Pure VS Tempered tuning

  • If I'm not mistaken tempered means 12 tone equal temperament, and pure means just intonation. I recommend you look up these terms as a full explanation is too much for one forum post. But in short, just intonation means tuning your instrument so that the relation between the pitches of any two notes (their fundamental frequencies) is always a rational number - n:m where both n and m are natural (whole) numbers. It is based on the natural harmonic series (look this up as well), and so, depending on which note you start tuning from, some chords and scales (modes) will have clear harmony, and others will sound really bad (wolf intervals). I think that the Virus readjusts the basic note of the tuning according to the notes you are actually playing (singers do that as well), so you will probably not come across any wolfs.
    Equal temperament means that all semitones are "evenly spaced" in the octave, so there is no need to change the tuning if you change key or mode, but the relation of pitches between almost all notes is an irrational number, meaning that this tuning misses the natural harmonic series by a little bit. This gives chords a little bit of a fuzzy detuned/chorused quality which I personally really like. In fact, I find just intonated instruments playing only the harmonious notes, dull sounding, boring even, but it is all a matter of taste. And also, untrained listeners can't really tell unless you stumble upon a really nasty wolf interval.

  • Apologies for resurrecting an old post, but I'd like to add a bit more info. I haven't seen many explanations of some of the various tuning systems out there, so I'll give it a go.

    Equal Temperament means that the ratio between the frequencies of any two notes of your choosing will be exactly the same as any other two notes, given that they are the same amount of semitones apart. The ratio between C and D is the same as between G and A - two semitones. Or the ratio between F and G# is the same as between D and F - three semitones. And so on and so forth...

    Where 'S' is semitone and 'n' is the note you are working from, the Chromatic scale uses this formula for working out +1 semitone:

    S = n * 2^(1/12)
    [Semitone equals note times two to the power one over twelve.]

    In the above formula, the '2' represents the octave (or doubling of frequency) and the '12' in the denominator is how many equal divisions you want in the octave (twelve notes). The '1' in the numerator represents the single note spacing, or semitone. You could even have this formula for cents: c = n * 2^(1/1200). It's the same deal, just 1200 equal spacings that correspond to the non-linear curve created by the formula. One semitone in that formula would be written as: S = n * 2^(100/1200) which, of course, resolves back down to: S = n * 2^(1/12).

    While the Chromatic scale works perfectly, I think some perfectionists probably didn't like the fact that a fifth doesn't resolve to a nice round number. You'd think it would be a multiplication factor of x1.5, but it isn't! It's actually: 2^(7/12) = 1.498307077...

    This is so close to 1.5 that I guess people thought they could implement something like Hermode Tuning (called "Pure" tuning on the Virus TI) to keep fifths sounding absolutely perfect as well as major thirds being exactly x1.333r and so on. For Hermode Tuning, 440Hz for middle A will have a fifth of exactly 660Hz instead of it being 659.2551138Hz like it is in the 12-note Equal Temperament scale.

    The only unfortunate downside to Hermode tuning is that it skews the mathematical formula. When you shift the fifth of A (or E) across from 659.2551138Hz to 660Hz, this has now essentially shifted the rest of the scale upwards. So now the octave of 440Hz isn't exactly 880Hz but more around ~882Hz... This is a BIG problem for music production. Every single chord now doesn't relate to any other chord in a mathematical or even a musical sense. While an individual chord may sound perfectly in tune and all harmonics of all frequencies are lined up precisely, the next chord you play may sound either too sharp or too flat vs the previous chord.

    Here's a list of the multiplication factors for all notes in any octave in the Chromatic (12-note Equal Temperament) scale:

    +0 semitone = x1.000000000 (root note)
    +1 semitone = x1.059463094
    +2 semitone = x1.122462048
    +3 semitone = x1.189207115
    +4 semitone = x1.25992105
    +5 semitone = x1.334839854 (Almost 1.333 recurring, but not quite)
    +6 semitone = x1.414213562 (Very interesting number in maths. This EXACT number is used a lot in electrical engineering.)
    +7 semitone = x1.498307077 (A fifth. Almost exactly 1.5)
    +8 semitone = x1.587401052
    +9 semitone = x1.681792831
    +10 semitone = x1.781797436
    +11 semitone = x1.887748625
    +12 semitone = x2.000000000 (octave)

    These ratios/multiplication factors are altered for Hermode tuning, which aims to get rid of audible note 'beating' when playing two notes. If you understand anything about two notes being played together, they form a third beat frequency that is a function of the difference between the two frequencies. 440Hz vs 660Hz will have a 220Hz beat frequency. 220Hz divides well into both 440Hz and 660Hz. With the Chromatic scale, however, the beat frequency between 440Hz(A) and 659.2551138Hz(E) is 219.2551138Hz. This frequency will cause audible beating, because it is not mathematically related to either the middle A or the E you are playing. It works great in a musical sense, but people nowadays tend to believe Nature itself is incorrect :rolleyes:

    EDIT: The edits are for spelling, grammar and styles/formatting purposes only.

  • This would be true if you were talking about pure sine waves - superimposed 440Hz and 660Hz sine waves do sound like a 550Hz sine wave ring modulated by a 110Hz sine wave, in other words, a 550Hz carrier AM modulated by a 110Hz beat (you can arrive at that by solving the equations {x+y=600; x-y=440}). But usually much richer timbres are used where all the harmonicas of one note interact with the harmonics of the other, resulting in a jumble of perceived effects, which I like to call fuzzyness.

    * but the effect I fear the most is us boring the hell out of the readers of the forum... ;)