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Maths, Music & Motion

Article from Music Technology, December 1993

Expert tips on fine-tuning your effects in exact time with your BPMs.

Need to get your effects in line with your time signatures and tempos? Bob Dormon is the operator of his pocket calculator...

Sophisticated tempo/time sig functions on a modern computer often demand careful calculations

Pythagoras claimed he could hear the music from the spheres - the sound produced by the planets orbiting in the heavens. Their movements are actually at relatively regular frequencies, albeit measured in days and years rather than seconds. We know the Moon takes 24 hours to orbit the Earth, and this cyclic motion can be transposed to a musical frequency. It is claimed that the Moon is in the key of A! I checked it out and its pitch is F#, so there you go.

Back on terra firma, you'll find that AC mains hums at around G, whereas in America it sounds sharper, because they run at 60Hz, not 50Hz as we do. I'm not suggesting that you should tune your synth to a dodgy amp but, like it or not, maths and music are inextricably linked.

Tempo & time signatures

Tempo is a measure of frequency. While frequency is normally expressed in Hertz (Hz) or cycles per second, have you ever thought about what 'bpm' (beats per minute) really means? 120 beats per minute means that you get a beat every half-second, and anybody with a digital delay knows that 500ms is the quarter- note (crotchet) delay time for that tempo... Or is it? Well, that depends on whether you're in 4/4 time or not. In 4/4 time, a quarter-note lasts a quarter of the length of a bar because 4/4 is four quarter-notes (or beats) in the bar, and remember we are talking about beats when we refer to tempo.

Confused? Let's check it out. If we've got 120 beats in 4/4 lasting a minute, then how many bars will that be? Well, if you set your sequencer or drum machine to that tempo, you'll find that at the end of bar 30, exactly one minute will have elapsed; after all, 4 (beats or quarter-notes) x 30 (bars) = 120.

Alternatively: 120 (beats per min) / 4 (beats in the bar) = 30 (bars)

I feel I should expand on the meaning of time signature. You may instinctively know what it's all about, but how does it actually relate to time with respect to the overall length of a piece of music? Before the days of hi-tech recording, different time signatures were chosen not only for emphasising the beat but to make score reading and writing easier. A piece would be written in, say, 7/4 if the tempo was slow, but in 7/8 if it needed to be faster. Performances benefitted from rooms or halls that complemented the orchestration.

But, with computers taking over as conductors these days, it is important that all the players are sitting comfortably, and that in particular goes for the effects devices that have all but taken over from acoustics. The Americans call bars 'measures' and when it comes down to it, a time signature is simply a way of measuring time, while the fractions 3/4, 7/8, 4/4 and so on are a way of calibrating the ruler you're measuring with. Basically, the bottom figure shows the graduations, while the top shows how many to measure.

Imagine a 4/4 time signature like this. Say the ruler measures in quarter-inch steps. So four of these would make one inch and you could call the 'inch' a bar in 4/4. Three steps would make three-quarters of an inch, so the 'bar' or 'inch' is now shorter. This is exactly what happens in time: a 3/4 bar lasts only three quarters of the time of a 4/4 bar running at the same tempo. A 5/4 bar lasts a quarter longer than a 4/4 bar at the same tempo.

So, we can now see that to get things happening in time, both tempo and time signature are part of the equation.

But, what about 6/8 and 7/8? Well, taking our ruler example again, 6/8 would be measured in eighths and there would be six of them.

In fact, if they're both at the same tempo, a bar of 6/8 lasts the same as one in 3/4; it's just the way you count or measure it that's different. In 6/8 you have to count twice as fast over the same period of time:

6/8: 1 2 3 4 5 6 1 2 3 4 5 6 1...
3/4: 1 2 3 1 2 3 1...

But hang on a minute. If I'm counting twice as fast in 6/8, then why isn't the tempo doubled? Good question. Most sequencers use crotchets as their yardstick for measuring beats, despite the fact that your click might be in quavers. So have a cup of tea and an aspirin and I'll tell you how your calculator can be your best friend in the studio.

"If you find your reverb lasts too long and is overhanging into the following bar, you can alter its length to fit precisely into the bar"

Delay times

Delay/echo units can be lifesavers when it comes to filling in the gaps during solos. Lead guitar, sax, vocals, anything can sound classy with a touch of delay on it, provided the timing is correct. Some happen by chance - older delays you just twiddle - but for accuracy you feed a regular bass drum or snare into your delay unit, making adjustments until the repeats sit right.

However, most of today's gear comes with jolly nice digital displays that you can adjust with millisecond accuracy (by the way, there are 1,000 of them in a second). The 'every home (studio) should have one' Yamaha SPX90 goes one better with a display in tenth-of-a-millisecond steps - you can just hear the difference, can't you...?

Anyway, it's mix time and you want to get a quarter-note delay on your lead vocal. You know what the bpm is, as you glance at your sequencer or track sheet; you did write it down, didn't you? So what next? If you're in 4/4 (and who isn't these days?) you simply divide 60 (ie. one minute) by the tempo (ie. the beats over one minute) and hey presto! There's your delay time in milliseconds. Here's an example:

A ballad
60/100bpm = 0.6 seconds or 600ms

A rave track
60/140bpm = 0.428783 seconds or 428.783ms

So long as you always remember to divide the bpm into 60, then you can't go wrong. What you can do now is put your result into the calculator's memory and start messing about. Try sub-dividing the time by two. This way, you get eighth-note repeats. Do it again and you get 16ths, and so on.

It's up to you whether you choose a quarter- or eighth-note delay - my golden rule is that there are no golden rules in creative recording - but have a listen to what the rich and famous do. Madonna has had a quarter-note delay on her voice on practically every track for the last ten years; all that seems to happen is that they vary the number of repeats! The Edge, being the master of all things echoey, often uses 8th note repeats to give his licks a sense of speed.

As you get into the smaller numbers (16ths, 32nds...), you cross the border into Elvis territory. Make sure you have only a couple of repeats at the most, and you too can be The King, Buddy Holly, or even John Lennon. Being dead isn't essential, though you might sell more records that way.

Okay, so what about other time signatures? How do you work out timings for them? Well, if you simply want them on the beat, you can use the same values as for 4/4, but you won't be able to call them quarter- or eighth-note delays as they'll be occurring (within the bar) as often as is specified by your current time signature. If you really want to get triplets happening in 4/4 or fours going over 3/4, then your best bet is to calculate how long the bar lasts and then divide that figure by whatever fraction you want to use.

So in 4/4 at 120 bpm a bar lasts two seconds (four beats per bar at 500ms each = 2 seconds). If you want a triplet delay, just divide two seconds by three:

2/3 = 666.666ms

In 3/4 at 120bpm, a bar lasts 1.5 seconds (3 beats x 500ms), so if we want straight fours over the top of that, then divide 1.5 seconds by four:
1.5/4 = 375ms

You can, of course, divide or multiply your results by two to double or halve the timing.

The more comprehensive the display on your digital effects, the easier it will be to program delays in sync with your music; this is Yamaha's new SPX990

"Madonna has had a quarter-note delay on her voice on practically every track for the last ten years"

Tempo & reverb time

Knowing the length of your bar for a specific tempo can be extremely useful. If you find that your reverb lasts too long and is overhanging into the following bar, then you can alter its length to fit precisely into the bar. Just use the above calculations - you'll see that not only can you have your delays set up perfectly in a few keystrokes, but you can also have your reverbs sitting pretty, too. If you're pressed for time during a session, then a minute on a calculator can lead you out of chaos.

If you want to go further, then sub-divide your delay time a few times so that it's in double figures, and try using the result as a pre-delay on your reverb. If you go into the early hundreds, then you can use the pre-delay timing to work sympathetically with a snare or vocal. The initial sound is left free from confusion and then, in perfect time, the reverb cuts in. You might want to subtract a long predelay time from your overall reverb time in order to keep things within the bar.

When I was an assistant engineer, I'd spend hours doing this kind of thing on Lexicon PCM70s - I'd even try to calculate appropriate room sizes by considering the speed of sound and the distance travelled within a particular delay time! How those winter evenings would just fly by...

Chorus, tremolo & auto-pan

That's nailed time. What about motion? Well yes, there are a few more things left in the rack that your trusty calculator can help you with. If you examine the parameters of your chorus program, with any luck you'll be able to adjust the frequency. This simply determines how fast or how slow the effect oscillates. Should it be appropriate, you can get your chorus, flanger or panner to oscillate or 'beat' in time with the tempo of your music. In fact, you can do this with anything that has a frequency control: tremolo, vibrato, even LFOs on synths.

So how do you get these magic numbers? Well, as I said right back in the beginning, the first step once you've calculated your delay time is to put the result into your calculator's memory. Now clear everything, and divide one by the memory. (You can bypass this if your calculator has a 1/x function.) The figure you now have is the frequency for that tempo.

Let's use our earlier examples:

A ballad
Delay calculations
60/100bpm = 0.6seconds or 600ms

Frequency calculations
1 (second) / 0.6secs (delay time) = 1.666Hz

A rave track
Delay calculations
60/140bpm = 0.428783 seconds or 428.783ms

Frequency calculations
1 (second) / 0.428783secs (delay) = 2.3333Hz

These frequency figures will work on every beat - which should sound great on tremolo and vibrato settings, but might be too often for use with an auto-panner. If you divide by two, then your panning will take place twice in the bar, going something like this:

Left - Centre - Right - Centre - Left etc,
1 2 3 4 1

And if you divide by two again, the pan will take a whole bar to go from left to right. If you know how long your bar is, then you can shortcut these calculations and just divide that figure into one. Using bar lengths is a useful way to determine frequencies for other time signatures, as you can pan across the bar, rather than on the beat.

For chorussing and flanging effects, having your oscillations (or sub-divisions thereof) beating in time can help make even the wildest guitarist sound thoughtful. This works well on backing vocals, basses, you name it, but how much you notice the effect will depend on your modulation depth, so don't overlook that little beauty.

With a little practice, you'll be able to set up your delays, reverbs, chorus and panning effects in a matter of minutes - provided, of course, that you know the tempo. So what do you do for a live band? Well, one of the quickest ways to get a guide tempo is to listen to the track and count the beats over 15 seconds, then multiply by four to get the beats per minute. I once had to do this in rehearsals, monitoring the live sound for Dave Gilmour and Roger Daltrey. Those boys love their vocal delays and I had them up and ready before you could say "See Me Feel Me".

And finally...

...We mustn't lose sight of the music. After all, each note has a pitch or frequency and I've often felt that certain tempos work better in different keys, as they are likely to be mathematically sympathetic. For instance, the tempo for A (after a few transpositions) is 103.125bpm, C is 122.625bpm. The trouble is that you'll probably want to play a few other notes, too...

It makes me wonder about old Pythagoras. If he really could hear the music of the spheres, then what chord would it be? I couldn't resist the challenge and calculated that it's a C#sus4b9b13 spanning 17 octaves. It's a massive sound if you transpose it, and if it weren't for the G#s of both Neptune and Uranus, then we'd have a Dmaj7 universe with just a touch of phasing as we glide between the planets... Cosmic or what?

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Publisher: Music Technology - Music Maker Publications (UK), Future Publishing.

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Music Technology - Dec 1993

Donated by: Chris Moore

Feature by Bob Dormon

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