11 note chain-of-minor-thirds scale
by David C Keenan, 9-Jul-1998
last updated 8-Jun-2001

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

Click on the following diagram to see it at full size.

Prompted by discussions with Dan Stearns and Graham Breed on the Alternative Tuning List, I discovered that an octave-based scale of 11 consecutive notes in a chain of minor thirds (approximate 5:6 ratios) has some useful properties, particularly if the minor thirds are widened very slightly (between 0.1 and 1.4 cents).

Melodically, it is a moment-of-symmetry (MOS) as described by Erv Wilson. That is, it has only two step sizes and has reflective symmetry (about D in the notation below). Of course any single chain of any interval will have melodic symmetry, so it's the "only two step sizes" that is significant here. It also has Myhill's property, which means that each interval class (number of scale steps) comes in exactly two sizes (cents). It is not a "proper" scale, but with step sizes of 181 cents and 68 cents, the largest step is only about 2.7 times the size of the smallest, so it is not too improper. When constructed as a subset of 19-tET it looks like this:

The numbers give the size of each scale step in 1/19ths of an octave. In a proper scale one does not find any interval that contains more scale steps than another while being smaller in size, such as Bb:Cb vs. Ab:Bb above. In a strictly proper scale an interval with more scale steps is always larger in size, not merely larger or equal.

Here are all the notes of 19-tET arranged as a chain of minor thirds.

This is actually a cycle of minor thirds in the case of 19-tET, but this sequence, with D as its centre, is used in this document as a standard notation for all chain-of-m3rds scales with 19 or fewer notes. Notice the hiccup every 6 links where a note has two names. This is necessary if all 5-limit (major and minor) triads are to be named consistently with their diatonic counterparts, since six minor thirds corresponds to a fifth.

If we omit the four notes Bb, Db, D#, F#, from our 11 note scale, what remains is a 7-note minor-third MOS that goes 1 4 1 4 4 1 4 in 19-tET. In the preferred keyboard mapping, this 7-note MOS maps to the white keys according to the letter part of the note-name not in parenthesis. The other four notes map to the black keys according to their names, and the G#/Ab key is not used.

Listen to this piece by Mats Öljare, in a 7-note minor-third MOS from 19-tET. http://www.angelfire.com/mo/oljare/images/skopnytt.mid

Graham Breed pointed out that a key signature of E#, G#, Ab, Cb, would allow these to be notated economically on standard staves. This key signature can then simply be ignored when playing on a keyboard using the standard mapping. When this key signature is omitted or ignored we have a non-diatonic notation with 7 naturals giving the following names for the chain of minor thirds.

Here's the standard keyboard mapping in Scala format, which corresponds directly to the above notation for the 11 note scale.

I was surprised to learn, thanks to Carl Lumma, that an 8 note chain is proper, although it has 3 different step sizes and so is not a MOS and does not have Myhill's property. There are no other proper chains with more than 4 and less than 15 notes. Nor are there any other MOS/Myhill's chains in this range apart from 7 and 11. With the above keyboard mapping, either Db or D# can be added to the white notes to make an 8 note scale.

Carl proposed that we ignore diatonic notation and try notations with 4, 8 or 11 naturals, but noted problems with all of them. He also suggested that due to the limitations of human short term memory noted by Miller in The magical number seven, plus or minus two, and Rothenberg's results regarding propriety, an 11-note scale may be heard as a proper 8 or 4 note scale. Carl later pointed out that, although the 8 note scale is proper, it has so many ambiguous intervals (is so far from strictly proper (has such low Rothenberg stability)) that its propriety is almost worthless and it will have melodic properties similar to those of improper scales. This still leaves the possibility that it will be heard as a 4 note "scale" (a single diminished 7th chord) with variations.

Note that every scale of interest here has 4 large steps (of between a wholetone and a minor third) and a variable number of much smaller steps (of about two thirds of a semitone). So, following Carl's suggestion, I propose the following 4-naturals notation for the chain of minor thirds.

Notice that this is entirely consistent with the diatonic-based notation above, since H corresponds to Ab. This should not be confused with the German diatonic notation where H is B and B is Bb. This notation suggests mapping the naturals B, D, F, H to the black keys Eb, Gb, Bb, Db respectively. Then, apart from H#, which does not appear, each natural has a sharp and a flat on white keys to the right and left of it. As before, the Ab key is not used. Here's this alternative keyboard mapping in Scala format.

To notate the above on a staff, the following 8-naturals scheme is more economical, but is no longer consistent with diatonic.

Ebb Gbb  Ab  Cb  Eb  Gb  A   C   E   G   B   D   F   H   B#  D#  F#  H#  Bx
               |       |       | 4-note SP MOS |   |   |   |
               |       |-------- 7-note MOS -------|   |   |
               |       |-------- 8-note Proper --------|   |
               |--------------- 11-note MOS ---------------|

To get 11 notes on the keyboard, one would need to map the notes ABCDEFGH to the mostly white keys EFGABCDEb in a shifted version of the standard mapping, or map the notes EGBD to the black keys EbGbBbDb in a shifted version of the alternative mapping. However it is probably preferable to change the notation so the note names match the keys in the standard mapping. Unfortunately this makes the 8th natural occur, somewhat unnaturally, between D and the E so that pitch order is no longer simply alphabetical order. I strongly recommend calling the additional natural "L" in this case. The logic behind this is that it allows ABCD to be considered as HIJK (restoring alphabeticality), and also avoids confusion with a notation that uses 11 naturals A to K.

Bbb Dbb  Eb  Gb  Bb  Db  E   G   B   D   F   A   C   L   F#  A#  C#  L#  Fx
               |       |       | 4-note SP MOS |   |   |   |
               |       |-------- 7-note MOS -------|   |   |
               |       |-------- 8-note Proper --------|   |
               |--------------- 11-note MOS ---------------|

Note that the above notation is consistent with the second-given notation that has 7 naturals, since L corresponds to D#.

The melodic properties alone are not particularly interesting. There are many such scales. It's the harmonic properties in conjunction with these melodic properties that make it interesting. Out of the 55 possible pairs of notes, 45 are acceptable approximations to 9-limit consonances. That's a connectivity of 82%. For comparison, the diatonic scale has a 9-limit connectivity of 18/21 = 86%. Paul Erlich pointed out that a weighted connectivity, where interval counts are weighted according to relative consonance (e.g. reciprocal of odd-limit), would be more meaningful. Of course I'm assuming something about what are maximum acceptable errors here. Namely about 18 cents in any interval, which can be achieved in the 1/6 kleisma tuning described below. When embedded in 19-tET this creeps up to 21.5c (in the 4:7).

The 11 note scale contains 5 major and 5 minor triads which can all be extended to rough 7-limit tetrads (augmented 6ths and minor diminished 7ths). For every note that is dropped from the scale we lose one major and one minor, so 8 notes gives two of each and 7 notes gives only one of each. Many other chords are possible, as shown with the lattice diagram above.

In a chain of minor thirds the following ratios are approximated by the following numbers of minor thirds (octave reduced).

So we expect that plus and minus 4, 8 and 10 links will produce dissonances. 4 and 8 links are indeed dissonant since they correspond to 1 and 2 small steps (chromatic semitones). -10 links, of which there is only one example in an 11 note chain, is similar in concept to the meantone wolf but just happens to give a more accurate 7:9 than does 9 links (equally accurate in 19-tET). This "wolf" 7:9 (more like a poodle) is not shown in the lattice above.

Note that 6:7 and 7:8 (4:7 inverted) are approximated by the same size interval in this scale, in the same way that 7:8 and 8:9 are in Paul Erlich's decatonics, and 8:9 and 9:10 are in diatonics (which is where the name "meantone" comes from). Here are the errors in three tunings of this scale that represent three different types of optimum. Click on a type to download the corresponding file in Scala format.

                       Generator            Error in interval (cents)            F#:Bb
Type of optimum        (cents)    2:3   4:5   5:6   4:7   5:7   6:7   5:9   7:9   7:9
-------------------------------------------------------------------------------------
Min max abs error      316.99       0  -1.4  +1.4 -17.8 -16.5 -17.8  +1.4 +17.8  -5.0
Min RMS error (10.6c)  316.49    -3.0  -3.9  +0.8 -19.4 -15.5 -16.3  -2.2 +13.3     0
Min max otonal         315.79    -7.2  -7.4  +0.1 -21.5 -14.1 -14.2  -7.1  +7.0  +7.0
 beat rate = 19-tET
 (0.34 of root freq)

Note that the first tuning above is essentially Just at the 5-limit. Regarding the second tuning, the RMS error is calculated in a generic 9-limit manner, using all 10 intervals. That is, it includes the 4:9 error (as twice the 2:3 error) and includes the 2:3 error twice (once as 4:6 and again as 6:9) despite the fact that these do not all occur in this scale. It ignores the "wolf" 7:9 from F# to Bb. The last tuning minimises the worst beat rates for all intervals in an approximate 4:5:6:7:9 chord (which actually only occurs as separate 4:5:6:7 and 5:6:7:9 chords). In this case 4:7 and 6:7 have equal worst beat rates at approximately 0.34 of the frequency of the root of the chord (the 4). Of course this is too fast to be perceived as a beat but is perceived as roughness. This is equivalent to weighting the 6:7 error at 1.5 times the 4:7 error. The generating minor third for this last tuning just happens to be the same as the 19-tET minor third (5/19 of an octave) to within 0.01 of a cent.

Even if you consider the above approximation to the 4:7, or all the ratios of 7, to be too rough, the scale is still significant for its large number of diatonic-type harmonies and the fact that they can be rendered essentially just (errors less than 1.4 cents).

Manuel Op de Coul pointed out that these tuning variations can be described as being generated by minor thirds widened by various fractions of a kleisma, in the same way that the various meantones are described as generated by fifths narrowed (or fourths widened) by various fractions of a syntonic comma. A kleisma is the difference between a just fifth (2:3) and an octave reduced chain of 6 just minor thirds (5:6), i.e. 5⁶/(3⁵.2⁶) or about 8.11c. So the first tuning above is exactly 1/6-kleisma and the generator corresponds to the 6th root of 3. Minimum RMS error occurs at approximately 1/10 kleisma and 19-tET is approximately 1/55 kleisma. A 1/5 kleisma tuning has just major thirds (4:5) and 1/7 kleisma has just minor 7ths (5:9).

After 19-tET, the next largest ET that might be said to contain this scale is 34-tET, but it corresponds to about 1/4 kleisma and the errors in its ratios of 7 are worse than 19-tET. Of those with lower errors, 53-tET is close to 1/6 kleisma and 72-tET is about 1/8 kleisma.

The unusual thing about this scale is that there are no chains of fifths. There is a long chain of minor thirds of course, and there are chains of many other intervals as you can see on the lattice above. We would need to extend the chain to 13 notes before a chain of two fifths appeared. After 11 notes, 15 notes is the next MOS and this is also strictly proper.

Here is the 9-limit lattice for the diatonic scale, for comparison.