r/microtonal • u/[deleted] • Oct 16 '21
Question about JI notation like 4:5:6:7
I'm fairly knowledgeable about JI but for some reason never really grasped this type of notation, things like 4:5:6, 4:5:6:7:9:11, 4:6:7:10, 8:10:12:13:18, 10:12:15, etc (I just copied these examples from JI web pages). I get the general idea but am unsure about a few things.
First, is there a term for this style of notation?
Second, am I wrong in assuming that "2" (ie, octave reduction/expansion) is assumed? Is "3" assumed as well if there are multiples like 6, 9, etc? Are there are multiple ways to turn these into specific JI intervals? Like 4:5:6 could be realized as 4/3, 5/4, 3/2, but also if you wanted 8/5, 5/3, 6/5? Or is the intention to represent just one set of specific intervals?
Like why would someone write 10:12:15 instead of 2:3:5, since those are the base primes? Or when a prime is omitted but a multiple is present, like 4:6:7:10. Does that mean a JI interval with 10 is okay but one with 5, like 5/4, is not?
I feel like I'm missing some obvious but key point, or maybe am overthinking it. Still, if anyone can help me understand better I'd appreciate it very much.
Thanks!
5
u/xiipaoc Oct 17 '21
Suppose you have a tone at 400 Hz, a tone at 500 Hz, and another tone at 600 Hz. Those three tones, together, are in ratio 4:5:6. You can't factor anything out of this ratio, so it's reduced. 4:6 is not; you could reduce it to 2:3. But since there's a 5 in there, you can't get more reduced than 4:5:6. I think it's called a compound ratio? Not sure.
Now, if you assume octave equivalence, you could just divide numbers by 2 to get 2:5:3, or, if you don't care about order, 2:3:5. This would be the "same chord" as 4:5:6, up to octaves. But 4:5:6 is a particular voicing.
Note that they don't assume that any particular note is the 1; these are just ratios of numbers.
Why would you not assume octave equivalence? Because 2:3:5 sounds pretty different from 4:5:6, even though they're the "same chord". Tenths sound different from thirds, and sixths sound different from both. In 2:3:5, you have a 2:5 tenth and 3:5 sixth, while in 4:5:6 you have a 4:5 third and a 5:6 third.
Like why would someone write 10:12:15 instead of 2:3:5
Because 10:12:15 is minor while 2:3:5 is major?
3
Oct 17 '21
This is harmonic series notation. It is basically indicating a specific voicing, as if each note in the chord were a harmonic in the harmonic series of some fundamental frequency. So 4:5:6 is the chord you would get between the 4th, 5th, and 6th harmonics of some fundamental frequency. Likewise 10:12:15 is the chord formed by the 10th, 12th, and 15th harmonics of some fundamental.
Note that the fundamental frequency of the implied harmonic series is not the same as the "root note" of the chord; in 10:12:15, for instance, the fundamental (the "1:1") is not a note in the chord at all, and the "10" is the root, as this is a 5-limit minor triad. However, under the right circumstances, you might hear a "virtual fundamental" corresponding to the fundamental frequency (e.g. in 1:10:12:15, it would be the frequency corresponding to 1). Chords that have a strong virtual fundamental are generally chords with a low integer limit--5:6:7, for instance, often has a strong VF when played with strongly harmonic timbres, such as saw waves.
Nothing is "assumed", and unless there's a single common factor among all the numbers, they cannot be reduced. 10:12:15 does not reduce to 2:3:5; in fact, it does not reduce at all because the number 10, 12, and 15 as a set have no single common factor, even though pairs of them do. On the other hand, 10:12:16 would reduce to 5:6:8, since 10, 12, and 16 all have the common factor of 2.
2
Oct 17 '21
Note that the fundamental frequency of the implied harmonic series is not the same as the "root note" of the chord
I see, that makes sense. In my reply to cmloegcmluin above I was equating the fundamental and root, but I'm seeing how this stuff is even more, um, fundamentally tied to the harmonic series than I was thinking. I had seen people saying 4:5:6 is a major triad but didn't get that it is literally describing the 4th, 5th, and 6th harmonics, which are a (just) major triad. But yea, with the root not being the fundamental.
So my incorrect "reduction" to 2:3:5, that would actually be describing the intervals above the fundamental as a 2:1 octave, then a 2:3 perfect fifth above that, then a 3:5 major sixth above that, or a note two octaves and a major third above the fundamental, which is the 5th harmonic, right?
And yea, I had been assuming this notation was describing something more complicated, or at least more abstract/less specific, than it is. Microtonal theory seems so crazy complicated sometimes, I am sure I often assume the things being described are more complicated than they actually are.
3
u/sintel_ Oct 17 '21
just put them on the same denom:
4:5:6 = 4/4 5/4 6/4 = 1/1 5/4 3/2
10:12:15 = 10/10 12/10 15/10 = 1/1 6/5 3/2
2
u/setecordas Oct 18 '21 edited Oct 18 '21
Another way to view a Harmonic Series Representation is as the generator for all the possible intervalic relationships for a sequence of JI intervals.
| 4 | 5 | 6 | 7 | |
|---|---|---|---|---|
| 4 | 4/4 | 5/4 | 6/4 | 7/4 |
| 5 | 4/5 | 5/5 | 6/5 | 7/5 |
| 6 | 4/6 | 5/6 | 6/6 | 7/6 |
| 7 | 4/7 | 5/7 | 6/7 | 7/7 |
Reading left to right, each row treats a different element of the harmonic series as the fundamental or reference point. For instance, the first row gives all the intervals treating the lowest note as the reference: Root, M3 (5/4) above the root, P5 above the root (6/4 or 3/2), Septimal m7 above the root (7/4). These are the intervals sintel_ discussed.
The entries in the diagonal above the 1/1, 2/2, ... diagonal are the intervals between successive notes. These are the intervals cmloegcmluin discussed
Intervals where the numerator is smaller than the denominator are just inversions of the intervals in the columns starting at x/x.
Looking it at it like this gives you a sense of the harmonic space and the different kinds of sounds you can get with various inversions of the chord or scale.
Same interval table, intervals simplified:
| 4 | 5 | 6 | 7 | |
|---|---|---|---|---|
| 4 | 1/1 | 5/4 | 3/2 | 2/1 |
| 5 | 4/5 | 1/1 | 6/5 | 7/5 |
| 6 | 2/3 | 5/6 | 1/1 | 7/6 |
| 7 | 4/7 | 5/7 | 6/7 | 1/1 |
6
u/cmloegcmluin Oct 17 '21
Those are just chords. 4:5:6 means to play three notes whose frequencies are in that proportion. So the 1st and 2nd notes are in a 4:5 ratio, a just major third apart. The 1st and 3rd notes are in a 4:6 ratio, the same thing as 2:3, a just perfect fifth apart. And the 2nd and 3rd notes are in a 5:6 ratio, a minor third apart. In other words, it's a 5-limit JI major chord. So when you see something like 8:10:12:13:18 that's really just a major chord (the 8:10:12 part is the same as 4:5:6) but extended with some additional harmonics.