Music by Numbers
An Overview by Mark Gregory
Music and mathematics go back a long way, long before Pythagoras, who is often credited with working out the first mathematics of musical scales. He probably borrowed something from the Sumerians whose philosophy intimately linked their numbering system with music into their view of the world. Like a number of civilisations they used 60 as their mathematical base. The remains of that 60 are still with us in our time divisions, minutes, seconds, still holding out against the decimal onslaught. The advantage of 60 is the number of whole numbers that it can be divided into. For expressing the relationship between notes as well as dividing up the day this remains very useful.
Ever since computers were invented they have been put to use for making music as well as doing sums. Early computers, or rather their operators, delighted in making up new music in the style of Bach. Somewhere I read that Bach had a specially mathematical approach to music, writing passages that worked backwards as well as forwards, or where the notation formed a visually pleasing pattern or symmetry on the staff.
These days music programs for computers allow for a great variety of possibilities. At the simplest level you can enter notes via the computer keyboard into a sort of database and then listen to the resulting tune. For example on the Macintosh I use a program called HyperCard which allows me to type in a string of numbers and will then play them back as a tune:
play Mandolin tempo 200 "59q 64q 64q 59q 64h 64e 66e 67q 66q 64q 64e 62e 59q"
The tune by the way is the first line of Brisbane Ladies and looks like this
This system matches up numbers with notes (maybe middle C being matched to 60 owes something to those Sumerians?):
| C | C# | D | D# | E | F | F# | G | G# | A | A# | B | C |
| 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |
A difference of 1 represents a semitone, a difference of 2 represents 2 semitones and so on.
This system also allows a representation of the note length so that the letter e stands for an eighth, h for a half, q for a quarter, w for a whole and s for a sixteenth.
Pretty wooden stuff compared to the real thing but it does allow you to hear much the same information as you get from the notation. And unlike a sound recording it takes up very little room. Maybe only a quarter second of a banjo plunk is sampled, and that sound is played over and over by the computer at different pitches and for different lengths of time. And you can use any such sound sample so a dog bark can become a musical chorus, or a monkish ooh or aah can sound out the tune.
Things you can do with computer music programs
- Hook
up a music keyboard to your computer and play the notes you want and then see what
you have played neatly arranged in notation you can edit and print
- Copy
and paste parts of a score just like you do with a word processor
- Transpose
your score to any key you like, create harmonies, hear parts played with
different instruments
- Scan music in using a scanner and listen to the
computer play it
- Create your own sounds to play music
- Hook up a microphone
and sing to your computer
and record and play back just like a tape
- Sing to your computer and
have your efforts turned into notes on a stave
- Change the pitch of a sequence
without affecting the speed and vice versa
- Play tunes backwards,
add echo, reverberation and weird effects
- Clean up scratches, wow and flutter,
edit out annoying coughs and splutters from previously recoded material
- Enter the strange world of computer generated music and tunes, fractal tunes for example.
Folk Song and Computers
Folklorists quickly understood the value of recording even on wax cylinders on the Edison hand wound machines with their huge horn you still see the dog listening to in the logo for HMV. Bartok lugged his around Hungary and Rumania and opened up a whole era of musical discovery. Percy Grainger tried to do the same in Britain and was able to replay a recording dozens of times till he was sure he had the notation right. One of the earliest Australian wax cylinders is of a Tasmanian Aboriginal singing some kind of ancestry story.
Today we have a whole world of music captured and preserved by ever more faithful sound recording technology, from every corner of the world, and sometimes it seems that this great store is being plundered and repackaged as so called "world music"
Grainger much later devised strange machines to create new sounds. Machines powered by air blown via a vacuum cleaner or by turning a vast scroll of wavy ridges that could be turned into sounds. The sort of organic sounds he was after were like the wind or sea, not confined by musical scale or notions of interval, but making use of glissando and slides. Sounds not unlike those speeded up recordings of whales communicating through the ocean. Machines with names like "Kangaroo Pouch Tone Tool". Visit the Grainger Museum at Melbourne University sometime and you'll get a better idea.
What Grainger was trying to do mechanically has since been done electronically and now with computing power to manipulate and create numberless effects (using numbers of course!). Some we hear in modern classical music, and a vast amount in muzac, and those awful mood recordings that Katoomba post office afflict their customers with! Some we hear in the world of pop with its "techno" and so on.
MIDI
I began by describing a simple method of getting strings of numbers to represent musical notation. This has evolved into a system known as MIDI (Musical Instrument Digital Interface) which allows electronic musical instruments to communicate with each other and be centrally controlled. With MIDI a whole orchestra of synthesisers, electronic keyboards and instruments can be controlled by a computer. MIDI is a standard set of instructions or messages that activate the instruments they are sent to. MIDI messages tell when a note should start and stop, what pitch it should be and how loud it should be. If a MIDI message is sent to an instrument programmed to sound like a sitar then that sequence sounds like it played on a sitar. The same message sent to an instrument programmed to sound like a banjo will sound like a banjo and so on.
You can copy a MIDI file to a disk and send it off by post. Since MIDI files are a set of instructions they dont contain sounds at all. The sound gets synthesised by the computer that you use to "read" the MIDI file. This means an awful lot of music can fit on a floppy disk, or be sent via the phone wires on the World Wide Web, or included in an email message to someone halfway round the world.
Sound Compression and Streaming
Compression is big in the computer world. Compressed files take up less disk space. You can compress text files or graphics files so they occupy much less room. To read them again they have to get decompressed or expanded. Sound files can also be compressed, but still they tend to be large. One minute of sound can take up 10 megabytes of space (about 8 floppy disks). So if I had recorded this article to send away on floppy disks it would require 100 or more disks. Sound is a disk or space hog. There are new compression methods however. The best I know of claims a 136 times compression. That translated to CD terms would mean something like 2000 songs on a CD, unbearable! In internet terms it means real sound is becoming a viable proposition.
In fact a large number of web sites are using it right now. Apart from compression sound files can use "Streaming". Streaming means that the sound file starts playing before it's all downloaded so the waiting time is much less.
Compressed and Streamed sound is getting so good that record companies and radio stations are using it to make their recordings and programs accessible over the internet. Up an coming bands do the same thing, obviating the expense of producing a CD or even tape. The most commonly used program for this is called RealAudio, but there's one called Liquid Audio and the one I prefer QuickTime.
Music Programs for your computer
There are dozens of different music programs ranging in price a from 10's of dollars to thousands. So its horses for courses I'm afraid.
I use one called MusicTime and it's fine for what I want to do although I have to tidy up the notation quite a bit with a graphics program to get to look the way I want. I literally build a score by placing notes on a stave, dragging them into place with the mouse. I could also use the keyboard if I learnt the letters and combinations to press. Alternatively I could hook up an electronic piano keyboard and input the music that way. Once the music is in place I can set things like the key and tempo and time signature. When I've finished the notation I can play it through to check if it sounds right save it and also export it as a MIDI file.