Paradoxical intervals
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As mentioned earlier, some intervals have a negative degree; they go “down” the scale to a higher pitch. For example, starting from the bluish 5th and going up a deep reddish comma (rryy-2 = 35¢) takes you down to the reddish 4th. The interval from a diminished 5th to an augmented 4th is a descending diminished 2nd that actually raises the pitch! The descending aspect is why the degree is written as minus two.
More strangeness: an octave sharpened by a deep reddish comma is a deep reddish augmented 7th, rryy7 = 100/49 = 1235¢. It's a 7th, not an octave, because it's the sum of two reddish 4ths. The sum of two bluish fifths is an octave flattened by a deep reddish comma, the deep bluish diminished 9th bbgg9 = 49/25 = 1165¢. Finally there's the deep bluish diminished 2nd, bbgg2 = bg5 - ry4 = 49/50 = minus 35¢. It's a diminished 2nd that's so far diminished, it's flatter than a unison. It's a descending negative 2nd that goes “up” the scale to a lower pitch.
The deep reddish comma is the nearest negative 2nd in 7-limit JI. Other negative 2nds include the sub, the yellow subcomma and the white comma. If you venture into double large ratios and triple colors, you'll find negative 3rds, negative 4ths, etc.For example, the sum of any two negative 2nds is a negative 3rd.
Negative intervals sound ascending but look descending on paper, as in the upper voice here:
Remember, the ratio and the cents are the reality. The quality and degree are the theory, and the theory is based on a somewhat arbitrary choice of steps per octave and keys per octave. As we'll see, in pentatonicism, the deep reddish comma is not negative.
Most negative 2nds are diminished 2nds, with a keyspan of zero, which means that at least they make sense on a standard keyboard. You go down the scale to a higher pitch on the same key. However there are also minor negative 2nds which go down the scale to a higher pitch on a lower key, and thus have a negative keyspan of -1. Intervals with a negative keyspan are called upside-down intervals. The nearest one in 7-limit JI that I know of is the greenish deep red minor negative 2nd, grrr-m2 = 1728/1715 = 64*27/35*49 = 13¢ = g1 + r1 - bb2 = class 11.
What's the musical significance of an upside-down interval? Consider the large deep greenish negative 2nd Lggrr-m2 = 19683/19600 = 39/16*25*49 = 7¢. It takes you from a yellowish minor 3rd 280/243 = 245¢ to a greenish major 2nd 81/70 = 253¢. Thus in A, ybC is slightly flatter than grB! Also, in the melody A – ybC – D, the A–C step is 7¢ narrower than the C–D step. Same with the melody gD – bF – gG.
Now, all these examples are quite contrived; one would have to modulate like crazy to use both yb3 and gr2 in the same song, the melodies seem unlikely, and a 7¢ difference in melodic step sizes is barely audible. Because of their small size and extreme remoteness, minor negative 2nds are not a practical problem in 7-limit JI. But as we'll see, 11-limit and 13-limit JI produce much less remote upside-down intervals.
Is there a way to avoid negative 2nds entirely? What if we defined 7/4 as an aug 6th? Then 7/5 would be an aug 4th, 10/7 a dim 5th, and the deep reddish comma would be a diminished 2nd. 15/14 would be a minor 2nd, just like 16/15, and the sub would be a unison. However, 12/7 would be a dim 7th, and the deep blue comma from 12/7 to 7/4 would be... a negative 2nd. It would be a double-diminished negative 2nd with a keyspan of 1!
Likewise, we could avoid the yellow subcomma being negative by defining 5/4 as a dim 4th. But not only would that be very counter-intuitive, it would make 81/80 a negative 2nd. Furthermore the white comma would still be negative. It would seem negative 2nds are unavoidable.
There are also diminished primes, which diminish the quality but raise the pitch. (Diminished primes which lower the pitch, such as gg1 = 24/25, are really descending augmented primes.) Diminished primes take you to the next lower key, thus they have a keyspan of -1, and are upside-down. For example, in the key of A, the yellowish-yellow 3rd C# is yybM3 = 175/144 = 338¢, but the greenish 3rd C is grm3 = 128/105 = 343¢. The minor is sharper than the major by the small green deep-greenish diminished prime sgggrrd1 = 6144/6125 = 2048*3/125*49 = 5¢. This makes C# slightly flatter than C, thus adding a sharp flattens the note. In the chord Ayyb,y5 = A – yybC# – yE, A–C# is slightly narrower than C#–E. Again, these examples are quite contrived (class 12 melody, class 7 chord); diminished primes are too small and remote to be a practical problem. sgggrrd1 = gggd2 - bbm2 = class 12 is the nearest 7-limit dim prime that I know of.
An octave sharpened by a diminished prime is a diminished octave that is sharper than 1200¢. An octave minus a diminished prime is an augmented octave flatter than 1200¢. The inverse of a diminished prime, e.g. LyyybbA1 = minus 5¢, is a descending diminished prime which augments the quality but lowers the pitch.
The sum of any two diminished primes is a doubly-diminished prime. Needless to say, these are extremely remote!
Every interval has a degree and a quality, which define its keyspan. Here's what conventional music theory has to say about these 3 concepts:
Table 12.1 – Degree, quality and keyspan
| 7 steps, 12 keys | double | dimin | perfect | augmt | double | |
| dimin | minor | major | augmt | |||
| unison | 0 | 1 | 2 | |||
| 2nd | 0 | 1 | 2 | 3 | 4 | |
| 3rd | 1 | 2 | 3 | 4 | 5 | 6 |
| 4th | 3 | 4 | 5 | 6 | 7 | |
| 5th | 5 | 6 | 7 | 8 | 9 | |
| 6th | 6 | 7 | 8 | 9 | 10 | 11 |
| 7th | 8 | 9 | 10 | 11 | 12 | 13 |
| octave | 10 | 11 | 12 | 13 | 14 | |
| 9th | etc. | |||||
Here's the same chart, expanded to include negative and upside-down intervals. Every interval now has an additional property, its sign, which is positive or negative. A negative interval has the same keyspan as the corresponding positive interval, but with the opposite sign. Thus if d3 = 2 semitones, -d3 = -2.
Table 12.2 – Degree, quality and keyspan
| 7 steps, 12 keys | double | dimin | perfect | augmt | double | |
| dimin | minor | major | augmt | |||
| neg. 4th | etc. | |||||
| neg. 3rd | -1 | -2 | -3 | -4 | -5 | -6 |
| neg. 2nd | 1 | 0 | -1 | -2 | -3 | -4 |
| unison | -2 | -1 | 0 | 1 | 2 | |
| 2nd | -1 | 0 | 1 | 2 | 3 | 4 |
| 3rd | 1 | 2 | 3 | 4 | 5 | 6 |
| 4th | 3 | 4 | 5 | 6 | 7 | |
| 5th | 5 | 6 | 7 | 8 | 9 | |
| 6th | 6 | 7 | 8 | 9 | 10 | 11 |
| 7th | 8 | 9 | 10 | 11 | 12 | 13 |
| octave | 10 | 11 | 12 | 13 | 14 | |
| 9th | etc. | |||||
Negative seconds are a mere annoyance; they make your note names run out of order, so that F# is sharper than G♭. The real headache is upside-down intervals. They make your keyboard run backwards! Unfortunately they seem to be inevitable no matter how you define your ratios, if you modulate far enough. Notice from the chart that not all negative intervals are upside-down and not all upside-down ones are negative. Negative refers to degree and upside-down refers to keyspan.
For prime limits lower than 7, the paradoxical intervals are more remote. The nearest negative interval in 5-limit JI is the yellow subcomma, Ly-d2 = 2¢ = 0 semitones = class 10. The nearest upside-down interval in 5-limit JI that I know of is the septuple-green comma, 78732/78125 = 4·39 / 57 = g7dd2 = 13¢ = -1 semitones = class 16. Note that it's not negative, because it takes you up the scale from C to D♭♭♭.
The nearest negative interval in 3-limit JI is the white comma, LLw-2 = 24¢ = 0 semitones = class 12. The nearest upside-down interval in 3-limit JI is the white subcomma, 353 / 284 = L8w-d76 = 3.6¢ = -1 semitone = class 53, a descending septuple-diminished sixth!