I wasn't planning to put this in. Beware, for if you're not a theoretically-trained musician, you probably won't understand it!
One could even take this argument as far as the musicians of the Renaissance took it: that the second simplest interval deserves second place, third third place, etc. If this is done, one of the possible results is what is called functional tonality. Obviously, the interval of the perfect (major) fifth directs the entire piece through harmonic motion, while the second most similar interval of the major second directs it through melodic motion (the minor second, a far less simple interval, is also used frequently in melodies, but, like the interval of major second, it is self-evident and unnecessary to be proven mathematically that it is good to use this interval in melodies). The intervals of minor and major third, the third and fourth most simple intervals, play the lesser role of determining what notes are used along with the fifth motion through the consonant intervals of harmonies. As inverting all these intervals gives all the possible diatonic and chromatic intervals, except the incredibly distant and very infrequent tritone, the system of functional tonality is an excellent way of acheiveng this element of musical perfection (the inversion of the minor second has no justificaiton whatsoever, but it is rarely used).
However, it is not necessarily the only way. Any form of music in which the intervals of perfect fifth, major second, minor third, and major third, are used in that order as elements of behavioral significance would be just as perfect in this manner. Why the thirds have to be used 'vertically' and the seconds 'horizontally' is beyond me.
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There is a hitch. Rousseau tested that theory, on a monochord probably.
5:6 is good
6:7 is terrible (I haven't heard, but Rousseau has)
on 7:8 I know not
BUT
4:7 makes a nicer 7th than 9:16 or 5:9
so 7:8 ought to be good
I assume you are talking about the last paragraph. I am not trying to give a mathematical justification for what sounds harmonious, as Pythagoras did, but I am trying to find out what music is (not sounds) good, regardless of our impressions of it. Perhaps we are meant to follow both our ears and our mathematics, but as Christians, we ought to know that what is true and good is not necessarily pleasant. Why should music be different?
I do believe that the very pleasantness of harmonious music has value, but that shall be discussed in "The Book, part..." well, I don't know what part.
"Hard food for Midas, I will have none of thee, nor none of thee, pale and common drudge {silver}...but thou, thou meagre lead, which rather thraten'st than dost promise aught, thy plainness moves me more than eloquence!"
---William Shakespeare "The Merchant of Venice"
In the moral realm, the good of virtue or duty may not always coincide with the lesser good of pleasure. In any other realm, pleasure is good.
The beatitudes are not about saying tears are good in themselves, but about saving the pleasure for heaven rather than getting it on earth.
Music is made either to honour God or to please the ear. A music that displeases the ear draws attention from prayer to music, and is thus not good to honour God with. Same applies to some kinds of musical pleasure. Music that is made merely to please the ear is of course good if it achieves it purpose to please the ear.
"As Christians we know that pleasure is not always good" should be kept where it belongs, in the moral domain, not be misapplied to everything else.
"As Christians we know that we must sometimes believe what appears not to sight" should equally be reserved for the Eucharist. It is no excuse to be heliocentric, just because that arrangement, contrary to the senses, is a theoretical possibility.
The genera of good are:
- honourable
- pleasant
- useful to procure the honurable and the pleasant
Beauty=what pleases immediately when perceived.
It differs thereby from the "sexually attractive", which pleases when seen as a potential medium of sexual pleasure, though both may coincide in the same woman.
In saying that "As Christians we ought to know what is true and good is not necessarily pleasant" we must remember the distinction between the pleasure itself and the thing with which it is associated, something you obviously understand. Because of this, we know that something could be pleasureable in its effects on us, but deficient in its essence. Likewise, something could be not pleasureable in its effects on us, but be completely non-deficient in its essence. Part of my case is that I have not yet proved it to be impossible for music to always be deficient when it is displeasurable, or non-deficient when it is pleasurable, but that all music that is not deficient must be organized around certain intervals, albeit not necessarily in a harmonious way. After all, one's pleasure in things, including nonharmonious musics, can change with time. I used to never like nonharmonious music, now I do. Therefore, for determinining the deficiency or non-deficiency in the essence of a piece of music, something more than my perception of pleasure is required.
I think there is a slight discrepancy between our uderstandings of Aquinas's definition of beauty. Let us look at it in the context of teh article.
FOURTH ARTICLE [I, Q. 5, Art. 4]
Whether Goodness Has the Aspect of a Final Cause?
Objection 1: It seems that goodness has not the aspect of a final
cause, but rather of the other causes. For, as Dionysius says (Div.
Nom. iv), "Goodness is praised as beauty." But beauty has the aspect
of a formal cause. Therefore goodness has the aspect of a formal
cause.
Obj. 2: Further, goodness is self-diffusive; for Dionysius says
(Div. Nom. iv) that goodness is that whereby all things subsist, and
are. But to be self-giving implies the aspect of an efficient cause.
Therefore goodness has the aspect of an efficient cause.
Obj. 3: Further, Augustine says (De Doctr. Christ. i, 31) that
"we exist because God is good." But we owe our existence to God as the
efficient cause. Therefore goodness implies the aspect of an efficient
cause.
_On the contrary,_ The Philosopher says (Phys. ii) that "that is to be
considered as the end and the good of other things, for the sake of
which something is." Therefore goodness has the aspect of a final
cause.
_I answer that,_ Since goodness is that which all things desire, and
since this has the aspect of an end, it is clear that goodness implies
the aspect of an end. Nevertheless, the idea of goodness presupposes
the idea of an efficient cause, and also of a formal cause. For we see
that what is first in causing, is last in the thing caused. Fire, e.g.
heats first of all before it reproduces the form of fire; though the
heat in the fire follows from its substantial form. Now in causing,
goodness and the end come first, both of which move the agent to act;
secondly, the action of the agent moving to the form; thirdly, comes
the form. Hence in that which is caused the converse ought to take
place, so that there should be first, the form whereby it is a being;
secondly, we consider in it its effective power, whereby it is perfect
in being, for a thing is perfect when it can reproduce its like, as
the Philosopher says (Meteor. iv); thirdly, there follows the
formality of goodness which is the basic principle of its perfection.
Reply Obj. 1: Beauty and goodness in a thing are identical
fundamentally; for they are based upon the same thing, namely, the
form; and consequently goodness is praised as beauty. But they differ
logically, for goodness properly relates to the appetite (goodness
being what all things desire); and therefore it has the aspect of an
end (the appetite being a kind of movement towards a thing). On the
other hand, beauty relates to the cognitive faculty; for beautiful
things are those which please when seen. Hence beauty consists in due
proportion; for the senses delight in things duly proportioned, as in
what is after their own kind--because even sense is a sort of reason,
just as is every cognitive faculty. Now since knowledge is by
assimilation, and similarity relates to form, beauty properly belongs
to the nature of a formal cause.
Reply Obj. 2: Goodness is described as self-diffusive in the
sense that an end is said to move.
Reply Obj. 3: He who has a will is said to be good, so far as
he has a good will; because it is by our will that we employ whatever
powers we may have. Hence a man is said to be good, not by his good
understanding; but by his good will. Now the will relates to the end
as to its proper object. Thus the saying, "we exist because God is
good" has reference to the final cause.
Aquinas says that beauty and goodness are based upon the same thing, the form (or essence), and their difference is that goodness consists in how the thing is desirable as an end (and, by extension, as the means to the end). He then proceeds to say that "beauty relates to the cognitive faculty; for beautiful
things are those which please when seen." He cannot mean sensible pleasure, for the senses and feelings associated with sensible pleasure, though they are important human faculties, do not percieve in any way the essence of a thing, but merely appearances and emotional associations that go along with them. The perception of essences is a matter of intellect. Upon seeing in this way that something is perfect or approaches perfection in some way, we ought to have the more spiritual pleasure proper to such a perception. This sort of pleasure is what I am concerned with at this time, not yet are my blog posts concerned with the other, sensible pleasure.
to Aquinas the eyes ARE a cognitive faculty, just as the ears also
taste and touch considerably less so
which is answer enough to your objection
"the senses and feelings associated with sensible pleasure, though they are important human faculties, do not percieve in any way the essence of a thing, but merely appearances and emotional associations that go along with them."
intellect does not mean primarily reflection to Aquinas (that being a default in human intellect)
though intellectual intuitions based unreflectingly on eyes and ears are partially falsified and in need of the default, nevertheless, they are the principles that reflection is built upon
learning to like unharmonious music is not a perfection in triumphing reflection over senses, it is a threat to the sense- and intuition-based principles of every sane reflection
Regardless of the sanity of the senses' common sense, it is theoretically true that music can behave according to the proper ratios in the proper hierarchial position without sounding pleasant. I do not see why there must be a connection between completeness of essence (and thus beauty) and the pleasure of the senses when one can have the spiritual pleasure of intellectual perception of said perfection while the senses experience displeasure. if there is no such connection in music, then when one experiences sensible displeasure, yet can justify intellectually what is going on using the intellect, then the pleasure upon being seen (this time, seen with the mind, not the senses) still exists.
Also, why did the ancients and midevalists think that simple ratios were better than complex ones? Does not Aristotle say that there is not actually any goodness in mathematics?
That being said, I do not believe that most disharmonious music actually has this unharmonious but correct relationship in its behavior.
Regardless of the sanity of the senses' common sense, it is theoretically true that music can behave according to the proper ratios in the proper hierarchial position without sounding pleasant. I do not see why there must be a connection between completeness of essence (and thus beauty) and the pleasure of the senses when one can have the spiritual pleasure of intellectual perception of said perfection while the senses experience displeasure. if there is no such connection in music, then when one experiences sensible displeasure, yet can justify intellectually what is going on using the intellect, then the pleasure upon being seen (this time, seen with the mind, not the senses) still exists.
Also, why did the ancients and midevalists think that simple ratios were better than complex ones? Does not Aristotle say that there is not actually any goodness in mathematics?
That being said, I do not believe that most disharmonious music actually has this unharmonious but correct relationship in its behavior.
6:7 is a bad ratio, if the sound of it is bad
Aristotle saying there was no goodness in mathematics, was speaking about the science of such, not about the realities
Before this conversation turns into a shouting match, let me see if I can clarify our positions for both of us.
First: The Modernist position (horrors!)
Modernist musical aestheticians believe that there are no standards of beauty because they are relativists for various reasons. You and I are certainly not that.
Second: My position
I believe that the provable standard for music is that any music that makes the perfect fifth the most important element in its behavior, the major 2nd second most important (and minor second in this spot as well in horizontal motion only), and the 3rds 3rd most important (with minor more important than major) is ideal. The reason I believe this is that these intervals have the two notes be closely related. The 5th is most close, etc, and this relationship needs to be illustrated for the ideal to be achieved. The reason this is ideal is because these relationships are part of the nature of music. The more the music achieves its nature, the more perfect is the form applied to matter to make the piece of music. This makes the music more good. It also makes it more beautiful, for when the intellect sees this perfection, it ought to have pleasure in it. I do not specify how this behavioral integration of these intervals ought to be achieved, and acutally go so far as to say that I cannot prove that the way called "Harmonious" (where dissonant intertvals are only played simultaneously under carefully controlled conditions) is more valid than ways that would be called nonharmonious.
Third: Hans Lundhal's position:
Your position is mostly the same as mine (though your reasoning seems to be more based on the Classical ideal of simplicity). Our main difference is that you do believe that it can be proven that harmoniousness is provably part of musical beauty.
This is your proof:
1. Upon hearing a dissonant interval, it is common sense to have a reaction of displeasure.
2. If something causes displeasure, it cannot cause pleasure.
3. Since beauty is defined as what pleases upon being seen, the displeasure makes beauty absent.
This is why I do not think your proof is as compelling of a proof as is necessary.
I do not disagree with point #1. What I do disagree with is point #2, that something that causes displeasure cannot cause pleasure as well. One can probably find a way to organize the music around the correct intervals but have it remain dissonant. As these constitute part of the ideal, one ought to have intellectual pleasure (as distinct from sensible pleasure ) once this perfection is seen. It is somewhat like praying: it (usually) produces anything but sensible pleasure (at least this is my experience, you may be much holier than I), but we know it is a beautiful thing to do because upon intellectual examination, we see that it is so.
I do not deny that what you say MAY be true (and I actually hope it is), for the person who commits the either/or fallacy may still be right.
You may actually like dissonance better than you think, by the way. Have you ever listened to any of the music of American composer Aaron Copland? I'm playing one of his dissonant pieces right now, and I can see even non-jaded musicians liking it. It's called "Four Piano Blues." I have not examined its intervallic orgainization.
What is the classical argument that simplicity is better than those messy ratios anyway? I understand that what we call harmonious has the simple ratios, but why (besides the sound) did the ancients actually find it better?
I have not said that dissonances cannot be incidental in causing pleasure: like a dissonance resolved the next beat or measure into a consonance, or even into another dissonance needing another resolution. When I call it bad, I mean bad for an end (final chord or leap) or for a prolonged exposure to same.
My point with Four Piano Blues (which I really ought to post a detailed analysis of) is that the dissonance is not treated in this manner. Another piece you might want to try is "Matins" by David Von Kampen (I just met him and heard this piecelast night). If this probably improperly treated dissonance doesn't increase the pleasure, I don't know what will. I bet even my mother (who has musical tastes a lot like you, although she has no clue of the math behindit) would even like it.
Would you like to continue our conversation/argument, should I post something else, or should I do both?
In Four Piano Blues, I bet dissonances are mitigated in another way: i e by adding thirds (consonants) to them. Otherwise it would be very unlike the Blues music from which it takes its name.
When I speak about the badness of uniquely dissonant music, I refer not to Copland, but to Alban Berg, composer of Wozzek.
Of course you can both continue the discussion here and post something else, I usually have no problem with parallel discussions, the technicalities are right now such that I have no problem following both.
As for improperly treated or properly treated dissonance, I consider any treatment of them that subjects them to consonance and make the consonances more exciting a proper treatment, even if not like the classical rules.
Like salt and vinegar are subordinate to flat and sweet ingredients.
In books one and two of De Musica (treating so far on rythm, whether verse, drum beat or dance) by St Augustine of Hippo the Magister and Discipulus do not limit proper rythmics to the already then classical metres.
So tell me, what counts as a proper migitation and what does not? Do quartal and quintal harmonies fulfill this description? (In my system they do, so long as the fifths and fourths are more important behaviorally than the seconds.) What about triads with nonresolved nonchord tones added? Where do you draw your line? Do any absolutes actually guide you? And what is the precise meaning of 'Subjecting dissonance to consonance?' It is upon such subtleties as these that I spend small but sizable portions of my waking hours.
Rest assured, I have about as low of an opinion of Berg as you do.
One of my early compositions back in ninth grade began with the chord:
CGDA or DACG (one minor third, two major seconds/ninths/minor sevenths but THREE fifths).
It ended with CGEC (two octaves between the Cs, all voices a fifth or sixth apart) after some variety of G7.
My music teacher played it, I liked it with my ears. He liked it at sight, and I think he must have thought of Copland.
If there are four consonant and only two dissonant intervals, and those not the sharpest dissonance, I find dissonance is duly subordinate. And in that case I find the chord harmonic, even if not in Rameau or Riemann.
All this nearly quarrel for a matter of nearly just terminology.
However, if most people like Copland when hearing, that argues that he composed harmonic music and does not invalidate that good music is always likeable by the ears, though the reverse may not be true.
Even so: bad music likable by the ears, is artistically good and "only" say morally, intellectually or so on bad.
Very interesting. I agree with you that this sort of note combination fits your description I will have to show this to my composition professor. However, does it fit my perfect-fifth dominant criteria? I know it has more fifths than anything else, but if you invert them, many of them become seconds. There is also no way to determine the root of the 'chord.' Not that a chord needs a root, but I don't know of any way that you could tell whether they were fifths or seconds and what was dominant without a root. And since I don't know how to find a root without a chord being all consonances (or a 7th chord) I don't see how my criteria could work except in music based on harmonious chords. Although I daresay that there may be a way. Do you agree with me yet? Or didI suddenly turn too conservative even for you :)?
I understand your comment about Copland too. Harmonic in your sense, not necessarily "hierarcial" in my sense (i think I misspelled something :). And yes, I agree that even music that I think is unsatisfactory has the merit of the pleasure it gives...if it gives any. I just prefer metaphysics to pleasure. It's more...fun. It's more other things too, but you know that as well as I do, I'm sure.
I did NOT invert them to CDGA, rest assured! But even that one contains two fifths, two fourths, one sixth and only two seconds. And in that one I cannot find a root either.
Palestrina and Lasso did not have a concept of root as distinct from bass.
DFB was ok, because minor third and major sixth are ok.
BDF was not, since minor third and minor fifth (or even major third and minor fifth) are not.
See my post: Why is the tritone bad
If Palestrina and Lasso had no concept of root, why is it so very easy to find the root in their chords? Did they understand it a way similar to how the baroque composers understood figured bass? As far as I can tell, even though they didn't understand chord roots, the roots sound like they progress by fifths or fourths anyway. Thus, I am satisfied.
Anyway, the problem I seemed to have with your composition is perhaps better illusrated in this way:
Take these notes played as a chord: CDEGA. Is C the root? You can progress up by fifths and get all the notes. Is E the root? You can progress up by fourths and get all the notes. Since you are allowing dissonance to be part of a chord, you might as well let any note be the root, omit F, and you can progress by seconds to get all the notes. Or, since chords can be in inversion, let any note be the root, and proceed by some other interval, omitting one note. This ambiguity (and I don't think I'm cheating, as inversions and omitted notes are obvious permitted practices in harmonious music that can be indulged in without spoiling the identity of a chord) makes me think that you must, to paraphrase C.S. Lewis "Organize by the golden thirds or not at all." And if a note cluster has no identity, how can it fit into the perfect fifth organizational scheme?
waaaaaay to many questions for me, I am stricken with a cold
Palestrina and Lasso obviously understood things in the figured bass way, confirmative.
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