the 0-1 range. This is because the slider object can't output floating point numbers (integers only). This kind of number trick is common on Max. We'll see more on the next lesson.
  2. the converted dB value is multiplied by -1 to see a positive number.

Download the patches for this lesson

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