In the past I’ve held some disparaging opinions of quarter tone music. So have some fairly notable microtonalists. Easley Blackwood has little to say in its favor, despite (or perhaps because of) having included a 24-equal piece in his “Microtonal Etudes”.

But I’ve modified my position. It seems to me that the nature of the thirds has a strong impact on the way westerners perceive a melody or a harmony: the most basic characterization of a chord or a scale is “major” versus “minor”, and the distinction is due to what size thirds are used where. So if you want to challenge a listener’s perceptions (and that appeals to me) an obvious way to do that is by using neutral thirds, halfway between major and minor, or, equivalently, half of a perfect fifth.

A just fifth is a frequency ratio of 3/2; half of this is sqrt (3/2) = 1.224745 = 350.98 cents. This is the frequency ratio for a… well, we can’t call it “just”, can we? But a neutral third that’s half a just fifth.

We can also consider the value halfway between the minor third, 6/5, and the major third, 5/4; it’s the geometric mean of the two, sqrt (6/5 * 5/4) = sqrt (6/4) = sqrt (3/2) — the same value, as we might have expected.

And we now might want a just interval, the ratio of two small integers, that approximates this. (When you think about it, this is using a rational value as a temperament of an irrational one — exactly the opposite of the usual sort of tempering!) The fairly obvious approximation is 11/9. (The approximations generated by continued fractions are 4/3, 5/4, 6/5, 11/9, 38/31, 49/40, 60/49…). 11/9 is 347.41 cents, 3.5 cents flat, which should be fine for our purposes. (Of course (3/2)/(11/9) = 27/22 is the interval left when 11/9 is taken away from 3/2 so is a 3.5 cents sharp neutral third.)

This is an 11-limit interval, but it uses no factors of 5 or 7. This suggests using the primes 2, 3, 11 as a basis for a lattice. Then, following Ehrlich (“A Middle Path Between Just Intonation and the Equal Temperaments, part 1”, Xenharmonikôn vol. 18, 2006, pp. 159-199), we can now look for optimal rank-2 tunings (i.e. ones generated by a period and a generator) that approximate this by tempering out suitable commas.

One such comma is 2^{-1} 3^{5} 11^{-2} = 243/242 = 7.1 cents. For Tenney optimization (TOP) the primes 2, 3, and 11 are tempered as follows:

ln (nd) = 10.98

cents(n/d) = 7.14

temper 2 by -cents(n/d) * ln(2) / ln(nd) = 0.45 cents => 1200.45 cents

temper 3 by cents(n/d) * ln(3) / ln(nd) = -0.71 cents => 1901.24 cents

temper 11 by -cents(n/d) * ln(11) / ln(nd) = 1.56 cents => 4152.88 cents

To verify, the 243/242 comma then becomes -1200.45+5*1901.24-2*4152.88 = 0.0.

We also can find periodicity blocks for this lattice using pairs of commas. If one comma is relatively large these can give rise to diatonic-like scales, and if both commas are small they can suggest equal temperaments. One possibility one finds is to use the commas 243/242 and 2^{-5} 3^{1} 11^{1} = 33/32 = 53.3 cents to give a periodicity block of 7 notes. With a non-parallelogram boundary, one realization of this is the scale 1/1 9/8 11/9 4/3 3/2 44/27 11/6 (2/1) which is just the set of notes contained in three neutral triads with roots in a chain of fifths.

121/108 | 121/72 | 121/96 | 121/64 | 363/256 | 1089/1024 | 3267/2048 | |||

44/27 |
11/9 |
11/6 |
11/8 | 33/32 | 99/64 | 297/256 | |||

32/27 | 16/9 | 4/3 |
1/1 |
3/2 |
9/8 |
27/16 | |||

512/297 | 128/99 | 64/33 | 16/11 | 12/11 | 18/11 | 27/22 | |||

4096/3267 | 2048/1089 | 512/363 | 128/121 | 192/121 | 144/121 | 216/121 |

Note that the first of these two commas is just the difference between two neutral thirds ((11/9)^{2}) and a perfect fifth (3/2). Also, the difference between the above two commas is 2^{-4} 3^{-4} 11^{3} = 1331/1296 = 46.1 cents, which could be used instead of either of the others to get the same block.

Another comma is 2^{12} 3^{-1} 11^{-3} = 4096/3993 = 44.0912 cents; in conjunction with 243/242 it gives periodicity blocks of 17 notes. Here is one:

121/108 | 121/72 |
121/96 |
121/64 | 363/256 | 1089/1024 | 3267/2048 | |||

44/27 | 11/9 |
11/6 |
11/8 |
33/32 |
99/64 | 297/256 | |||

32/27 | 16/9 |
4/3 |
1/1 |
3/2 |
9/8 |
27/16 | |||

512/297 | 128/99 | 64/33 |
16/11 |
12/11 |
18/11 |
27/22 | |||

4096/3267 | 2048/1089 | 512/363 | 128/121 | 192/121 |
144/121 |
216/121 |

With 4096/3993 and 1331/1296 one gets a 15 note block, e.g. as above but missing 121/72 and 144/121. Using 4096/3993 and 33/32 one gets a less than interesting 2 note block.

Further afield, another comma is 2^{-17} 3^{2} 11^{4} = 131769/131072 = 9.18177 cents. Aside from 243/242 this is the simplest comma of under 40 cents. Used with 243/242 it gives periodicity blocks containing 24 notes:

1331/1296 | 1331/864 | 1331/1152 | 1331/768 |
1331/1024 | 3993/2048 | 11979/8192 | 35937/32768 | 107811/65536 | |

121/81 | 121/108 | 121/72 |
121/96 |
121/64 |
363/256 |
1089/1024 |
3267/2048 | 9801/8192 | |

88/81 | 44/27 | 11/9 |
11/6 |
11/8 |
33/32 |
99/64 |
297/256 |
891/512 | |

128/81 | 32/27 |
16/9 |
4/3 |
1/1 |
3/2 |
9/8 |
27/16 | 81/64 | |

1024/891 | 512/297 | 128/99 |
64/33 |
16/11 |
12/11 |
18/11 |
27/22 | 81/44 | |

16384/9801 | 4096/3267 | 2048/1089 | 512/363 | 128/121 | 192/121 |
144/121 | 216/121 | 162/121 | |

131072/107811 | 65536/35937 | 16384/11979 | 4096/3993 | 2048/1331 | 1536/1331 | 2304/1331 | 1728/1331 | 2592/1331 |

Given the small sizes of the commas generating the 24-note blocks, it is not a surprise that 24-equal temperament gives a nice approximation to our lattice. So here is some sort of justification of 24-equal — in terms of a 5-less, 7-less, 11-limit lattice.

Since the 15- and 17-note blocks use larger commas I would not expect 15-equal or 17-equal to work as well. In fact 17-equal contains good neutral thirds: 1200*5/17 = 352.9 cents. (So does any equal temperament in which the perfect fifth is represented by an even number of steps, of course.) But it does not support other 11-limit intervals as well; for instance, 11/8 = 551.3 cents and 1200*8/17 = 564.7 cents. So if neutral thirds are the motivation, 17-equal is good; if 5-less, 7-less 11-limit is the motivation, 17-equal isn’t so good.

How does this work in practice? Well, surprisingly and a bit annoyingly (to me at least), these neutral thirds work well harmonically but I’m not so sure they work melodically. That is, to my ear, an 18:22:27 triad isn’t strongly major or minor. But I tried using the above 7-note scale to play, for instance, “Frere Jacques” — so famously used in a minor mode by Mahler in his First Symphony — and it didn’t sound neutral; it sounded decidedly minor. Maybe because I’m used to hearing 400-cent major thirds?

Or maybe it’s a sort of audio Necker cube, sounding minor sometimes or major other times. I’ve spent a little time listening to thirds of different sizes and 360 cents sounded mostly minor while 370 sounded mostly major, but not necessarily very consistently.

After writing the above I finally went back and re-read, for the first time in decades, Charles Ives’s essay ‘Some “Quarter-Tone” Impressions’. He doesn’t say much about melody but has this to say about triads:

Chords of four or more notes, as I hear it, seem to be a more natural basis than triads. A triad [using quarter-tones], it seems to me, leans toward the sound or sounds that the diatonic ear expects after hearing the notes which must form some diatonic interval [the fifth, C–G]. Thus the third note [a tone halfway between E and D sharp] enters as a kind of weak compromise to the sound expected — in other words, a chord out of tune. While if another note is added which will make a quarter-tone interval with either of the two notes [C–G] which make the diatonic interval, we have a balanced chord which, if listened to without prejudice, leans neither way, and which seems to establish an identity of its own.

That makes some sense to me; and maybe similar ideas apply to melody. If we simply replace E, A, and B with half-flat versions in the C major scale, they function melodically as “a kind of weak compromise to the sound expected” — a melody out of tune.

Ives goes on to speak favorably of the tetrad C–D#s–G–A#s (I’m using f for half flat, s for half sharp, since the proper symbols aren’t so readily available), which I would rather spell as C–Ef–G–Bf. The C–Bf interval (about 1050 cents) corresponds to frequency ratio 11/9 x 3/2 = 11/6, so the tetrad is 18:22:27:33.

### 12–note scales and keyboard mapping

The 2, 3, 11 lattice doesn’t seem to yield any useful 12-note periodicity blocks. Not that one necessarily needs a 12-note scale, but if you’re mapping the 7-note scale to the white notes on a conventional keyboard, you need something to map the black notes to.

It seems the simplest thing is to note that of the seven white notes, four are a set of consecutive fifths (F, C, G, D) while the other three are another (Af, Ef, Bf). So one might as well get the other five notes by extending one or both of these series. And each note in the F-C-G-D series (except maybe the last) probably ought to get a neutral third over it, in which case you’d need to split the twelve notes evenly between the two series. That’d be nice… but it doesn’t quite work if you want to assign C, D, Ef, F, G, Af, Bf to the white notes, for convenience in playing.

The next fifths in the F-C-G-D series are B♭ (down from F) and A (up from D), but there is only one key between Af and Bf so we can’t use both. Nor can we start from F and go up to A and then E, because there is no key between Ef and F. So if we use six notes in this series they have to be E♭, B♭, F, C, G, D. These are all assigned to their conventional keys.

Now the neutral thirds above E♭ and B♭ are Gf and Df, which we assign to the G♭ and D♭ keys. Now what? The next neutral third downward is Cf and the next upward is Fs, neither of which goes with the one remaining black key (conventionally A♭). A♭ is the next downward fifth in the first series, though, so that’s the best choice for that key. We end up mapping:

Conventional mapping | C | D♭ | D | E♭ | E | F | G♭ | G | A♭ | A | B♭ | B |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Neutral mapping | C | Df | D | E♭ | Ef | F | Gf | G | A♭ | Af | B♭ | Bf |

Not perfectly elegant but probably the best solution if you want to keep the whole C neutral scale on the white keys. Of course if you eliminate that restriction you can choose any twelve notes you like.

### 2, 3, 11 Details

It’s rather interesting that both 3/2 at 702.0 cents and 11/8 at 551.3 cents are close to multiples of 50 cents; that means every interval close to the center of the (2, 3, 11) lattice is a near multiple of 50 cents and so is well approximated with a 24-equal division of the octave.

Cents values for just lattice (powers of 3 across, 11 up/down):

-4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | |

4 | 597.5 | 99.4 | 801.4 | 303.3 | 1005.3 | 507.2 | 9.2 | 711.1 | 213.1 |

3 | 46.1 | 748.1 | 250.0 | 952.0 | 454.0 | 1155.9 | 657.9 | 159.8 | 861.8 |

2 | 694.8 | 196.8 | 898.7 | 400.7 | 1102.6 | 604.6 | 106.5 | 808.5 | 310.5 |

1 | 143.5 | 845.5 | 347.4 | 1049.4 | 551.3 | 53.3 | 755.2 | 257.2 | 959.1 |

0 | 792.2 | 294.1 | 996.1 | 498.0 | 0.0 | 702.0 | 203.9 | 905.9 | 407.8 |

-1 | 240.9 | 942.8 | 444.8 | 1146.7 | 648.7 | 150.6 | 852.6 | 354.5 | 1056.5 |

-2 | 889.5 | 391.5 | 1093.5 | 595.4 | 97.4 | 799.3 | 301.3 | 1003.2 | 505.2 |

-3 | 338.2 | 1040.2 | 542.1 | 44.1 | 746.0 | 248.0 | 950.0 | 451.9 | 1153.9 |

-4 | 986.9 | 488.9 | 1190.8 | 692.8 | 194.7 | 896.7 | 398.6 | 1100.6 | 602.5 |

Deviations of lattice from 24-equal values:

-4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | |

4 | -2.5 | -0.6 | 1.4 | 3.3 | 5.3 | 7.2 | 9.2 | 11.1 | 13.1 |

3 | -3.9 | -1.9 | 0.0 | 2.0 | 4.0 | 5.9 | 7.9 | 9.8 | 11.8 |

2 | -5.2 | -3.2 | -1.3 | 0.7 | 2.6 | 4.6 | 6.5 | 8.5 | 10.5 |

1 | -6.5 | -4.5 | -2.6 | -0.6 | 1.3 | 3.3 | 5.2 | 7.2 | 9.1 |

0 | -7.8 | -5.9 | -3.9 | -2.0 | 0.0 | 2.0 | 3.9 | 5.9 | 7.8 |

-1 | -9.1 | -7.2 | -5.2 | -3.3 | -1.3 | 0.6 | 2.6 | 4.5 | 6.5 |

-2 | -10.5 | -8.5 | -6.5 | -4.6 | -2.6 | -0.7 | 1.3 | 3.2 | 5.2 |

-3 | -11.8 | -9.8 | -7.9 | -5.9 | -4.0 | -2.0 | 0.0 | 1.9 | 3.9 |

-4 | -13.1 | -11.1 | -9.2 | -7.2 | -5.3 | -3.3 | -1.4 | 0.6 | 2.5 |