Sunday 18 April 2010

Minor Seventh Equal Temperament

Many scales have been defined by propagating a certain interval up or down until an multiple-octave interval (or an approximation thereof) from the root is arrived at. Each of the intermediate notes is then transposed through all of the octaves to fill in all of the notes of the scale. Equal Temperament is defined like this, using an interval of a "fifth" where seven octaves equal twelve "fifths". Meantone Temperaments tried to even out the error between in tune fifths and in tune thirds by propagating with flattened fifths. There are many interesting just (and non-just) intervals that can be used to create scales in this manner.

I'm particularly fascinated by the minor seventh at the minute. I came to a greater understanding of the thirds by creating scales propagated on those intervals, so perhaps I will get somewhere propagating a scale using the minor seventh? The problem is that the equally tempered minor seventh is powerful and fascinating because the brain reads it as both 7/4 and the complement of 9/8. You cannot have a just version of the equally tempered minor seventh, because the minor seventh is a product of equal temperament. (An approximation would be possible, but what would be the point in abstracting it one further step?)

So let's define a scale by propagating the equally tempered minor seventh. What happens? You get 12-TET normal everyday standard equal temperament. This is true for any equally tempered interval. In fact, you could define notes intervals in any given equal temperament as those notes which when propagated produce each other. This is because the temperament is equal. That, at least, is interesting in itself.

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