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Thread: The explanation of black key pentatonics?

  1. #1
    Registered User ragasaraswati's Avatar
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    The explanation of black key pentatonics?

    What is the mathematical reason that the black keys make a pentatonic scale and specifically a major pent. a tritone away from the white keys C major scale, and a minor pent. a tritone away from the A minor scale? Thanks.

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    Er, I'm not sure I understand your question. The black keys make a pentatonic scale because there are 5 of them. The fact that they make the major pentatonic scale is because it has to be called something. The minor pentatonic scale is simply a mode of the major pentatonic in the same way that the minor scale is a mode of the major. They're only a tritone away from their 7 note counterparts if you choose the starting note that is 6 notes away.

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    Registered User ragasaraswati's Avatar
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    You didn't understand my question. Of course the remaining 5 notes would make a scale with 5 notes. The beautiful thing is that they speciffically make the "natural" anhemitonic, atritonic pentatonic scale. As for the other symmetry, they are actually a tritone away by design, it's not a matter of choosing intervals. Look at the piano layout for yourself - C major and Gb major pent; A minor and Eb minor pent. Cheers.

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    Well if you take a chromatic scale (H-H-H-H-H-H-H-H-H-H-H-H) and subtract out the major scale (W-W-H-W-W-W-H) on which the white keys are based, what you're left with is W-m3-W-W-m3, which can be looked at as a major or minor pentatonic scale or mode thereof (depending on what you choose as the root), if you want to...I don't think there's any more "why" to it than that, it's just how the intervals happen to work out.
    Last edited by walternewton; 05-26-2015 at 02:46 AM.

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    Registered User ragasaraswati's Avatar
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    Nothing in music is a coincidence. It has to be a deeper reason for the scale matrixes of complementarity to be the way they are.

    For example I just realised why a pentatonic scale is so widespread and so safe, without tense/wrong notes. It comes from acoustical rules, mathematics IOW. The reason is that it's the 5-limit scale with the maximal number of notes that it's also anhemitonic and atritonic. So it embodies the balance of having enough notes for melodic material but also minimal dissonance. You can do a little thought experiment; the moment you add another note you must create a semitone and in all but two cases a tritone.

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    Registered User ragasaraswati's Avatar
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    So, what I'm searching is the first principle(s) of this symmetry that says the moment you form a diatonic scale from the chromatic scale the remainder forms a pentatonic scale. And also their distance is maximal, a tritone, which is the square root of 2.. it goes on and on.
    Last edited by ragasaraswati; 05-26-2015 at 03:23 AM.

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    ragasaraswati,

    It's a natural consequence of set theory. The major scale (Forte 7-35) is the most maximally even way of selecting 7 notes from 12. Its 5 note complement (Forte 5-35) is, necessarily, the most maximally even way of selecting 5 notes from 12. The same applies to:-
    8 note scales (the octatonic) and its complement (the diminished 7th) and
    9 note scales (Forte 9-12) and its complement (the augmented triad).
    The 6 note whole tone scale is self complementary.
    If you spread x things over y as evenly as possible then, y-x things spread over y will also be as even as possible.

    I still think that the, apparent, tritone gap is a simple consequence of the way we choose to list the intervals in the major scale and the pentatonic scale. We have to start somewhere (on what is inherently a continuum) for the purposes of teaching music. From a scalar point of view there is nothing special about C-C and Gb-Gb.
    Last edited by nuffink; 05-26-2015 at 11:30 AM.

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    Registered User ragasaraswati's Avatar
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    Thank you very much. Can you provide some links as well for those "Forte" things?

    The tritone interval means they are maximally separated though. But I disagree the major (C-C) and the major pent. (Gb-Gb) scales have nothing special. Really? For starters, the major scale is the only 7-note scale with no minor or tritone intervals. The major pent. is the only mode of the anhemitonic-atritonic pentatonic scale that has a major triad on its tonic. I'm sure there are many other symmetries like that.
    Last edited by ragasaraswati; 05-26-2015 at 11:58 AM.

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    Sorry ragasaraswati, I should have made myself clearer.

    First the Forte stuff... http://www.mta.ca/faculty/arts-lette...ction/toc.html

    I'm not saying that there is nothing special about the Major scale. What I'm saying is that from a scalar point of view there's nothing special about C-C as opposed to say D-D or F-F etc. We learn the intervals TTSTTTS from convention. As the scale is a continuum we could have easily settled upon TSTTTST.

    As for your statement that "the major scale is the only 7-note scale with no minor or tritone intervals" I'm afraid that's just wrong. The Major scale contains 2 minor seconds, 4 minor thirds and 1 tritone.

  10. #10
    Registered User ragasaraswati's Avatar
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    Nice link.

    Quote Originally Posted by nuffink View Post
    I'm not saying that there is nothing special about the Major scale. What I'm saying is that from a scalar point of view there's nothing special about C-C as opposed to say D-D or F-F etc. We learn the intervals TTSTTTS from convention. As the scale is a continuum we could have easily settled upon TSTTTST.
    And there are styles that settle to dorian, Santana comes to mind. But the greater tonic force in the case of diatonic chord progressions comes from the major/ionian scale. It's convention in small part but in large part it's tonal gravity, especially in harmonic movement. Next comes natural minor/aeolian. You are wrong thus that there is nothing special about modes. My point becomes crystal clear if you suggest for example that we might as well settle upon STTSTTT (locrian), the weakest diatonic mode that lingers on the verge of unmusicality. So the symmetrical properties of the diatonic scale is not the whole story, the choice of the tonic/mode is equally important.


    Quote Originally Posted by nuffink View Post
    As for your statement that "the major scale is the only 7-note scale with no minor or tritone intervals" I'm afraid that's just wrong. The Major scale contains 2 minor seconds, 4 minor thirds and 1 tritone.

    I was not clear, I was reffering to intervals made with the tonic. In that case there are no minor or tritone intervals formed between the major/ionian scale's tonic and remaining notes.
    Last edited by ragasaraswati; 05-26-2015 at 12:39 PM.

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    ragasaraswati,

    I did emphasise (using italics) that I was talking from a scalar, not a modal, point of view. I accept your point about the ionian.

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    Registered User ragasaraswati's Avatar
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    I understand. Though in practice there is no democracy, emphasized by the very few votes that leave locrian outside the parliament each time.

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    I believe that these things are learned. You hear the Locrian as unmusical because it's rarely used rather than because it's inherently unmusical. I don't believe there is a league table of modal consonance. You state that the Aeolian is the second strongest mode, well not to me it isn't.
    Ask a theorist as to why the Ionian became the principle starting point of western tonal music and they'll confidently talk about the V-I cadence. Ask them about why it works and they'll, somewhat less confidently, talk about the leading tone movement. Ask them why the dozens of other possible leading tone progressions aren't considered cadential and they'll start to panic. Ask them why the blues V7-IV7-I7 works as a cadence and they'll wander off muttering.

    These are learned, culturalised things. They have almost nothing to do with mathematics except in the broad abstract way that set theory deals with them. A classic example is people's insistence on talking about the harmonic series as the basis of western harmony. It was once, we ditched it when we moved to 12tet. It's no good talking about consonance when our very notes are not tempered. In psycho-acoustics "a miss is as good as mile" really does apply. If we could somehow play a 12tet minor third to a medieval listener they wouldn't recognise it as remotely consonant.

    I say this as someone who's deeply into music theory - much of it is rubbish.

  14. #14
    Registered User ragasaraswati's Avatar
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    Quote Originally Posted by nuffink View Post
    I believe that these things are learned. You hear the Locrian as unmusical because it's rarely used rather than because it's inherently unmusical.
    They are learned by natural acoustical means first, then by culture. The Locrian case is upside down, it's rarely used because it's unmusical in comparison to a plethora of other scale options in the first place. Lack of P5, dark and with suicidal tendencies. What percentage of music is made with it? 1 every 1000 songs, every 10000?

    Quote Originally Posted by nuffink View Post
    I don't believe there is a league table of modal consonance. You state that the Aeolian is the second strongest mode, well not to me it isn't.
    There is, from bright to dark it goes from Lydian to Locrian. Lydian has no avoid notes while Locrian's tonic is the avoid note of itself. Listen to Terminal Frost - Pink Floyd which Lydian tonality acts like a glue for all notes to coexist in consonance. The same mood with Locrian is impossible. More like Terminal Death.

    In a diatonic progression the tonal gravity is strongest for Ionian followed by Aeolian, ceteris paribus (without rhythmic/accent/duration considerations etc). The strongest cadence of Ionian is with penult V7 to I while the strongest cadence of Aeolian is softer with penult V(7) to i. The tonic of Ionian is a resolution note while Aeolian's tonic isn't. That's why Aeolian cadences are softer and not as decisive as Ionian's. Even in the absence of the strongest cadences for each one their "weight" is nonetheless carried by music. Of course all this is known at least intuitively by a listener or a composer. You can drift in parallel from Ionian to Aeolian and back it's far easier to end an Aeolian piece on the relative Ionian/major, the opposite, to end a Ionian/major on the relative Aeolian/minor is not all that convincing, it's the equivalent of a sad end, in an otherwise upbeat movie, instead of a happy end. This ties also with the ubiquitness of the Picardy third. The opposite of that must be extremely rare. Don't forget how many classical age minor scales we have (3) and how many major (1). Again, to say it's tradition and nothing else misses the point because for something to become cliche it has to be good in the first place.
    What is the second strongest diatonic mode for you in the context of an elaborate (no vamps) diatonic chord progression?

    Quote Originally Posted by nuffink View Post
    Ask a theorist as to why the Ionian became the principle starting point of western tonal music and they'll confidently talk about the V-I cadence. Ask them about why it works and they'll, somewhat less confidently, talk about the leading tone movement. Ask them why the dozens of other possible leading tone progressions aren't considered cadential and they'll start to panic. Ask them why the blues V7-IV7-I7 works as a cadence and they'll wander off muttering.
    It came out from musica ficta. Musicians started altering the (modern names) Lydian and Mixolydian modes to be tonally enhanced, which turned them into the Ionian mode which was smoother and tonally more decisive than the other major modes. As a sort of musical evolution it survived as the fittest of the three in the context of more and more adventurous harmony. The V-I cadence is used less than 50% in pop music written in Ionian and still the tonic is clear. This is because Ionian's leading tones imply that cadence all the time

    The blues progression is easy to explain if you see it from the correct angle. The traditional modes most close to blues are Mixolydian and Dorian. Nothing to do with Ionian (that maj7 is not bluesy) or Aeolian (m6 is not bluesy). The I7 is the Mixolydian I and the IV7 is the Dorian IV. The V7 is just the classical influence and merge of blues with classical cadences. In each case the seventh overtone is implied of each chord's b7.

    Quote Originally Posted by nuffink View Post
    These are learned, culturalised things. They have almost nothing to do with mathematics except in the broad abstract way that set theory deals with them. A classic example is people's insistence on talking about the harmonic series as the basis of western harmony. It was once, we ditched it when we moved to 12tet. It's no good talking about consonance when our very notes are not tempered. In psycho-acoustics "a miss is as good as mile" really does apply. If we could somehow play a 12tet minor third to a medieval listener they wouldn't recognise it as remotely consonant.

    I say this as someone who's deeply into music theory - much of it is rubbish.
    They have everything to do with mathematics. Octave equivalence, P5, triads etc. Also, the harmonics series is still the basis of almost all global harmony, not the least of 12tet. The notes are not pure but they are close enough. The people critisizing ET are blowing it out of proportion. I wonder why detuned supersaws or guitars with a chorus effect exist. They must be very psycho-acoustically harsh.
    Actually the ET minor third is closer to the JI one than is the case with the major third. The average medieval listener would not perceive a mistuned minor third, though he might distinguish a slight beating of the major third.

    I'm deeply into music as well as math. I don't know what you've been reading (maybe old theory?) but I've found music theory most of the time codifying aspects of music I've found intriguing since before I knew how to count. Ratios and symmetries, evolution and compromise, beauty and grotesque. Music is a language as Victor Wooten says. I'm proud to be living in a society that many of its rules it has discovered. I'd rather take some rubbish here and there if I can find some gold coins in it.
    Last edited by ragasaraswati; 05-26-2015 at 05:02 PM.

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    Lots and lots of fascinating subjective stuff there ragasaraswati. Very little actual maths. In fact, none.

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