Subliminal Extacy
#03
01 апреля 2001 

Just Intonation: Making music come to life
Just Intonation: Making music come to life By Gasman / RA "...The way I used to type in the notes was directly into the assembler editor as pairs of numbers, without labels, so I used to be able to tweak the numbers up and down to make the chords 'ring' properly."  Music god Tim Follin, speaking on comp.sys.sinclair With all the sound trackers available on the Spectrum today, musicians can create their compositions without worrying about the way their notes are translated into sound waves. However, if you have some knowledge of the science involved, you can achieve a sound quality higher than anything produced by a tracker. Before the age of computer music, this science belonged to the exciting world of piano tuning. No, really. You see, there's more than one 'correct' way to tune a piano. Most people would find it hard to tell the difference between them, but each method has precise details which conflict with the others. The only thing they all agree on is the octave. Take any note you like, double the frequency, and you get the same note an octave higher. Double it again and you get another octave. We're now faced with the problem of dividing the octave up into 12 notes. (Yes, it is 12, and not 8: C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B.) The obvious thing to do is have them equally spaced: just multiply the frequency by 'wibble' to get to the next note, and once you've multiplied by 'wibble' 12 times, you reach twice the frequency of the original note  precisely one octave above. A quick bit of calculator prodding shows that 'wibble' is 2^(1/12), which is approximately 1.0594631. The big advantage here is that you don't have to start from C. If you start with B and then do the multiplying, you hit the same sequence of notes. This is good news for piano tuners, because it means that you can get a Bach fugue in D minor sounding just as good as a Mozart sonata in G major. This method is called 'equal temperament', and it's used in everything from saxophones to synthesisers. Unfortunately, this scheme results in poor quality sound. To explain why, let me tell you about Pythagoras. After a hard day of proving stuff about triangles, Mr P would chill out with his Fender Stratocaster (hmm, you have to use your imagination a bit here). He discovered that the coolest sounds came when the frequencies of the notes had a nice simple ratio between them. For instance, if two notes are an octave apart, they have a 2:1 ratio, which is about as simple as it can get. On the other hand, 1.0594631:1 is not a nice ratio at all, and as a result, equal temperament doesn't sound as nice as it should. 2500 years of music later, it's been decided that a 'nice' ratio is one where the numbers are all multiples of 2, 3 and 5. This is known as 'Fivelimit Just Intonation', because you're limited to the prime numbers up to 5. You can get other systems by choosing a different limit, but they're a bit pointless and they just confuse the issue. Trust me on this, okay... So, sticking to fivelimit just intonation, these are the ratios to go for: C  1: 1 C#  25:24 D  9: 8 Eb  6: 5 E  5: 4 F  4: 3 F#  45:32 G  3: 2 Ab  8: 5 A  5: 3 Bb  9: 5 B  15: 8 C  2: 1 Of course, if you want more than one octave, you can move up or down one octave by doubling or halving the ratio. A word of warning though  THESE RATIOS ONLY WORK IF YOUR TUNE IS IN THE KEY OF C! The ratios aren't actually connected to the note names themselves; I've just chosen C as a starting point. To illustrate this, take the sequence of notes C F Bb. We'll say that the frequency of C is 10OOHz; it doesn't matter what you choose. Working from the ratios above, we get: C=10OOHz, F=13ЗЗHz, Bb=18OOHz. Now, let's do the same thing in the key of F: the notes turn out to be F Bb Eb. We now know that F should be 13ЗЗHz. However, we can't just read the values from the table now, because F is the starting point, not C. And this time, Bb really means 'five semitones above the starting point', and the ratio for that is 4:3. So, it should be calculated as 1333 x (4/3), which works out as 1778Hz. Likewise, Eb is ten semitones above F, and this turns out to be 24OOHz. So, we get: F=13ЗЗHz, Bb=1778Hz, E=24OOHz. Hang on a moment... Bb was 18OOHz a minute ago, and now it's 1778Hz! No, it's not a mistake. Both values can be correct, depending on the key your tune is written in. Obviously this is impossible on a piano, because the Bb key can only be tuned to one note  so the piano tuner has to compromise and use equal temperament instead. On the Spectrum, there's no such problem, because we can produce any frequency we want (well, near enough). The first step is to transpose your tune to the key of C (either C major or C minor). Next, decide which frequency to use for C. Anything will do, so choose one that doesn't make your tune too high or low. Finally, use the handy table of ratios to calculate the frequency of each note. It's as easy as that! Well, that's not the full story, because you now have to write your own routine to play the music  but that's the easy part, because you've already done most of the work. Anyway, if it's good enough for Tim Follin, why would you want to disagree...
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