Harmonic errors in double-chain-of-equal-fifths tunings
by David C Keenan, 26-May-1999
last updated 28-Nov-1999

http://users.bigpond.net.au/d.keenan

More alien landscapes? Click on a chart to see the full sized version.

Errors in 5-limit intervals (explained below)

Errors in 7-limit intervals (explained below)

This paper is a sequel to Harmonic errors in single-chain-of-equal-fifths tunings. Please refer to that paper for a detailed explanation of the charts and the terminology used below.

We consider here those octave-based tunings that consist of two chains of equal fifths a half-octave apart. The closed members of this set consist of those ETs having either (a) two cycles of fifths with an odd number of fifths in each or (b) one cycle with an even number. Among the ETs with tolerable fifths and no more than 72 divisions, the former are 10, 14, 34, 38, 54, 58, 62 and the latter are 12, 22, 26, 32, 40, 42, 46, 50, 56, 62, 64, 70.

Remember that these charts are somewhat arbitrary since the number of fifths (or fourths) allowed to approximate the 4:5 and 4:7 intervals is limited to some (more than) reasonable value. For the single-chain charts of the preceding paper the limit was ±14 fifths. For the double-chain charts above I have chosen to allow ±8 fifths, and of course the half-octave may or may not be used. One should not forget that different choices are available and so these are not the double-chain-of-fifths charts.

For example, -8 fifths (8 fourths) plus a half-octave allows a good approximation to 4:7 in the region of the spectrum from about 703 to 705 cents, including 58-tET and 46-tET. This greatly exaggerates the usefulness of tunings in this region for harmony involving ratios of 7. In the 707 to 711 cent region, including 22-tET, -2 fifths gives a reasonable approximation to a 4:7. Consider a scale consisting of 10 consecutive fifths a half-octave away from another 10 consecutive fifths (a total of 22 notes per octave). In the 704 cent region we would only have 6 intervals of 4:7 while in the 709 cent region we would have 18 such intervals, although it should be noted that the errors are nearly three times as great.

We can recognise 5 regions of interest at: 694, 702, 704/706, 709 and 715 cents (±1 cent). The 704/706 refers to the fact that the minimum for this region moves considerably between the 5-limit and 7-limit cases. Note that four of these are essentially the same as those we found for the single chain tunings (within a cent), but we have lost the meantone region near 697 and gained a new region at 704/706.

Here are the numbers of fifths and half-octaves required to approximate lone odd-number ratios up to 11, for each of these five fifth sizes. "+h" indicates that a half-octave was used in addition to the fifths. Those for ratios of two odd numbers can be found by subtraction. e.g. In the vicinity of 704 cents a 5:7 is best approximated by -8+h - (-2+h) = -6 fifths, and a 5:6 by 1 - (-2+h) = 3-h = 3+h (the sign of the half-octave is irrelevant). Note that the 4:9 must be twice the number for the 2:3 if consistency is to be enforced, even though better approximations may exist.

Note that most of these regions can do no better than to represent the 5:7 as the half-octave (the orange plateaus at a height of 17.5 cents). The 704 and 715 cent regions do better than this, but with other costs.

We can see that double-chain in the 706 cent region (near 34-tET) is an even better choice for 5-limit accuracy than a single chain in the 697 cent (meantone) region (near 50-tET and 31-tET). However the 706 cent double-chain will give two less major triads and two less minor triads than a meantone with the same number of notes.

For 7-limit, double-chain in the 709 cent region (near 22-tET) is not the most accurate but certainly has the most tetrads per note by far, due to the extreme compactness of its tetrad on the chains. Here is what its otonal (major) tetrad looks like on a double chain.

Here is the next most compact, which is the meantone tetrad.

Meantone is considerably more accurate at the 7-limit than the 22-tET double-chain region but it will always have fewer tetrads per note. In the 22-tET region a 12 note scale can have 6 otonal (major) and 6 utonal (minor) tetrads. The following optimised 12-note meantone scale still only has 3 of each.

If you wish to see the errors in the individual intervals, or experiment with things like weighting the errors differently for different intervals, extending the results to 9-limit or 11-limit, allowing more or fewer fifths to approximate the intervals, or using triple or quadruple chains, you are welcome to download the Excel 97 spreadsheet I used to generate the charts for both single and double chain tunings.
http://users.bigpond.net.au/d.keenan/Music/ChainOfFifthsTunings.xls.zip (94kB)
However I don't guarantee it will be easy to understand.