Harmonic errors in doublechainofequalfifths tunings
by David C Keenan, 26May1999
last updated 28Nov1999
http://users.bigpond.net.au/d.keenan
More alien landscapes? Click on a chart to see the full sized version.
Errors in 5limit intervals (explained below)
Errors in 7limit intervals (explained below)
Introduction
This paper is a sequel to Harmonic errors in singlechainofequalfifths tunings. Please refer to that paper for a detailed explanation of the charts and the terminology used below.
We consider here those octavebased tunings that consist of two chains of equal fifths a halfoctave 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 singlechain charts of the preceding paper the limit was ±14 fifths. For the doublechain charts above I have chosen to allow ±8 fifths, and of course the halfoctave may or may not be used. One should not forget that different choices are available and so these are not the doublechainoffifths charts.
For example, 8 fifths (8 fourths) plus a halfoctave allows a good approximation to 4:7 in the region of the spectrum from about 703 to 705 cents, including 58tET and 46tET. This greatly exaggerates the usefulness of tunings in this region for harmony involving ratios of 7. In the 707 to 711 cent region, including 22tET, 2 fifths gives a reasonable approximation to a 4:7. Consider a scale consisting of 10 consecutive fifths a halfoctave 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.
The charts
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 5limit and 7limit 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 halfoctaves required to approximate lone oddnumber ratios up to 11, for each of these five fifth sizes. "+h" indicates that a halfoctave 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) = 3h = 3+h (the sign of the halfoctave 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.

2:3 
4:5 
4:7 
4:9 
8:11 
694 cents 
1 
4 
4+h 
2 
6 (or 7+h) 
702 cents 
1 
8 
8+h 
2 
6 (poor) 
704/706 cents 
1 
2+h 
8+h 
2 
6 (poor) 
709 cents 
1 
2+h 
2 
2 
5+h (or 6) 
715 cents 
1 
7+h 
2 
2 
5+h (poor) 
Note that most of these regions can do no better than to represent the 5:7 as the halfoctave (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 doublechain in the 706 cent region (near 34tET) is an even better choice for 5limit accuracy than a single chain in the 697 cent (meantone) region (near 50tET and 31tET). However the 706 cent doublechain will give two less major triads and two less minor triads than a meantone with the same number of notes.
For 7limit, doublechain in the 709 cent region (near 22tET) 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.
 7  4 6 
 5    
or
 5    
 7  4 6 
Here is the next most compact, which is the meantone tetrad.
 4 6   5      7 
Meantone is considerably more accurate at the 7limit than the 22tET doublechain region but it will always have fewer tetrads per note. In the 22tET region a 12 note scale can have 6 otonal (major) and 6 utonal (minor) tetrads. The following optimised 12note meantone scale still only has 3 of each.
DbAb  F C G D A E B F#  D#A#
Further investigations
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 9limit or 11limit, 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.