User:Robertinventor/Regular tunings and temperaments

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This is a draft for a lede for the Regular temperament article. It follows Wikipedia's guideline that the lede of an article shouldn't be a teaser for the article, but rather should be a short mini article in its own right, describing the main points to be covered and expanded on in the article in a self contained way.

There are many other ways we could introduce the material. If this lede is thought good, or some rewrite of it, the main question would be whether we can have a single article called "Regular tuning and Temperament" or whether this is too much for one article and we should split it into two separate articles, the first on "Regular tuning" and the second on "Regular temperament". If we do it as two articles then the reader would probably have to read the article on "Regular tuning" first before they read the article on "Regular temperament" if completeley new to the subject.

It's a tricky problem I think how to present this, as there is a lot to present in a single lede. On the other hand, if we assume too much or spread this over several articles then it may not be so easy for a newbie reader, as they have to go back and forth between two or more articles until they have all the concepts clear in their minds. Perhaps another alternative is to have a shorter lede summarizing the basic points then the rest of the material split over sub sections, but the problem then is - will the reader understand the lede if they don't yet fully grasp the basic concepts it needs to state what a regular temperament is? I expect this to be a matter of some debate on the talk page. This draft lede is around 1412 words. By comparison the Van Gogh article lede, to take an example of one with a longish lede, is around 510 words. Can it be done with a third of this number of words? If not, what is the way ahead?


The topic of regular temperaments is one of the most central in modern microtonal theory. It requires a number of ideas:

  • Generators of a scale
  • Just intonation
  • Musical equivalence
  • Tempering a musical interval
  • Vanishing commas
  • Graphing a tuning in 1D, 2D and higher dimensions with "lattice diagrams".

This lede will introduce each of those ideas briefly. For a more detailed treatment see Paul Erlich's article in Xenharmonicon.[1]

First, the idea of a generator. Usually scales and tuning systems are built up by combining musical intervals. This can be done in many ways, the whole tone C D can be treated as one of the generating intervals, or you can get it by stacking together two fifths C - G then G - D then dropping down by an octave.

<score vorbis="1"> { \time 5/4 c' g' d d' }</score>

You can construct the same tuning from different choices of generator. Let's take an example: an equal temperament pentatonic scale C D E G A may be obtained by combining together ascending whole tones like C - D and minor thirds like E G.

<score vorbis="1"> { \time 5/4 c' d' e' g' a' }</score>

Alternatively, you could use the ascending fifth C - G and the descending fourth such as G - D

<score vorbis="1"> { \time 5/4 c' g' d' a' e' }</score>

Any of these choices, and several others, can generate all the notes of the scale. The intervals you use to make a scale are known as its "generating intervals". The pentatonic scale needs a minimum of two generators to construct all its notes starting from a start note or the 1/1 such as C, though there are many ways you can choose those two generators.

It's common to treat the octave as a musical equivalence, since notes an octave apart sound "the same" to humans. We use equivalences like this all the time without noticing. For instance in tuning theory, notes at the same pitch are treated as equivalent irrespective of their volume or duration. Tremelo is also ignored as well as pitch vibrato and small variations in pitch and pitch variation during the note. Timbre is also ignored - notes on all musical instruments at the "same pitch" are treated as equivalent even though it may sometimes be quite a complex matter to explain how this works, e.g. for bell sounds. Also except in specialist studies we ignore the stretched harmonics and stretched octaves of a piano tuning. In a similar way, in tuning theory musicians often treat notes an octave apart as equivalent too, especially when discussing temperaments.

Under octave equivalence, the pentatonic scale only needs one generator, the fifth (or alternatively, the fourth as a descending interval). You can then get all the notes of the pentatonic scale using an ascending fifth plus octave equivalence. To illustrate it, here is the pentatonic scale as ascending fifths, with the notes outside the octave dropped back into the octave through octave equivalence: <score vorbis="1"> { \time 5/4 c' g' d d' a a' e e' }</score> Less commonly, musicians use tritave equivalence, so that notes are treated as the same if they are a tritave apart (interval 3/1). The Bohlen Pierce scale is an example. There are other possible choices of equivalence as well such as the 3/2. It's the same approach though. Your choice of equivalence can convert a scale with two generators into a scale with one generator plus an equivalence relation.

Many musicians will be familiar with at least some examples of linear tunings. These include

  • Pythagorean tunings, connected by a chain of pure fifths
  • The twelve equal scale with a fifth of 700 cents
  • Meantone tunings connected by a chain of tempered fifths smaller than 700 cents
  • Schismatic tunings connected together by fifths in between 700 cents and the pure fifth of around 702 cents.

Less familiar examples include the phi tunings which are based on the golden ratio. The golden ratio is the interval hardest to approximate by pure ratios, and these tunings don't have any direct connection with fifths, fourths, thirds or any other familiar musical interval.

To explain the notion of a regular temperament we also need to present just intonation tunings. These use intervals between notes in the harmonic series. For instance the musical interval between the second and the third harmonic is a pure fifth, and as a frequency ratio it is 3/2 so 3/2 is a just intonation ratio. Other familiar just intonation ratios include the major third 5/4, and the minor third 6/5. 7/5 also would count as a just intonation interval (it doesn't have to be an interval between successive harmonics) and indeed any ratio between small whole numbers is more broadly called a "just intonation interval".

A temperament is a tuning which approximates some form of just intonation, often by tempering the intervals so that one of the musical commas vanish. For instance a meantone tuning tempers the fifth, and possibly also the major third, in such a way that a stack of four fifths, such as C G D A E leads to an interval that is a major third above the first note, here the C - E, up to octave equivalence.

If you don't use any tempering, but use the pure musical intervals from the harmonic series 3/2 and 5/4, then a stack of four fifths takes you to 81/64, a very sharp "major third" which is sharper than 5/4 by the syntonic comma of 81/80. So it's impossible to have both the fifth and the major third pure and preserve the usual structure of a diatonic scale. Quarter comma meantone preserves the purity of the major third by tempering the fifths. Other compromise meantone scales will temper both the major third and the fifth to get the musical intervals to fit together.

Just intonation tunings don't temper the 5/4 or the 3/2, but treat them as distinct intervals. If you do this, then the tuning will need two separate generators apart from the equivalence relation.

Just intonation 5 limit lattice. Only two of those intervals are needed to generate the tuning, for instance the major thirds and the fifths. Because there is a third consonance connecting across the diagonal of the unit cell, such tunings are often shown using a triangular grid like this. The entire lattice in this case consists of dovetailed major chords (pointing upwards) and minor chords (pointing downwards).

One way to display tunings is to use a lattice diagram. The idea is to draw lines in different directions for each of the component musical intervals. In the diagram shown to the right, movement by one step to the right raises the pitch by 5/1 and movement by one step diagonally upwards to the right raises the pitch by 3/1. When the number of intervals is more than two, then the diagram may be understood as showing a lattice in three or more dimensions.

In diagrams like this, then the octave is usually not shown. One way to understand this mathematically is that the octave is an "equivalence relation". We think of notes at multiple octaves apart as equivalent for the purposes of these diagrams. So there is no need to show all the octaved notes. They are all "the same note" in this context.

Regular tunings are classified by rank. The rank includes the octave as a generator and doesn't treat it as an equivalence. So a tuning with generators 5/4, 3/2 and 2/1 would have rank 3.[1] However the tuning can be shown using a two dimensional diagram by using octave equivalence. For this reason they are often referred to by dimension as well, referring to the dimensionality of its lattice diagram.

It's common to refer to a scale itself as "two dimensional" when its lattice diagram is 2D up to octave equivalence, and so on. Such a scale has two generators for its lattice diagram, but it has three generators when we are required to generate the notes in every octave separately. Usually the equivalence relation for a given scale, such as the octave or tritave is treated as fixed, but its other generators may often be chosen in many different ways, all of which generate the same pattern of pitches though organized in different ways.

Any interval such as 15/8 which can be got by combining those intervals will be somewhere in the lattice. As with the pentatonic example, then there are many ways to choose the two generators for its lattice diagram. For instance the same lattice can be generated using

  • 5/1 and 3/1
  • 5/4 and 3/2
  • 6/5 and 3/2
  • 6/5 and 5/4

amongst many other possibilities.

A scale with a two dimensional lattice diagram (up to octave equivalence) normally, but not always, has rank 3 once you treat the octave as a generator along with the others. If it has a one dimensional lattice diagram, it usually has rank 2 and so on. However twelve equal or any equal temperament has a one dimensional lattice diagram - just a straight line. But it only needs one generator, the semitone. Of course you can use two generators, such as the fifth and the octave, to generate twelve equal and in some ways that's a better way to think about it. However, the rank is defined using the minimum set of generators, and in that sense, it is rank 1.

The same can happen in 2D. For instance, divide the octave into three equal parts (the equal tempered major third) so one generator of 400 cents, and as a second generator use a step of 15 cents. It's got a good approximation for 5/4 as 400 cents - 15 cents. It has a two dimensional lattice diagram. But the octave is not needed as a generator so it has rank 2, not 3.

Tuning theorists therefore often work with "Octave specific" latices - in other words they will show the lattice, or if they show it in 2D they do that on the understanding that this is a single slice in a larger lattice rather than using octave equivalence.

As with the one dimensional case of the phi tunings, this generalizes to tunings that have any pair of generating intervals, which need not be thought of as approximating just intonation intervals in any way. The 2D regular tunings that approximate harmonic series intervals are amongst the ones of especial interest to musicians of course.

Looked at this way then a meantone scale is then a result of tempering the 5/4 or the 3/2 or both in such a way as to make the tempered 81/80 "vanish" - you want the result of sequence of tempered fifths C G D A E to be exactly the same as the tempered major third from C - E. So in this system the tempered 81/80 transforms into the unison, as the two versions of the note E are identical. In quarter comma meantone the fifth is tempered and the major third kept pure. In other meantone scales both are tempered.

The 7/4 can also be treated as another generator leading us to a three dimensional just intonation lattice and higher dimensions are also possible using intervals such as 11/8 and 13/8 which can't be obtained using any of the previous intervals.

We can now explain the concept of a regular temperament. It is similar to the idea of a linear temperament but involves two or more tempered generating intervals. So for instance you can start from a three dimensional lattice involving 3/2, 5/4 and 7/4 and then by tempering you can define the tempered 7/4 in terms of tempered 3/2s and 5/4s so eliminating it and reducing the dimension of the lattice, for instance from 3D to 2D. Typically this is done by transforming one or more of the various musical "commas" or small intervals into the unison, so causing the comma to "vanish" as with the example of 81/80 for the meantone tunings. Sometimes several commas are tempered at once reducing the dimension by more than one e.g. from 4D to 2D or from 3D to 1D.

With this background, we can now define a regular temperament as a regular tuning in which the generating intervals are interpreted as tempered just intonation intervals. Usually the result is that one or more musical commas is tempered to the unison and so vanishes.

As with a regular tuning, often one of the intervals used to generate the tuning is treated as a musical equivalence, and is not shown in the diagrams or treated as a generator. The most common choice of interval for a musical equivalence is again the octave.

Though many regular tunings are also used as regular temperaments, some are not intended to be used in this way at all. The phi tunings are a good example of regular tunings that are not intended as regular temperaments.

  1. 1.01.1 Erlich, Paul (Summer 2006). "A Middle Path Between Just Intonation and the Equal Temperaments Part 1" (PDF). Xenharmonikôn, An Informal Journal of Experimental Music,. 18.