musical thoughts

This is a sort of a basic theory topic compared to my usual discussions, but it’s an interesting question asked on one of the theory baords on Orkut, and I decided to repost my reply here for those interested.

The question, paraphrased, is:

Is there an explanation to why the tones are named major second, major third, perfect fifth etc with respect to the ratio of frequencies?

11/10, 10/9, 9/8, 8/7 have all been described as the interval of a major second.

How do we tell a major second from a minor third? When does a major second stop and a minor third begin? Is there something about the ratios or their cent values that I’m missing?

The theory answer:
Within our lifetimes, pretty much all Western music has used an equal temperament system rather than just intonation like you describe. The names are in a sense arbitrary, but they refer to the classification of pitches into a scale system.

I learned the rules like this, but it’s not a very functional way to look at it:

• If the top note is in the key signature of the bottom note, and the bottom note is in the key signature of the top note, the interval is perfect. Only unisons, fourths, fifths, and octaves may be perfect.
• If the top note is in the key sig of the bottom note, and the bottom note is not in the key sig of the top note, the interval is major.
• If the bottom note is in the key sig of the top note, but the top note is not in the key sig of the bottom note, the interval is minor.
• If neither note is in the other’s key sig, you have a diminished or augmented interval.

Those ratios of whole numbers don’t really enter into music theory anymore except in a historical sense.

The psychoacoustic answer:
There’s a limit to the human ear’s ability to discriminate between two frequencies. Modern theories of pitch perception suggest our minds allocate a pitch to a range (”it’s one of those C# notes”) rather than perceiving frequency on a continuous scale. An interval is just two of those pitches perceived simultaneously (”it’s one of those C# and one of those A notes”). So our ability to discriminate pitch affects interval perception. Incidentally, the finest ability to discriminiate pitch falls within the frequency range of the human voice.

When I suggested that the explanation above wasn’t functional, I meant that you’ll get a lot more good out of investigating the way notes are ordered into scales. To even apply the rules I list, you have to know the key signatures for every note, and make a decision about whether a note is in the scale. (This is not a tremendously useful approach.)

It’s probably more fruitful to consider a major scale as being built of the following intervals with respect to its root:
perfect unison
major second
major third
perfect fourth
perfect fifth
major sixth
major seventh
perfect octave

And if you guessed that you could change all the occurrences of “major” to “minor” in the list above and get a minor scale, I have to disappoint you. (It couldn’t be *that* simple.) The natural minor scale is built of these intervals with respect to its root:
perfect unison
major second
minor third
perfect fourth
perfect fifth
minor sixth
minor seventh
perfect octave

To expand a little on the equal temperament thought above, before this development, the interval of a major third which occurs between the I and III scale degrees was a different ratio than the major third that occurs between the V and VII scale degrees in a major scale. After equal temperament, these are the identical ratio (which is four semitones, or 1.25992. The just intonation system would require a major third equal 5/4, or 1.25000. The difference is perceptible, but western ears have accomodated the discrepancy.

The perfect fifth is 1.49831 under equal temperament, really close to the just interval 1.50000. There’s a chart of all the intervals and their relative deviations here.