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Thread: Need help understanding the neapolitan chord

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    Senior Member EdwardBast's Avatar
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    Quote Originally Posted by millionrainbows View Post
    Analogies don't work for me. What functions in what keys are you specifically referring to?

    Beethoven's use of it in the Apassionata sonata seems to be to be the most clear use.

    The deviation from the diatonic scale (ii-V-I) to a chromatic alteration (bii-V-I) is why it is 'special case.'
    The Gb major chord in the first theme of the Appassionata doesn't have a Neapolitan function. It's just a lurch to Gb major confirmed by its by own dominant and vii°7/V (enharmonically spelled). It's a different animal.

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    Quote Originally Posted by millionrainbows View Post
    I'm glad you know what you're talking about. Is there something about diminished chords in this? Please expound at length.
    Diminished chords of 12 equal require 648/625 to be tempered, so augmented fourth and diminished fifths are equivalents, but these aug/dim intervals are not the same as these found in diatonic scale, because they are also altered by syntonic comma. We get diminished temperament. Only 12 equal is ideal for 5-limit diminished temperament with any reasonable amount of notes per octave.
    648/625 is 128/125 (minor diesis) x 81/80 (syntonic comma).

    We can actually generate all the enharmonic equivalences by using just two basic intervals -> major third and perfect fifth and heavily subdividing and multiplying the pythagorean and minor diesis comma that we get when we multiply them enough times -> 4 times for major third/minor sixth, 12 - for the fifth/fourth.

    If you want to consider 12 equal as "septimal" tuning, you need another one-> 131072/117649 = 187.044561 cents.


    Diminished chords of meantone are three minor thirds and one augmented second, this means that 126/125 is tempered (81/80 = 225/224 x 126/125).

    In non-meantone/non-126/125 tempered, we get by stacking minor thirds - 1262.565148 cents = major diesis +1 octave. This major diesis is equated to a step in 19 equal, so 25/24:648/625 gives "kleisma" comma.


    We can easily extend 12 equal in 5-limit to meantone (19 equal, 31 etc) or diaschismic (34, 46 etc), or schismic (41, 53 etc). All these three are temperaments even if the last one sounds like pythagorean tuning and we need hundreds of notes for it to diverge from pythagorean just intonation in a meaningful way.

    Here are some "tritones" (augmented fourths)
    1: 25/18 568.717426 classic augmented fourth
    2: 1024/729 588.269995 Pythagorean diminished fifth
    3: 45/32 590.223716 diatonic tritone

    Differences found between them.
    32805/32768 1.954 cents schisma
    2048/2025 19.553 cents diaschisma
    81/80 21.506 cents syntonic comma, Didymus comma

    All these three temperament sound better (for harmony) than regular 12 equal and you can translate 12 equal music that is written using the right accidentals, if you want.

    We can extend 12 equal to higher-limits like 7-limit or 11-limit etc.
    For example:
    81/80 = 225/224 x 126/125, so septimal meantone = 31 equal is good -> augmented fourth is becoming equivalent to 7/5 tritone, diminished fourth becomes septimal "major" third of 9/7.
    128/125 = 64/63 x 126/125 = 27 equal, so septimal augmented, it's good scale, but for 7-limit, 12 equal is better in 5-limit
    2048/2025 = 64/63:225/224 = 22 equal (basically 34-12 = 22; 34 is good for 5-limit, this one for 7)


    The optimal (in terms of consonances) generator for 5-limit is actually not a perfect fifth/fourth.Perfect fifths/fourths means that we are lifting pythagorean (3-limit) to 5-limit. (So, ignoring octaves (number 2), we use irrational numbers that are between numbers that can be written in the form A^3 and Z^3 x Y^5. In terms of purity meantone<diaschismic<schismic<whatever comes next after we divide the octave in more than 600 parts.)

    It is actually a minor third (major third is lifting 2.5 subgroup to 5-limit, so numbers that are in the form A^5 vs numbers that are Z^3 x Y^5). Kleismic is good for minor thirds (so we use irrationals that are between different numbers in the form of A^3 x Y^5) and this is form of microtemperament.

    Or 3 x 5 = 15, so 15/8 (16/15), but this will gives "cluster" scales, if we want any sort of purity.
    Here is one such scale, generated by diatonic semitones/major sevenths:

    https://www.youtube.com/watch?v=DpKXQR2PtNg

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    Quote Originally Posted by EdwardBast View Post
    The Gb major chord in the first theme of the Appassionata doesn't have a Neapolitan function. It's just a lurch to Gb major confirmed by its by own dominant and vii°7/V (enharmonically spelled). It's a different animal.
    You're thinking about something else. The WIK example I cited (with sound) is F minor, not Gb major.

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    Senior Member EdwardBast's Avatar
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    Quote Originally Posted by millionrainbows View Post
    Key of C:

    EdwardBast has already confirmed that a German +6 chord is equal to a tritone substitute:
    No, I observed that jazz theory generally doesn't distinguish between the two but that classical theory regards them as quite different because the active intervals resolve differently. This thread isn't about jazz theory.
    Last edited by EdwardBast; Aug-11-2020 at 20:07.

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    Quote Originally Posted by EdwardBast View Post
    No, I observed that jazz theory generally doesn't distinguish between the two but that classical theory regards them as quite different because the active intervals resolve differently. This thread isn't about jazz theory.
    I'm very disappointed in that answer, Edward. You should be setting an example of the 'new paradigm' of music theory which is inclusive.
    Instead, you're being divisive.

    What's wrong, you don't like jazz music? Why not? Is it for the same reasons as Adorno?
    Last edited by millionrainbows; Aug-12-2020 at 12:53.

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    I'm often not sure where to draw the line between classical and jazz piano music. Sometimes when I'm listening to Brahms, I'll forget what it is and think for a few seconds that it's Bill Evans. And I know Bill Evans' music through and through, used to listen to him play live in Boston, have done many transcriptions, etc. Is it possible to develop a music theory system that does justice to both, and allows future musicians to more clearly understand the connections between them, on paper as easily as through the airwaves? Who knows. Maybe certain information just has to go straight from the pianist's inner ear to the listener's inner ear in order to be appreciated. Music theory is on some level, after all, an art form about an art form, and maybe there will always be something lost in the translation. Or maybe the problem just hasn't been solved yet.

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    Quote Originally Posted by Wes Lachot View Post
    I'm often not sure where to draw the line between classical and jazz piano music. Sometimes when I'm listening to Brahms, I'll forget what it is and think for a few seconds that it's Bill Evans. And I know Bill Evans' music through and through, used to listen to him play live in Boston, have done many transcriptions, etc. Is it possible to develop a music theory system that does justice to both, and allows future musicians to more clearly understand the connections between them, on paper as easily as through the airwaves? Who knows. Maybe certain information just has to go straight from the pianist's inner ear to the listener's inner ear in order to be appreciated. Music theory is on some level, after all, an art form about an art form, and maybe there will always be something lost in the translation. Or maybe the problem just hasn't been solved yet.
    In saying this, you represent the "new paradigm" of music theory of the future, which is inclusive of jazz ideas, and is not academically rigid. Congratulations.

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    Senior Member Torkelburger's Avatar
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    MR, do you know the reason why the tritone dominant can substitute for the dominant in jazz?

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    I think that this argument could evolve into more of a discussion if certain stipulations were laid down as to the overall system one is talking about. Reading over the thread, I don't see where the table was properly set for what could actually be a pretty substantial back and forth, because I can see two sides to this coin fairly clearly, having been raised somewhat equally on classical and jazz harmony.

    First I'd like to stipulate enharmonic equivalency for the purposes of this discussion, and 12-tone ET system (my rules for this post).

    Next I'd like to stipulate that we are in the present tense--you can walk over to the piano and play the Moonlight or Brahms or Evans right now, and we aren't looking at history, we're listen to sounds, right now, at the piano. Okay, thanks.

    There is a valid system behind what MR is saying, and it is the system I teach and developed over several decades. I believe it was first discovered by Erno Lendvai, who proported to find what he called the Axis system buried within Bartok and Kodaly's music. I'm not sure why it never took off--possibly because Lendvai developed a sort of "golden ratio fetish", and most of his book is (IMHO) wasted on some sacred geometry stuff that is, while interesting, more the stuff of stoned hippies listening to Debussy and seeing rainbows oozing from the speakers.

    Getting back to the cool part of Lendvai's theory: Like Reimann, he assigns T, D, and S designations to the various diatonic chord degrees, but they are not all the same as Riemann's, and he assigns a T, S, or D designation to all 12 chromatic tones of the scale. He does away with mediant, submediant, supertonic, leading tone designations. Scale degrees I, bIII, bV, and VI are Tonic. Degrees IV, bVI, VII, and II are Subdominant. Degrees V, bVII, bII, and III are Dominant.

    So to my point about setting the table, I think it's fair to say that within Lendvai's system MR has a point. I don't think he's made the point, but I think he has one, and is just not illuminating it very well. I have been guilty of the exact same thing many times--thinking that what I have written illustrates my point perfectly, only to read it over the next morning and go "what?".

    So within the Axis system it is not only reasonable but absolutely correct to state that the N6 chord, being a bII, does indeed have a Dominant function, because that is merely stating the obvious--it's true by definition within the system. And it makes sense to jazz musicians who recognize the 4 ways that Dominant chords tend to resolve as being more equal. (Those 4 root motions being down P5 (authentic), down m2 (tritone sub), up M2 ("back door"), and down M3 (as illustrated by Schumann's Kinderscenen #1, bar 12, The Beatles "I Want To Hold You Hand", etc.).

    So you're thinking "but it doesn't resolve to I, it resolves to V", and that is sometimes true, but N6 also quite often resolves to I6/4, in which case it makes perfect sense as a D to T progression (that then usually goes V : I, the more usual D to T progression.) What about the case where N6 resolves to V? We see it as a D to D tonality shift, but not a Functional shift.

    There is a lot more to say about this system, but I will save it for another time.

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    Senior Member EdwardBast's Avatar
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    Quote Originally Posted by Wes Lachot View Post
    So you're thinking "but it doesn't resolve to I, it resolves to V", and that is sometimes true, but N6 also quite often resolves to I6/4, in which case it makes perfect sense as a D to T progression (that then usually goes V : I, the more usual D to T progression.) What about the case where N6 resolves to V? We see it as a D to D tonality shift, but not a Functional shift.
    Wrong on two counts. The I6/4 chord in these situations is not (or only rarely) tonic in function. It is generally a dominant with ongoing unresolved non-harmonic tones. You're confusing taxonomy with function. Where N6 resolves to V it is subdominant. The clue that it's a functional shift is the word resolve.

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    Edward,

    The I6/4 chord may be said to "function" as a Tonic in many textbooks--we all know that, presumably. What I am stating is that in the Axis system we don't jump through hoops to explain why a I chord "functions" like a V chord. Rather, the sounds are judged at face value. A I chord functions as a I regardless of inversion. What if I said a I chord "functions" as a III because it's in 1st inversion? This is the sort of thing that the Axis system seeks to clarify. I'm not saying that it should be your or anybody's cup of tea, just explaining the system for those who may not be familiar with it.

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    I would expect Edward's answer to be something along the lines of "But we're not discussing axis theory."
    Last edited by millionrainbows; Aug-15-2020 at 10:55.

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    Quote Originally Posted by millionrainbows View Post
    I would expect Edward's answer to be something along the lines of "But we're not discussing axis theory."
    My reason for bringing the axis system into this particular conversation was merely to try to rescue the thread from what seemed to me to be a less than meaningful discussion. I thought injecting some fresh ideas into the conversation would be a good thing, and as MR noted above, it is a "new paradigm". This does happen to be a theory which unifies classical and jazz functional harmonic thinking, for those who are looking for such a theory going forward. It doesn't negate other theories--they are after all only theories--but it is a very helpful theory in and of itself.

    I am interested in hearing what Edward has to say, since he seems to have offered some very thoughtful perspectives in a number of threads. He may not have been aware that there is a system out there that does attempt to unify classical and jazz theory, when he made his statement above that they were two different things (paraphrasing). Or maybe he was aware--only he can tell us that.

    For those interested in a further discussion of Neapolitan 6 chords, I started a new thread titled "Beethoven's Razumovski 3 Aug6 to N6 Magic Trick". It's cool how he used these chords as a sort of Swiss army knife to make his way through modulations to and from distant key areas.
    Last edited by Wes Lachot; Aug-16-2020 at 09:15. Reason: tone, clarity, and content

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    Quote Originally Posted by Wes Lachot View Post
    My reason for bringing the axis system into this particular conversation was merely to try to rescue the thread from what seemed to me to be a less than meaningful discussion. I thought injecting some fresh ideas into the conversation would be a good thing, and as MR noted above, it is a "new paradigm". This does happen to be a theory which unifies classical and jazz functional harmonic thinking, for those who are looking for such a theory going forward. It doesn't negate other theories--they are after all only theories--but it is a very helpful theory in and of itself.
    I always go by what my ear tells me, and I now realize this makes me a "harmonic" thinker.

    Others here seem content to stick strictly to axiomatic CP theory, and prefer not to even question this. A Neapolitan chord has a root on bII, end of story...no "why" or how it may have been derived.

    Woodduck was complaining about my idea of how such a chord could be "generated" or created from a diminished context. When basic terms and ideas of mine are questioned in such an inflexible way, I am not inspired to go into depth.

    I am interested in hearing what Edward has to say, since he seems to have offered some very thoughtful perspectives in a number of threads. He may not have been aware that there is a system out there that does attempt to unify classical and jazz theory, when he made his statement above that they were two different things (paraphrasing). Or maybe he was aware--only he can tell us that.
    Good luck with that.

    For those interested in a further discussion of Neapolitan 6 chords, I started a new thread titled "Beethoven's Razumovski 3 Aug6 to N6 Magic Trick". It's cool how he used these chords as a sort of Swiss army knife to make his way through modulations to and from distant key areas.
    Yes, this "Swiss army knife" has been in use for a while, even sooner than some would have us believe.

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    Senior Member Woodduck's Avatar
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    Quote Originally Posted by millionrainbows View Post
    I always go by what my ear tells me, and I now realize this makes me a "harmonic" thinker.

    Others here seem content to stick strictly to axiomatic CP theory, and prefer not to even question this. A Neapolitan chord has a root on bII, end of story...no "why" or how it may have been derived.

    Woodduck was complaining about my idea of how such a chord could be "generated" or created from a diminished context. When basic terms and ideas of mine are questioned in such an inflexible way, I am not inspired to go into depth.



    Good luck with that.



    Yes, this "Swiss army knife" has been in use for a while, even sooner than some would have us believe.
    I think we are all (or at least those of us who haven't given up and gone off to watch a soap opera or something) waiting to see the process by which the Neapolitan is "derived" or "generated" from a diminished chord. What does this generation or derivation look like in practice? Were there intermediate forms used during the transition from diminished chord to Neapolitan? I'm confused, in part, because it's clear to me that a diminished chord containing the flatted second scale degree can't have a function similar to that of a Neapolitan.

    I understand harmonic relationships by ear, not by theory, of which I am not a thorough student. Can you help my ear in understanding how a Neapolitan can derive from a diminished chord?
    Last edited by Woodduck; Aug-18-2020 at 22:16.

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