Ordering Scale Tones By Resolution

Discussion in 'Playing and Technique' started by ♫♪♫, Jun 29, 2008.

  1. ♫♪♫

    ♫♪♫ Member

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    Ok, from most to least stable (resolved?) chord tone/scale degree, order the following:


    For Major scales:

    Root
    Third
    Fifth
    Seventh
    Ninth
    Eleventh (or #11 I guess?)
    Thirteenth



    For Minor scales:

    Root
    Third
    Fifth
    Seventh
    Ninth
    Eleventh
    Thirteenth





    For major scales, from most to least stable scale degree, I would say (from just playing around with the scale)...

    1) Root
    2) Fifth
    3) Third
    4) Seventh
    ?) Ninth
    ?) Eleventh
    ?) Thirteenth

    So, the tonic is obviously the most resolved note. The fifth is next, and yeah, third is probably next, and then I am not too sure after that.
     
  2. kimock

    kimock Member

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  3. JamonGrande

    JamonGrande Supporting Member

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    Steve,
    Interesting link. Couple of quick thoughts on it:

    Based upon his applet, there is a slight change to the graph past the perfect 5th in equal temperament. Thus reciprocal intervals do not produce the same effect (which I can definitely accept, m2-M7 for example).

    Extending this logic, we should however see a mirror image of the function if we were to compare intervals from C1 down, no? Further more, I would expect a gradual "flattening" of the response the further one got away from the base frequency. Thus there is no octave equivalency based upon his function.

    With that in mind, I'm troubled that he would extend his findings into a harmonic system that is not octave (or voicing) specific. It almost seems to challenge the validity of a beat/frequency based premise.

    I prefer the first part of his findings to the later, but I'll look over it some more to clarify my understanding of it.

    joe
     
  4. dewey decibel

    dewey decibel Supporting Member

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    I'm not sure what you're trying to establish here. Resolution doesn't have so much to do with the notes is a chord, as it has to do with where you're coming from. For instance, you have the root as being the "most resolved" note, but if the note before it was a 2nd it's not as strong a resolution as say, a 4th to a 3rd.
     
  5. kimock

    kimock Member

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    That would be a critical bandwidth issue, right?
    Absent from the OP's "Scale tones by resolution" the effect of register, illustrated here within the octave.
    "Slight change from et" could be read from either side of the graph, not picking a winner by any stretch of the imagination, but this is just another example of how man and nature split re: the overtone series.

    Well, if you accept the idea that the interval has an affect at all, then its inverse should have the opposite affect. . .doesn't work like that exactly.
    As regards resolution, the overtonal inversion is going to be the more resolved interval, probably because it's individual pitches sum to the tonic.
    Like this. . http://users.rcn.com/dante.interport/winckel.html
    So yeah, there you go.


    I wish I had time for this. There is no octave equivalency, period.

    Nah, you're reading to much into that I think. The notion of "beats and frequency" enter the picture as measurable qualities of physical vibration; that's yardstick stuff.

    This is the same thing for me as your first observation; you can read it from either side of the graph.
    Does our musical system of tones negate the overtone series, or does the overtone series cancel the octave equivalency program?
    Neither, obviously!

    It's way too much stuff no matter how you look at it.
    I pretty sure that nobody can even answer the question "what is music" in the first place, so it's difficult to know what info is applicable.
    Furthermore, it seems like the "science" folk do very little outside the pure tone relationships (low primes), and the musicians generally ignore both the math and the perception issues and just have fun. Nobody is even looking at the fun factor yet!
    Subwoofers maybe?
    The best you can do is to agree to accept what seems physically true that also agrees with your own aesthetic, and keep an open mind about the stuff that falls on either side of the line, right?
    Wish I had more time for this, but I'm on the bus tomorrow, won't have a break for a month at least. I refuse to travel with a computer, too much distraction. . .:argue

    later gator,

    peace
     
  6. JonR

    JonR Member

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    This is all down to context. A single note is neither "stable" nor "unstable". (Although I guess you could argue it's "perfectly" stable, being its own tonal centre!)

    OK, you are talking about the context of a major scale. But the context in practice (of any single note) can still be melodic, harmonic or both. The root (tonic) is not a stable note if heard against the dom7 chord!

    IOW, you should be looking at the "stability" (consonance) of INTERVALS - pairs of notes.
    Start with intervals with the tonic, by all means. (The link kimock posted may be of assistance here.)

    The basic hypothesis (largely born out by experiment) is that consonance relates to frequency ratio: simpler ratios = more blending of overtones = more perceived consonance.
    If we base the order of simplicity of ratio on a figure achieved by multiplying the 2 figures (which seems reasonable), we get this list, taking the 13 possible intervals from unison to octave:

    1. Unison = 1:1
    2. Octave = 2:1
    3. Perfect 5th = 3:2
    4. Perfect 4th = 4:3
    5. Major 6th = 5:3
    6. Major 3rd = 5:4
    7. Minor 7th = 7:4
    8. Minor 3rd = 6:5
    9. Tritone = 7:5
    10. Minor 6th = 8:5
    11. Major 2nd = 9:8
    12. Major 7th = 15:8
    13. Minor 2nd = 16:15

    The complication here is that intervals in actual music are based on equal temperament - not the neat ratios shown here.
    Eg, the ET minor 7th is closer to ratios of 16:9 or 9:5 than it is to 7:4, even tho they are both theoretically more dissonant than 7:4. Likewise, there's a couple of other options for the tritone: 10:7 and 17:12.
    (17:12 is not a very sensible one, as it introduces a factor - 17 - we are not using for the other intervals, which all derive from 2, 3, 5 and 7. Even 7 is a controversial factor to introduce. The original system developed from Pythagoras's ideas only worked with factors of 2 and 3 - which meant a major 3rd had to be 81:64!)

    You should find this site interesting
    http://www.purveslab.net/seeforyourself/index.html?6.00
    It lays out those 13 intervals in order from unison to octave, and lets you test your own perception of their relative consonance. When you're done, it displays - for comparison - the median results from scientific literature.
    That order (the way most people tested sort the intervals) is as follows (I've kept the ratio-simplicity rank no from the above list, for comparison):
    1. Unison = 1:1
    2. Octave = 2:1
    3. Perfect 5th = 3:2
    4. Perfect 4th = 4:3
    6. Major 3rd = 5:4
    5. Major 6th = 5:3
    10. Minor 6th = 8:5
    8. Minor 3rd = 6:5
    9. Tritone = 7:5 (or 17:12 or 64:45? see below)
    7. Minor 7th = 7:4
    11. Major 2nd = 9:8
    12. Major 7th = 15:8
    13. Minor 2nd = 16:15

    No doubt the minor 6th's increased perceived consonance is down to hearing it as an inverted major 3rd. (Eg, a B-G minor 6th can be heard as an inverted G-B major 3rd.)

    Unfortunately, Purves doesn't say which (or how many) results he's drawn his data from.

    Also - crucially - the intervals he's using in his test seem to be pure ratios - as far as I can judge from my software (which will get to within 2 or 3 cents of actual value). So some of them will sound quIte different from intervals you ae used to hearing in actual music (never mind the synthesized tones used).
    Eg, his minor 7th (sample no.11) is close enough to a 7:4 ratio - the top note is a good 30 cents flat of an ET pitch.
    This might explain why I (at least) heard his minor 7th as more consonant than the median value. I did the rest twice, putting it at #4 first and #7 second (I spent more time listening closer 2nd time). But the median level (where most people think it should be) is #10. Maybe that result comes from judging the equal tempered interval?

    His major 3rd, too, is around 14/15 cents flat, matching the 5:4 ratio.

    With his tritone, the upper note is about 8 cents sharp of ET. This matches neither the 5:7 ratio (18 cents flat) nor the 10:7 (18 cents sharp). It's very close to 17:12 - but we should maybe not consider that one. It's also close to 64:45, which is calculated by taking a perfect 4th and adding a semitone. Ratio-wise, that means 4:3 x 16:15 = 64:45.

    If this subject (perception of consonance) interests you, there've been plenty of scientific studies, some of which are available to read online (at least in abridged form):

    http://www.humdrum.org/Music829B/notes.html
    http://whatismusic.info/developments/StatisticalStructure.html
    http://www.psychology.mcmaster.ca/ljt/trainor_tsang et_al_2002.pdf
    http://www.medicalnewstoday.com/articles/111295.php
    http://www.stage3music.com/lies/liesreviews.html
    http://www.sohl.com/mt/maptone.html
     
  7. JonR

    JonR Member

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    [Part 2...]

    When it comes to CHORDS, of course :rolleyes: , things get a bit more complicated... :eek: :worried :jo
    Even a simple triad contains 3 intervals: root-3rd, root-5th, 3rd-5th. (And that's just in root position; voice it as an inversion, or in an open voicing, or double up notes in other octaves, you get different intervals, eg a 4th between 5th-root, or a 6th between 3rd-root.

    Generally (in chord theory) we only study the impact of intervals (and extensions) relative to the root - hence 3rd, 5th, 7th, etc. But in counterpoint (the original theory of harmony), every interval matters.
    A 7th chord (4 notes) has 6 intervals. A 9th chord (5 notes) has 10!
    These all interact to produce chord "quality" or "character".

    Eg, a Cmaj7 chord contains a highly dissonant C-B maj7 interval. But that's softened a great deal by B's relationship to E and G. So we get quite a "sweet"-sounding chord.
    So you can't say (eg) an interval of a major 7th is "always" dissonant. Only on its own does it have a consistent sound you could say was "rough", "tense" or "unstable". It doesn't have that effect on a maj7 chord, because the stability of the triad outweighs it.


    However, you can - and maybe should, at least for a breath of refreshing simplicity! - reduce the whole thing of stability/nonstability within a key to a TONIC vs DOMINANT duality.

    TONIC = I, vi or iii chords
    DOMINANT = V or vii chords

    (IV and ii get classed as "subdominant", but as they are not totally stable - relative to I - you can file them under "dominant" in this two-way system.)

    So you then have (1) a set of "tonic" (stable) chords, against which the other scale notes - as extensions - will (mostly) act as contrast, upsetting the stability - and (2) a pair (at least) of "dominant" (unstable) chords, against which other scale notes may either increase the instability, or represent the stable chord the dominant chord is (by implication) moving towards.

    Eg,
    C-E-G = stable I triad. Notes B, A and D can work as "extensions", supporting the stability, but generally only if voiced high in the chord. And even when used in support, these notes make the chord more "vague", less firm: still "stable", but with some ambiguous colour.
    F is the only note that really works against the stability of I (and also against the iii Em and vi Am).

    G-B-D-F = unstable dom7 chord.
    (You need the F to create the dissonant B-F tritone. A G triad on its own is perfectly stable. Even in relation to a C chord, it may not act as a dominant. One might hear G as I and C as IV...)
    A and E can act as extensions, enriching the chord, enhancing its tension (G9 and G13).
    C is the only note that really works against a G7 chord - because, of course, it represents the tonic that G7 is opposed to.
    Even so, we can use C as a sus4 on G7, if the B is removed. The C then acts as a useful tension of a different kind - rather like combining a dominant and subdominant chord.
     
  8. JamonGrande

    JamonGrande Supporting Member

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    Ha! Yeah, I can dig what you are saying. About the whole thing. Interesting for sure, but nothing I'll lose sleep over.

    I've had similar issues with music cognition folks I've worked with/read/heard. So much of the research conducted so far takes a highly reductionist approach to the idea of music, and produces a lot of "exceptions" once you view it within "real music context". In these cases, musical examples come from western concert music, to the exclusion of almost every other type of music, including other western forms like blues, rock, country, etc...

    THere is a growing movement to include affect as a function of perception. Last year Judith Becker (from U. Michigan) gave an interesting talk about perception and musical affect in religious experiences, and there were a host of others who's names are escaping me. Each was trying to build an "emotional" function into their perceptual theories. For me, it always seemed tacked on. I have a hard time believing that any given musical or perceptual gesture will always lead to the same feeling. I subscribed to what is known as "enacted", "situated" and "distributed" forms of cognition, which place more emphasis on context than on some irreducible perceptual "truth". But I try not to be a skeptic about the other stuff.

    With that said and to return to the OP, depending on the context, every note can have a tension or resolution effect. We can simplify the problem space down to one or two tone clusters, but that seems to take some of the fun out of it, no?

    Have a great tour Steve, be healthy and play well!

    joe
     
  9. Ed DeGenaro

    Ed DeGenaro Supporting Member

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    You didn't study with Dick Grove did you? :) This certainly reminds me of his approach.
     

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