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Tuning Your Breakbeats


Breakbeats and loops form the backbone of many a dance track, but should you need to change tempo the retuning of samples can be a headache. Matt Fletcher explains how to make it easy.

If you're one of the many who use breakbeats in your music, the following scenario may be all too familiar to you. You get a drum track, involving many separate sampled elements, just right when, suddenly, someone (always the one who is paying) asks you to make the track a bit faster or a bit slower. You start to change all the individual tunings of the samples. But are you sure you are not, in some subtle way, changing the groove?

This nightmare scenario seemed to happen to me once a week, and I was always in doubt as to whether the groove had actually changed. If I retuned the samples by ear — a painstaking process — the finished result always seemed to sound different. But because of the new tempo there was no way of determining whether the differences were a simple result of the change of speed, or because some of the samples were no longer tuned correctly.

Isn't there an easy way of knowing what tuning correction to apply? Of course there is, and that's what this article is about — but before we get down to the practical details, we need a little background.

PYTHAGORAS AND ALL THAT STUFF...



Many years ago, long before sampling had been invented, scholars pondered on the nature of music. More specifically, they pondered on scales, harmonics and overtones. Why was it that when they tuned an instrument perfectly it didn't sound right in every key?

Pythagoras thought that tunings, the relationships between notes, could be expressed as whole number ratios — such as 3/2 for the fifth above the tonic. This meant that if middle A was 440Hz, middle E (a perfect fifth above A) was 660Hz (440x3/2). By finding a whole number ratio for each semitone in an octave, you could derive a scale. Of course, given the infinite number of integers, there was no end to the number of scales that could be derived. We refer to this type of tuning as just intonation, tuning that is based on the natural laws of harmony as understood by the human ear. An example of this can be seen in the harmonic overtones of a plucked string — these are dependant on whole number ratios because only whole number cycles can resonate on the string. Just intoned tuning produced minor scales, major scales, mean scales and so on, and they all worked fine — except for one problem. The instrument could be played in only one key. The solution to this problem remained elusive until a satisfactory compromise tuning was eventually produced.

This solution, developed around 300 years ago, was called equal temperament. It was a compromise because there is a trade-off between the ability to freely change key and the richness of the harmonic overtones found in intervals played while using just intonation. However, it is still in use today, present in all forms of Western music. It uses, as the name suggests, a fixed an equal ratio between any two adjacent semitones (with just intonation, the ratio between adjacent semitones would depend on exactly which notes in the scale you chose).

Using this tuning, changing key became possible, allowing musicians and composers into a fresh new area of expression (but, on the other hand, opening the way to such crimes against taste and common decency as the key change in 'I Will Always Love You').

To achieve this, equal temperament tuning uses a fixed ratio of 12√2 (the twelfth root of 2, =1.059). The form of this number is explained when you consider the number of semitones in an octave (12), and remember that notes an octave apart have their frequencies in the ratio 2:1 (doubling the frequency produces an octave rise in pitch). In other words, if you multiply 12√2 by itself 12 times, you get 2 — doubling the frequency.

APPLYING THIS TO BREAKBEATS



If we have a breakbeat that sits perfectly in a track at 96bpm, can we tune it up to cope with a new tempo of 118bpm, confident that we haven't affected its groove? With the help of a little maths that uses the principles of equal temperament, and the magic number 12√2, yes we can.

First, however, let's look at a slightly different problem. Suppose we have a breakbeat running at 104bpm, and we transpose this up three semitones. What is its new bpm? Using the ideas mentioned earlier, we can generate a simple formula to find this. The formula uses 12√2 to calculate the 'tempo jump' of the breakbeat. The sampler treats a breakbeat just the same as any ordinary tone or pitched sample — it uses equal temperament tuning. If maths and the whole number thing is not your bag (like, you're a musician, after all) don't worry — I'll give you a formula into which you can simply punch the numbers using a calculator.

Essentially the old bpm needs to become faster so the breakbeat will still fit. We can express this in a relatively simple equation:

OLD BPM x (12√2)3= NEW BPM [a 3-semitone jump]

Here we are using the principles of equal temperament to calculate the jump of three semitones by multiplying the 'magic number' 12√2 by itself three times. This gives us the frequency ratio of the three semitone interval. We can now substitute our actual values into the equation to find the new tempo.

104 x (12√2)3 = 123.678

So the new tempo is 123.678bpm. If you have a sequencer that can set tempos to one thousandth of a bpm, such as Cubase, Creator or Notator, you can check this result. Tune a breakbeat to 104bpm, so it matches exactly, and transpose its tuning up by three semitones. Now set the sequencer to the new tempo of 123.678bpm. If you've done everything correctly the breakbeat should fit perfectly.

Although this is an interesting result, it is not particularly helpful. It is unlikely that you will ever need to transpose a sample by a fixed amount to a non-specified tempo. The only such case I can think of is in a remix where a vocal needs to come up to a specified key, and by doing so a new tempo must be found.



"If maths and the whole number thing is not your bag (like, you're a musician, after all) don't worry — I'll give you a formula into which you can simply punch the numbers."


What we would really like to do, however, is to be able to calculate the degree of transposition required when we alter the tempo. To do this properly, we need to know the answer to one crucial question: What is the tuning resolution of the sampler? For instance, the Akai S900 and S950 have 16 steps between each semitone, which Akai calls fine tuning, and other manufacturers produce machines with resolutions of 64 steps per semitone. For the purposes of this article I will use the S1000/S1100 tuning resolution of 100 steps per semitone, which is far superior to the S900 and more suitable for this kind of work.

100 steps per semitone means that the S1000 has 1200 tuning steps in each octave, so determining the relationship between each of these tiny intervals is a value that when multiplied by itself 1200 times yields 2 (octave doubling again). This can be expressed mathematically by writing:

x1200 = 2

The solution of this equation is easy enough to find by taking the 1200th root of each side:

x = 1200√2

To find our new tuning we actually need to know how many 'x's are multiplied together to equal the ratio of the old bpm to the new bpm. We shall use the variable a to represent this amount. We can now make an equation that combines everything we know about tunings, bpms, life, the universe and everything:

(1200√2)a = new bpm/old bpm

This equation is specific to the S1000/S1100 tuning resolution — if you do not own one of these machines you can substitute for the value 1200 the number of steps per octave found in your particular machine.

The way to solve this equation is by using logarithms. I will use natural logs (based on the value e [=2.718]) which are represented in equations and on calculators by the symbol ln. Again, it is not necessary to understand what this means as long as you are capable of pressing the correct button. If you don't have the ln function on your calculator, it is perfectly OK to use the log function, as long as you remember to change every occurrence of 'ln' to 'log'. Now, taking logs of both sides (note that 'new' and 'old' refer to the new and old bpms),

a.ln(1200√2) = ln(new/old)

which gives us

a = (ln(new/old))/ln(1200√2)

Finally, this can be simplified further to provide our last equation:

a = (1200/ln2).ln(new/old)

In words, the amount of tuning required when a tempo is changed from one bpm to another is given by the natural log of the ratio new/old multiplied by 1200/ln2. In the above equation the ratio 1200/ln2 is a constant — it is always the same — but only for the S1000/S1100. Because this value will vary from sampler to sampler it is useful to remove this value and replace it with a constant, f (the Fletcher constant!). This further simplifies the tuning equation and will make use of the formula faster and more convenient. The simplified formula is:

a = f.ln(new/old)
[a = transposition amount]

Listed below are some common values for f as they relate to different tuning resolutions.

Steps per semitone f
16 277
32 554
50 866
64 1108
100 1731


The values given for f are expressed as integers, but should prove to be accurate enough in almost every application (in fact, for the more mathematically inclined, checking the results shown for 16, 32 and 64 will reveal that they are almost exactly right). When using the equation, your calculator will almost certainly not give you a whole number. Round the answer to the nearest integer in order to find what parameter to use in your sampler.

Here is a brief example: the breakbeat is 'sitting' perfectly at 118bpm on a S1100, and needs to be speeded up to 130bpm. What is the transposition amount? Substituting these values into the equation gives us:

a = ln(130/118)x1731 = 167.647

This value should be rounded up to 168 (which in fact corresponds to a retuning of 1 semitone, 68 cents on the S1000/S1100). The value for a should only be rounded down if the decimal part of the number is less than 0.5. One nice feature of the formula is that if the new tempo is slower than the old, the value for a will be negative, automatically telling you to tune downwards.

There are many uses for this technique apart from the obvious quick retuning. If you take the time to accurately tune a break when you first take it, and write this tempo down, you can very quickly slot any sample into a new production, safe in the knowledge that it is correctly tuned. This can save an enormous amount of time and frustration, leaving you free to get on with the more important jobs: like Being Creative.

Finally, if all this maths stuff seems a bit heavy when all you want to do is to write the next Kylie single (or, more probably, not), just spend five minutes with your sampler and calculator until you get used to the ideas discussed and the little button marked 'ln'. Maths and music are inextricably entwined. Experiment, and have fun.


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Previous Article in this issue

Rack Attack!

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World Beating?


Sound On Sound - Copyright: SOS Publications Ltd.
The contents of this magazine are re-published here with the kind permission of SOS Publications Ltd.

 

Sound On Sound - May 1993

Feature by Matt Fletcher

Previous article in this issue:

> Rack Attack!

Next article in this issue:

> World Beating?


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