# Equal temperament

Practically speaking: Does equal temperament only mean arranging a set of intervals (scale) such that it can be repeated (similarly t o what is defined in 12 tone equal temperament as an octave)?

Continuing in a practical vein: Do non-equal temperament scales essentially do away with the concept of "repeating scales"?

Are all non-equal temperament scales just intonation scales. In other words, if I were to construct a scale made up of progressive (low to high) random sounds (and those sounds didn't by some wild coincidence result in an 'octaved' scale) would that by definition be a just intonation scale?

• I'm not sure what you mean by "repeating" scales; most (non experimental) note systems repeat in octaves, i.e. if a given note is in the system, then the note an octave above and below it is in the system too. This is true for 12TET, just intonation, various meantone tunings etc.
– Dave
Sep 26, 2016 at 21:00

No, no, and no.

• Equal temperament means you take some given interval – usually the octave – and divide it into an integer number of equal steps (in log-frequency domain). This obviously implies that everything repeats, but that alone doesn't characterise an ET tuning.

• It's also perfectly possible for a non-tempered scale to repeat. An example is the Fibonacci scale: perfectly regular and repeating, but not based on equal divisions of a given fixed interval.
If by “repeat” you actually mean that it's completely “translation invariant”, then the statement is kind of true: a scale that's symmetric under any interval transposition is necessarily either equal-stepped or infinitely dense.

• If any non-tempered scale were by definition just intonation then that term would be pretty useless. No, just intonation is generally taken to mean the intervals are tuned to integer frequency ratio. You could create a just-intonation scale with random numbers, but not by choosing random independent frequencies – rather you would need to choose a single base frequency and random rational numbers that specify the intervals between the note's scale.

Technially, if you choose random frequencies with integer Hertz number then the intervals are in fact all rational, but still couldn't really be called just intonation because the numbers are far to high. Generally only small fractions are considered (3:2, 5:4 and compounds thereof, more rarely 7:4, in more exotic genres possibly 11:9 or 13:8): what's important that the integer ratio can actually be heard, by means of coincident overtones.

Equal temperament, historically, specifically refers to dividing the octave into 12 notes equally. This allows you to play in different keys without sounding too out of tune. Before equal temperament, tuning was based more on how the notes of the scale, and out of the scale, relate to the tonic of the instrument. This is just intonation.

So since these two terms refer to specific things in western classical music I would think it would be confusing to use them to describe other tuning systems.

So by default, if a scale is not equal temperament it does not mean it is just intonation.

If you did want to use these terms you might want to qualify them, like "new equal temperament". Or something like that.

• What's the tonic of a pipe organ? Oct 1, 2016 at 6:41

Conceptually, you could have a equal-interval scale that never "repeats" a note. For example, suppose the pitch difference between each note is 103 cents not 100 (103 happens to be a prime number).

Historically there were many practical non-equal temperament scales that repeated notes in each octave - for example the various mean-tone temperaments.

A just intonation scale has pitches which are exact fractional multiples of each other (e.g. 3/2, 5/4, 5/3, etc) and in practice the fractions much be made from fairly small integers, otherwise you can't recognise the "just intonation" by ear - and my personal definition of "music" is basically "sounds that are intended to be understood by listening to them", not by some abstruse theoretical argument about them.

You could argue the position that any possible interval can be approximated as accurately as you like by a fractional ratio between big enough integers, therefore all scales (both equal and non-equal temperament) are really just intonation scales, but I think that's just playing a mathematical game so far as music is concerned.

• 103 being prime is (largely) irrelevant -- this tuning system would repeat notes every 1200 octaves, but a 49-cent step would also repeat every 1200 octaves. In practice, the way the ear works (physics and maths) means that any scale will repeat notes every octave. Sep 25, 2016 at 4:38
• When you say: “in practice the fractions much be made from fairly small integers, otherwise you can't recognize the "just intonation" by ear” do you (also) mean ‘otherwise you can't distinguish the sounds by ear. If not, why? Sep 25, 2016 at 19:03
• Your ear can distinguish any sufficiently different frequencis, but it can't "hear" them as a particular (harmonising) interval unless the ratios is one of small integers, so that for example a major 6th is a 5:3 ratio. ~~~~ Sep 27, 2016 at 4:41
• @BrianChandler I'm quite sure that a frequency ratio of 8409/5000 will also be perceived as a major sixth, as will 9999/5999. Oct 1, 2016 at 6:40

The temperaments used at Bach's time were experimental (and not only "just interval scales ") with an eye on creating satisfying chords and intervals .The experiment mainly was to divide and distribute the Comma around the scale in different ways. There are dozens of temperament scales.

The Equal Temperament is amazingly accepted as normal by the majority even though the dischords and ugly intervals are many and various. The more you listen for them the worse it gets. I hear bad intervals more than out of tune chords (personally). I like to imagine the earliest Stoneage attempts at scales and chords using lumps of bone /stone / wood which made pleasant chords when played together. This process would soon have revealed an octave and then the other satisfying combinations which would lined up to make an early scale. That's basically where scales started. Just use that as a background idea before education and technology arrived. So with Equal Temperament any scale is better than none.

Equal Temperament is a Mathematical division of an octave using logarithms. A natural scale that a singer would use would not match the same frequencies. Higher or lower scales in Equal Temperament sound pretty much the same apart from the pitch. Basically it's an unnatural construction to make Piano Manufacturing less complicated. It's strange that such an unpopular subject as Mathematics has overwhelmed the Art of Music so effectively.

• It is not strange that mathematics has "overwhelmed" the art of music: mathematics is about patterns, and is the best way to explain the patterns that make music. "Popularity", by the way, has nothing to do with whether things are true or not. Sep 25, 2016 at 17:47