Harmonic errors in singlechainofequalfifths tunings
by David C Keenan, 4Nov1998
last updated 19Jul2000
http://users.bigpond.net.au/d.keenan
Alien landscapes? Click on a chart to see the full sized version.
Errors in 5limit intervals (explained below)
Errors in 7limit intervals (explained below)
Some definitions
In this document, "singlechainofequalfifths tunings" refers to those octavebased tunings whose notes within each octave may be considered to be generated by a single chain of uniform fifths, whether the chain be closed (called a "cycle of n fifths" or "ntone equal temperament") or open (e.g. the various meantones). A "fifth" here is any usable approximation to a 2:3 frequency ratio. An ntone equal temperament (ntET) is a tuning where the octave is divided into n equal divisions. The tuning commonly called simply "equal temperament" is thus 12tET. A "tuning" is here distinguished from a "scale", where a scale is the set of notes (per octave) actually chosen for use in some situation, whereas, in the case of an open tuning, the number of notes in the tuning is potentially infinite. So a scale is here a subset of a tuning. This is similar to the relationship between a mode and a scale.
We assume here that it is desirable for tunings to provide good approximations to smallwholenumber frequency ratios for purposes of harmony. All notes are assumed to be repeated at perfectly tuned octaves (1:2) and therefore factors of two may be ignored or included without changing any of the following results.
Excluded tunings
Singlechainofequalfifths is a very large and significant class of tunings, but note that not all equaltempered scales are members of this class, since some consist of 2 or more closed chains of fifths, equally spaced within the octave. Well temperaments are not in this class because they have more than one size of usable fifth. Just tunings are not in this class because their intervals (other than fifths) are not generated or approximated by chains of fifths at all. Nonoctave tunings are also not included here.
The range of fifth sizes
Each set of four charts above, shows on a horizontal axis, a spectrum of generalised fifths (approximations to 2:3) ranging in size from 685 to 721 cents, in increments of 0.2 cents. The spectrum is split into four sections for convenience of display. I have also shown the approximate position of all the relevant equal tempered tunings (ETs) up to 72. If an ET doesn't appear here, it's because its fifths are either:
much too narrow (2, 4, 9, 11, 16, 23) or
much too wide (1, 3, 6, 8, 13, 18) or
it has more than one chain of fifths:
2 chains: 4, 6, 10, 14, 24, 34, 38, 44, 52, 54, 58, 62
3 chains: 15, 21, 36, 51, 57, 66
4 chains: 20, 28, 48, 68
5 chains: 25, 35, 60
6 chains: 30, 72.
Incidentally, the fifth size for a multichain ET is the same as that for the ET obtained by dividing by the number of chains. e.g. 72ET has the same fifth size as 72/6 = 12ET. Those with multiple chains may have more going for them than their fifth size, but I do not address this here. Similar charts could be produced for each class of multichain open tunings, which would include these ETs. Twochain tunings are addressed in a sequel entitled Harmonic errors in doublechainofequalfifths tunings.
I have also shown on the horizontal axis the approximate position of some historical meantone tunings and Pythagorean tuning. In fact these charts were inspired by Margo Schulter's meantone spectrum chart.
http://www.medieval.org/emfaq/harmony/pyth5.html#6.2
Wolves
Note that fifths outside the approximate range of 692 to 712 cents (max 10 cent error) are often considered unusable wolves based on various people's perception of their consonance, but there is no sharp cutoff. I wish to propose a different definition of wolves that comes from the mathematics of chains of fifths. From 714.9 cents up to 720 cents (5tET), a chain of 6 fifths produces a better 2:3 than the fifths making up the chain. Similarly from 689.3 cents down to 685.7 cents (7tET), a chain of 8 fifths produces a better 2:3 than the fifths making up the chain. This provides a handy functional definition of a wolf fifth for our purposes, namely any fifth for which some chain of less than 11 (±) gives a better approximation of 2:3. This corresponds to any fifth outside the range 689.3 to 714.9 cents, i.e. any fifth with an error outside 12.7 to +12.9 cents.
Familiar regions
Some authors use the term meantone to refer to all of the tunings considered in this paper. I object to this on historical grounds and use it to refer only to the region between 691.5 cents and 700.0 cents. Although this is still broader than the historical usage it is the region where 4 fifths remains the best approximation to a major third (4:5), within a 12 note scale.
On a similar basis I also refer below to the Pythagorean region as between 700.0 and 705.8 cents and the superPythagorean region (around 22tET) as between 705.8 and 711.8 cents.
The errors
The vertical axis represents the magnitude (absolute value) of the harmonic errors in cents. That is, it does not show whether the error is positive or negative (wide or narrow) only its size. The harmonic error in an interval is here defined as the difference in cents between its width in just intonation (a smallwholenumber ratio of frequencies) and the nearest approximation obtained by stacking the given size of fifths or fourths (fifths in reverse) and eliminating octaves.
If there was no limit to the number of fifths or fourths we could stack to approximate a given ratio we could approximate any ratio as closely as we liked with any size of fifth, but there are practical limits. I somewhat arbitrarily chose a limit of ±14 fifths for the approximations to 4:5 and 4:7. Note that this is not a limit on the number of notes in a scale, but if a scale contains more than 15 consecutive notes in a chain of fifths, or spans more than 14 such fifths whether or not all their endpoints appear in the scale, then it might contain a better approximation to some ratio than that shown on the charts. However it will be very difficult to make use of an approximation that appears in only one or two positions and is so many fifths distant.
For example, 14 fifths (14 fourths) allows an approximation to 4:7 in the Pythagorean region of the spectrum. This greatly exaggerates the usefulness of Pythagorean tuning for harmony involving ratios of 7. Increasing this number beyond 14 would introduce new valleys into the charts, but with even less practical utility. However one should not forget that such a choice is available and so these are not the singlechainoffifths charts.
In 1/4comma meantone we get a good approximation to a 4:7 ratio between Bb and G# (an augmented sixth), this is equivalent to stacking 10 fifths, BbFCGDAEBF#C#G#. This is not as useful an approximation as the one we get in the superPythagorean (22tET) region where only 2 stacked fourths (2 fifths) are needed. If our scale were limited to 12 consecutive notes from the chain, then in the meantone region we'd have only two 4:7's while in the superPythagorean region we'd have ten of them. In the Pythagorean region we'd have none.
In the meantone region we get a good approximation to the just major third (a 4:5 frequency ratio) by stacking only 4 fifths, e.g. CGDAE. In the Pythagorean region 8 fourths is best. In 12tET these two approximations are equivalent. In the superPythagorean region 9 fifths is best. In 17tET 8 fourths is the same as 9 fifths. This is why I have used 12tET and 17tET as the boundaries of the Pythagorean region.
The charts
The first set of four charts shows only 5limit intervals while the second set extends this to 7limit. Limit here refers to the highest odd number that appears in any ratio.
The colours are intended to be mnemonic since primary colours are used for the odd numbers: 3blue, 5yellow, 7red, and so ratios between odd numbers are the colour of the mixed paints; e.g. 3:5 (shown in superparticular form as 5:6) is green.
The error spectra for the different intervals are overlaid so it is clear at a glance where the good tunings lie, i.e. in the valleys. How low the valley goes is important, but it is also important that the valley not have steeply sloping sides. The slope of a line indicates how many fifths or fourths are needed to obtain the approximation to that interval. If it's steep it needs many, if it's shallow it needs few.
The dashed line is the RMS (root of mean of squares) combination of the errors in all the intervals on a given chart. This is an alternative to the maximum absolute value as given by the overlaying of the absolute error spectra.
Consistency is enforced in favour of the lone odd numbers (2:3, 4:5, 4:7) by the expedient of calculating the errors in the other intervals by subtracting the (signed) errors of the relevant pair of odd numbers. This may be less than optimum in some cases (but only near peaks).
If one's intention is purely to maximise 7limit consonance in a singlechainofequalfifths tuning, the main thing one learns from this is that there are really only three fifthsizes (±1 cent) that are of interest, namely 697, 702 and 710 cents, although 714 cents (37tET) and to a much lesser extent 694 cents, have some possibilities.
Here are the numbers of fifths required to approximate lone oddnumber ratios up to 11, for each of these five fifth sizes. Those for ratios of two odd numbers can be found by subtraction. e.g. In the vicinity of 697 cents a 5:7 is best approximated by 10  4 = 6 fifths, and a 5:6 by 1  4 = 3 fifths. Remember, factors of 2 on either side of the colon ":" do not affect the result. Note that the 4:9 must be 2 fifths if consistency is to be maintained.

2:3 
4:5 
4:7 
4:9 
8:11 
694 cents 
1 
11 
9 
2 
6 
697 cents 
1 
4 
10 
2 
13 
702 cents 
1 
8 
14 
2 
11 
710 cents 
1 
9 
2 
2 
6 
714 cents 
1 
14 
2 
2 
11 
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, or allowing more or fewer fifths to approximate the intervals, 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.