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Thread: Why are keys a fifth away most closely related?

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    Senior Member millionrainbows's Avatar
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    Default Why are keys a fifth away most closely related?

    You know, the circle of fifths. Why does it go in fifths, adding sharps? Furthermore, why are there 12 keys?
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    Senior Member tdc's Avatar
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    The circle of fifths goes that way I would assume, because that is the order sharps accumulate as occurring in the major scales. The interval of a fifth is a perfect consonance, the V chord in a given key contains the leading tone as its 3rd, which creates a kind of a magnetic attraction between the chords V and I.

    Our modern day system of tuning has at its roots the mathematics of Pythagoras, it has been altered slightly to allow for modulation to other keys while still sounding relatively in tune in all keys. I think it is interesting that these intervals of sound are based on naturally occurring mathematical acoustical concepts. It just so happens that the number of notes in a diatonic scale (7) and the number of notes in the chromatic scale (12), are also significant astrological numbers relating to lunar and solar cycles of time.

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    Senior Member EdwardBast's Avatar
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    These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:

    The obvious . . . :

    12 keys because 12 tones. 12 tones because of Pythagoras et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.

    . . . and the actual explanation:

    The whole history of western music and western music theory.
    Last edited by EdwardBast; Jul-12-2019 at 00:27.

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    Senior Member Larkenfield's Avatar
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    Because each fifth is the dominant fifth of the next key in the cycle. It's a cycle or circle that jazz musicians often practice because it takes them through the 12 keys or scales. The bridge to I've Got Rhythm is based on the circle of fifths as just one example, though in general the entire circle is usually not utilized. It's an often used chord progression worth learning in every key and it’s often been found in classical music.
    Last edited by Larkenfield; Jul-12-2019 at 13:59.
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    Senior Member JeffD's Avatar
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    I might be wrong, but I don't think you are looking for a music theory answer. There are tons of websites and videos that explain all that, anyway, and you would have found them if you were looking for them, I am sure.

    I think you are looking for the answer to a more fundamental question, like: Did these "standards" get picked arbitrarily, or is their something truly fundamental about them, something physical, and not arbitrary.

    The interval of the fifth is a pair of notes with a frequency ratio of 3:2. This can be shown on an oscilloscope, a taught string, etc. The ear drum responds this way as well. It is the first natural overtone, depending on how you count things.

    Emotionally all folks of all cultures can recognize the ratio of the fifth, and identify a fifth above a give note accurately. And the two together sound pleasing to most all people. Aside from the unison and the octave, the fifth is considered the most consonant of harmonies.

    Other harmonies are considered less consonant by most listeners, and different cultures start to rank the different intervals as to pleasantness etc., differently depending on what they are used to.

    Dividing the octave into 12 tones is somewhat arbitrary, since the piano was well tempered.

    From that basic stuff we get all the keys. There is a video of Leonard Bernstein showing this on the piano, and it is fascinating. All the keys and their progression follow mathematically from these fundamentals.

    Bernstein explains it much better than I can. But the interval of the fifth leads directly to all the keys and all the 12 division of the scale: https://www.youtube.com/watch?v=Gt2zubHcER4


    Watch the whole thing, its only five minutes but it is really great. 3 minutes in is the most relevant to your question.
    Last edited by JeffD; Jul-12-2019 at 05:59.
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    Quote Originally Posted by millionrainbows View Post
    Furthermore, why are there 12 keys?
    Are there really only 12 keys?

    19 equal is even better, traditional harmony would sound slightly more consonant (would be worse for power chords and extended jazz chords, based on stacking fifths/fourth, but this is true for any meantone tuning). The best small, non-meantone equal temperaments in 5-limit are 34 and 53 equal. 53 is basically 5-limit just intonation (the only notable comma, tempered in it is the schisma).
    If you want to modulate the diatonic scale in non-meantone tuning or just intonation, you get one additional chroma that is not mentioned in simple school books, so when modulating a fifth up - let's say from C major to G major - you have to notate two accidentals, because of the 81/80 difference appearing:

    0: 1/1 0.000000 unison, perfect prime
    1: 9/8 203.910002 major whole tone
    2: 5/4 386.313714 major third
    3: 4/3 498.044999 perfect fourth
    4: 3/2 701.955001 perfect fifth
    5: 5/3 884.358713 major sixth, BP sixth
    6: 15/8 1088.268715 classic major seventh
    7: 2/1 1200.000000 octave

    x 3/2 =

    0: 1/1 0.000000 unison, perfect prime
    1: 9/8 203.910002 major whole tone
    2: 5/4 386.313714 major third
    3: 45/32 590.223716 diatonic tritone
    4: 3/2 701.955001 perfect fifth
    5: 27/16 905.865003 Pythagorean major sixth
    6: 15/8 1088.268715 classic major seventh
    7: 2/1 1200.000000 octave


    27/16:5/3 = 81/80 - 21.506290 syntonic comma, Didymus comma, it's tempered in all meantone systems, so all theoretical Western music, but not in traditional Indian or Arabic, or Chinese music.

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    Quote Originally Posted by EdwardBast View Post
    These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:

    The obvious . . . :

    12 keys because 12 tones. 12 tones because of Pythagoras et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.

    . . . and the actual explanation:

    The whole history of western music and western music theory.
    But WHY are there ducks!?

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    Keys are "close" because they have a lot in common. Take C and G: all the notes are the same except one: F becomes F#. In the same way C and a-minor are close (parallel is the term) since in the ascending natural minor mode they have exactly the same notes - just different starting and ending points. C and G-flat have little in common (just two notes) and are distant.

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    Senior Member millionrainbows's Avatar
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    Quote Originally Posted by tdc View Post
    The circle of fifths goes that way I would assume, because that is the order sharps accumulate as occurring in the major scales. The interval of a fifth is a perfect consonance, the V chord in a given key contains the leading tone as its 3rd, which creates a kind of a magnetic attraction between the chords V and I.

    Our modern day system of tuning has at its roots the mathematics of Pythagoras, it has been altered slightly to allow for modulation to other keys while still sounding relatively in tune in all keys. I think it is interesting that these intervals of sound are based on naturally occurring mathematical acoustical concepts. It just so happens that the number of notes in a diatonic scale (7) and the number of notes in the chromatic scale (12), are also significant astrological numbers relating to lunar and solar cycles of time.
    Quote Originally Posted by EdwardBast View Post
    These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:

    The obvious . . . :

    12 keys because 12 tones. 12 tones because of Pythagoras et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.

    . . . and the actual explanation:

    The whole history of western music and western music theory.
    Edwardbast, you don't seem to understsnd that I ask these questions so that I can see for myself the answers that are given. This is the internet, and it's cut-throat.

    I've had the Pythagoran connection questioned before:

    (Pythagoras) didn't do anything like closing the circle at 12, there is no good reason to believe he did. It would mean nothing to a Greek music theorist to talk about such things. If you want to believe legends are literally true then how about Heracles? Or Beowulf?
    Last edited by millionrainbows; Jul-13-2019 at 21:28.
    "The way out is through the door. Why is it that no one will use this method?"
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    "In Spring! In the creation of art it must be as it is in Spring!" -Arnold Schoenberg

    "We only become what we are by the radical and deep-seated refusal of that which others have made us." -Jean-Paul Sartre

    "I don't mind dying, as long as I can still breathe." ---Me

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    Senior Member EdwardBast's Avatar
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    Quote Originally Posted by millionrainbows View Post
    Edwardbast, you don't seem to understsnd that I ask these questions so that I can see for myself the answers that are given. This is the internet, and it's cut-throat.

    I've had the Pythagoran connection questioned before:
    Pythagoras et alia.

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    Quote Originally Posted by ECraigR View Post
    But WHY are there ducks!?
    Where do the lousy ducks go in winter?

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    Quote Originally Posted by ECraigR View Post
    But WHY are there ducks!?
    Ask Groucho.



    (Best I could find. 36 minute mark.)

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    Quote Originally Posted by EdwardBast View Post
    Pythagoras et alia.
    Yeah, I know what "et al." means; I do crossword puzzles, too.

    Explain yourself, then. Where did "12 notes" come from, if not Pythagoran principles? You already said it: Pythagoras. it's too late to go back on that. Exactly what "hair" are you splitting?
    Last edited by millionrainbows; Jul-14-2019 at 00:23.
    "The way out is through the door. Why is it that no one will use this method?"
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    "In Spring! In the creation of art it must be as it is in Spring!" -Arnold Schoenberg

    "We only become what we are by the radical and deep-seated refusal of that which others have made us." -Jean-Paul Sartre

    "I don't mind dying, as long as I can still breathe." ---Me

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    Senior Member millionrainbows's Avatar
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    Before you can use "notes in common" and "closely related keys," you have to explain where the notes came from. Scales are "generated," they don't just appear. Academic theorists rarely think about how scales are "made."

    So where did the 12-note division of the octave come from? What procedure, idea, or concept?
    Last edited by millionrainbows; Jul-14-2019 at 00:28.
    "The way out is through the door. Why is it that no one will use this method?"
    -Confucious

    "In Spring! In the creation of art it must be as it is in Spring!" -Arnold Schoenberg

    "We only become what we are by the radical and deep-seated refusal of that which others have made us." -Jean-Paul Sartre

    "I don't mind dying, as long as I can still breathe." ---Me

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    Senior Member millionrainbows's Avatar
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    Before you can use "notes in common" and "closely related keys," you have to explain where the notes came from. Scales are "generated," they don't just appear. Academic theorists rarely think about how scales are "made."

    So where did the 12-note division of the octave come from? What procedure, idea, or concept?

    Also, this shows how people take things as "givens" and can't explain them when asked.
    Last edited by millionrainbows; Jul-14-2019 at 00:32.
    "The way out is through the door. Why is it that no one will use this method?"
    -Confucious

    "In Spring! In the creation of art it must be as it is in Spring!" -Arnold Schoenberg

    "We only become what we are by the radical and deep-seated refusal of that which others have made us." -Jean-Paul Sartre

    "I don't mind dying, as long as I can still breathe." ---Me

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