Alan Carruth wrote: ↑
Mon Jun 05, 2017 9:29 pm
"Since pretty much everyone agrees that temperaments are a compromise - which obviously means that there is some deeper tuning standard that they depart from - we never really get as far as discussing what the notes should ideally be."
The ideal that is being departed from is 'pure' intervals, with no beats.
Yes I think that's often the background assumption, and I'm sure some truth it it, but my point is that we never get down to details. It can't be entirely true because some intervals (a seventh, say) will beat however you tune them.
The reason I think it is interesting to get into the details is that the intervals that do need to be pure are, as I see it, the basic building blocks of music, with the others being derived from them. Just like the distinction between root position chords and inversions, I think this can help us understand the whole phenomenon of music.
For example, many people say that to make a minor chord you start with a minor third and put a major third on top, giving you the fifth. However if you take your minor third from the harmonic series (19/16), you find that when you go up a major third (5/4) you do not end up with a perfect fifth (3/2). You can fudge it and say the minor third must therefore be 6/5, but 6/5 is not from the harmonic series (denominator not a power of 2) and does beat, so I don’t think that is a legitimate move. What this means as I see it is that not all of these intervals can be constitutive of the chord, and that this is a fact of nature rather just a practical problem (unlike the comma issue you raise). I think that if we were to get into tuning and experiment properly we would find that the minor chord that sounds just right (even if we are not used to hearing it, which is telling) is made by going up a perfect fifth and then down a major third. That suggests that the fifth, not the third, is the basis of the triad and that the attention given to stacks of thirds in the theory of harmony is misplaced. This then gives us a principled basis for saying that added notes beyond the seventh are not chord tones (rather than the accident of history theory) and also suggests that Rameau was right to say that six chords are functional – the orthodox theory ignores these completely and reanalyses them as having a different root, largely because they don’t fit with the idea that chords are built by stacking thirds.
The problem is that 3 and 2 are incommensurable: there is no way you can multiply a number by three and divide it by two any number of times and get back to the original number. The difference between the frequency you started with and the one you ended up with is the comma, and the whole issue of temperament is figuring out what to do with the comma.
I would rather say that because people think that temperament is about the comma, they never get deep enough into it to cast any light on how music really works. The comma is just a practical problem - it arises when we try to put the sharps and flats on the same key. If you follow the series of fifths from Gb to A#, you never get back to the same note - you're not supposed to. The problem only arises if you get as far as F# and assume you are supposed to be back where you started. What this is telling us is that Gb and F# are not supposed to be exactly the same note, and you have to continue up the series to get the sharps. If you had a fixed pitch instrument capable of playing all the notes from Gb to A# there would be no comma issue, so the problem is just the practical one that such an instrument would have lots of keys / frets, and would be difficult to play. At the same time, because this forces us into fudging the intervals, it masks the question of what the notes are really supposed to be.
With that in mind I only partly agree when you say:
The compromise is not something that people chose to do out of shear cussedness; it's necessary. That's why you either get to listen to pure intervals, or modulate keys. You can't have both.
amezcua wrote: ↑
Tue Jun 06, 2017 12:03 am
Update for Rasputin . Ipso Facto used the word fantasy when answering my question about key colours .
OK, I found that post:
ipso facto wrote: ↑
Mon Oct 17, 2016 2:31 pm
amezcua wrote:In the process musicians discovered the various emotional reactions and effects produced by using "Well Temperaments". They started by trying to root out negative effects and ended with many positive effects.
A-ha - so that's where you are coming from. I think this is a fantasy, but if you believe in it then that explains your interest in alternative tuning systems.
That is not saying that key colours are a fantasy - the fantasy is that they have positive effects. As I said above, I don’t think it gets better than being in tune – the effect of tempering an interval is always negative.
But can you honestly say that any major scale on a guitar sounds the way a scale should sound? That`s a good basic start apart from key colours .
Yes, I do think a major scale in ET sounds very close to the way it should. A perfect scale IMO is just a series of perfect fifths put into a single octave (using perfect octaves for the conversion, obviously). The fifths in ET are only very slightly out.
That said, I suspect that for you this is equivalent to asking whether the tonic chord of that scale sounds the way it should. For me that is a different question, but I appreciate that most people believe that chords are built out from scale tones, which would make it the same. The tonic chord is acceptable to my ear but is much improved if the third is flattened (I can do this experiment on my keyboard by switching temperaments). This is why I said above that some violinists will play scales in one tuning (pythagorean or just fifths) but arpeggios in another (using just thirds for the chord thirds), and that I thought they were on the right track. The implication is that the thirds of chords do not really come from the scale, but are so close to the nearest scale tone that we can generally get away with using the same pitch for both – that again is a practical compromise and not a fundamental problem with the mathematics of music.