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

  1. #16
    Senior Member millionrainbows's Avatar
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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.
    This "cute" reply shows how people take things as "givens" and can't explain them when asked.
    "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

  2. #17
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    12 tones in just intonation (simplest and most consonant 5-limit interpretation) and equal temperaments, plus deviations in cents and scale pattern in equal temperaments:

    0: 1/1 0.000000 unison, perfect prime
    1: 16/15 111.731285 minor diatonic semitone
    2: 9/8 203.910002 major whole tone
    3: 6/5 315.641287 minor third
    4: 5/4 386.313714 major third
    5: 4/3 498.044999 perfect fourth
    6: 45/32 590.223716 diatonic tritone
    7: 3/2 701.955001 perfect fifth
    8: 8/5 813.686286 minor sixth
    9: 5/3 884.358713 major sixth
    10: 9/5 1017.596288 just minor seventh
    11: 15/8 1088.268715 classic major seventh
    12: 2/1 1200.000000 octave
    |
    12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 11.4284 c. M: 17.5963 c. Twelve-tone Chromatic (1/11-comma)
    19: 2 1 2 1 2 1 2 2 1 2 1 2 P M ME SD: 10.6924 c. M: 21.8027 c. Genus diatonico-chromaticum (if we use minor whole tone and augmented fourth, 19 comes way more accurate)
    22: 2 2 2 1 2 2 2 2 1 3 1 2 P D3 SD: 9.4792 c. M:-18.7673 c.
    29: 3 2 3 1 3 2 3 3 1 4 1 3 P SD: 11.4957 c. M:-16.8865 c.
    31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.3773 c. M: 11.1447 c. Genus diatonico-chromaticum
    34: 3 3 3 2 3 3 3 3 2 4 2 3 SP D3 SD: 5.0594 c. M:-9.7763 c.
    41: 4 3 4 2 4 3 4 4 2 5 2 4 SP SD: 4.7936 c. M:-6.7940 c.
    53: 5 4 5 3 5 4 5 5 3 6 3 5 SP SD: 1.1527 c. M: 1.5445 c. Genus diatonico-chromaticum
    118: 11 9 11 7 11 9 11 11 7 13 7 11 SP SD: 0.3350 c. M: 0.6471 c.

    Arabs have theories about 17, 22 and 24 tones musical systems. It is more interesting how did they came with them.
    Indians have a theory about 22-"shrutis".
    Babylonians are the oldest culture, developing 12 tone system, way before Greeks and Pythagoras. It was probably related to their calendar and numerical system.
    Anyway, it's easy to see that lower overtones of simple ratios will form 12 tone system, if we reduce them to octave. Stacking fifths is another way (3/2 = 5/4 x 6/5, so we will always have some approximations to just major and minor sixths and thirds in this scale). Another way: transposing 7-note diatonic scale and tempering any two of these commas: syntonic, major diesis (known as the difference between 4 minor thirds and octave) , minor diesis (difference between 3 major thirds and octave), diaschisma (difference between the two diatonic tritones; these diatonic tritones are different from augmented fourths and diminished fifths, and septimal tritones or 11th harmonic and its inverse, so there is a whole sea of intervals that sound roughly like "tritones"). (These commas are not musically irrelevant, if you want to notate what actually - let's say- a real violin player performs.)

    Another approach using statistical physics and lattice structures:
    https://advances.sciencemag.org/cont...-kISaq6bKNWxqo

  3. #18
    Senior Member millionrainbows's Avatar
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    Quote Originally Posted by BabyGiraffe View Post
    12 tones in just intonation (simplest and most consonant 5-limit interpretation) and equal temperaments, plus deviations in cents and scale pattern in equal temperaments:

    0: 1/1 0.000000 unison, perfect prime
    1: 16/15 111.731285 minor diatonic semitone
    2: 9/8 203.910002 major whole tone
    3: 6/5 315.641287 minor third
    4: 5/4 386.313714 major third
    5: 4/3 498.044999 perfect fourth
    6: 45/32 590.223716 diatonic tritone
    7: 3/2 701.955001 perfect fifth
    8: 8/5 813.686286 minor sixth
    9: 5/3 884.358713 major sixth
    10: 9/5 1017.596288 just minor seventh
    11: 15/8 1088.268715 classic major seventh
    12: 2/1 1200.000000 octave
    |
    12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 11.4284 c. M: 17.5963 c. Twelve-tone Chromatic (1/11-comma)
    19: 2 1 2 1 2 1 2 2 1 2 1 2 P M ME SD: 10.6924 c. M: 21.8027 c. Genus diatonico-chromaticum (if we use minor whole tone and augmented fourth, 19 comes way more accurate)
    22: 2 2 2 1 2 2 2 2 1 3 1 2 P D3 SD: 9.4792 c. M:-18.7673 c.
    29: 3 2 3 1 3 2 3 3 1 4 1 3 P SD: 11.4957 c. M:-16.8865 c.
    31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.3773 c. M: 11.1447 c. Genus diatonico-chromaticum
    34: 3 3 3 2 3 3 3 3 2 4 2 3 SP D3 SD: 5.0594 c. M:-9.7763 c.
    41: 4 3 4 2 4 3 4 4 2 5 2 4 SP SD: 4.7936 c. M:-6.7940 c.
    53: 5 4 5 3 5 4 5 5 3 6 3 5 SP SD: 1.1527 c. M: 1.5445 c. Genus diatonico-chromaticum
    118: 11 9 11 7 11 9 11 11 7 13 7 11 SP SD: 0.3350 c. M: 0.6471 c.

    Arabs have theories about 17, 22 and 24 tones musical systems. It is more interesting how did they came with them.
    Indians have a theory about 22-"shrutis".
    Babylonians are the oldest culture, developing 12 tone system, way before Greeks and Pythagoras. It was probably related to their calendar and numerical system.
    Anyway, it's easy to see that lower overtones of simple ratios will form 12 tone system, if we reduce them to octave. Stacking fifths is another way (3/2 = 5/4 x 6/5, so we will always have some approximations to just major and minor sixths and thirds in this scale). Another way: transposing 7-note diatonic scale and tempering any two of these commas: syntonic, major diesis (known as the difference between 4 minor thirds and octave) , minor diesis (difference between 3 major thirds and octave), diaschisma (difference between the two diatonic tritones; these diatonic tritones are different from augmented fourths and diminished fifths, and septimal tritones or 11th harmonic and its inverse, so there is a whole sea of intervals that sound roughly like "tritones"). (These commas are not musically irrelevant, if you want to notate what actually - let's say- a real violin player performs.)

    Another approach using statistical physics and lattice structures:
    https://advances.sciencemag.org/cont...-kISaq6bKNWxqo
    That's nice little exposition on five-limit systems. But what do you mean by "stacking fifths?" What can this method of generating an octave-division otherwise be described as? Has it anything to do with Pythagoras or Pythagorian-derived principles?

    Are scales and octave divisions "created" and derived from principles and/or procedures, or do scales "just exist?" Should we question why there are 12 notes? Should we try to find this out, and why? or Why not?

    Did scales "evolve historically," and are they inextricably interwoven with the practices of music throughout history, and so, "this is just the way it turned out?"

    Where do we draw the line between accepting things as "given," or looking at things in more detail, which seems to be frustrating for many academic thinkers?
    Last edited by millionrainbows; Jul-14-2019 at 12:58.
    "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

  4. #19
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    "What can this method of generating an octave-division otherwise be described as? "

    I'm pretty sure that we can find some academically correct definition in professional music theory journals or sites like Xenwiki (in the mathematical section - they use abstract algebra for their definitions, while some guys like D.Lewin or G.Mazzola would probably use category theoretical language).


    "Are scales and octave divisions "created" and derived from principles and/or procedures, or do scales "just exist?" -

    You are going into philosophical territory. For mathematicians or platonists such constructions probably "exist".

    "Did scales "evolve historically," and are they inextricably interwoven with the practices of music throughout history, and so, "this is just the way it turned out?"

    - I can see current popularity of 12 equal coming from mass commercial production of pianos in 19th century (I doubt piano manufacturers wanted customers to have out of tune intervals, making some keys unplayable), it is well known that meantone was the dominant musical tuning for several centuries in Western world.

    Today: anyone can buy isomorphic keyboard or guitar, designed to play in 19, 22, 24, 31 (or fretless).
    Anyway, 12 equal has more compositional resources than 12 notes meantone - we will need more than 12 keys to reach all enharmonics in meantone, leading to systems like 19 or 31 microtone, or in the case of some stuff by Liszt or Stravinsky use non-meantone 12 note tunings (there are various such options that will give more in-tune augmented, diminished or Messiaen scales, but regular diatonic will be worse than in meantone). Or septimal scales for blues and barbershop music.

    What we will never get with only 12 tones - arabic music - the closest thing to is 12 notes out of 17 equal and only one of the permutations of this scale is close enough to oriental intervals to sound "exotic".

    The minimum size of closed tonal system for oriental music with neutral thirds is 17 tones. 17 equal, 17 notes out of 22 (has neutral seconds, but not neutral thirds, so only some arabic/turkish scales work), 17 out of 24/26/27/29/31 equal are good enough.


    "Where do we draw the line between accepting things as "given," or looking at things in more detail?" -

    Understanding how sound works may give us new compositional resources, so we better not accept elementary theory as given.

    Timbres with distorted harmonic series like metal bars, based on frequency or phase modulation, or waveshaping synthesis techniques, or detuned partials (additive synthesis) can sound better in unusual music systems than in more harmonic equal temperaments like 12, 19 etc. (Still, very inharmonic timbres will never work well (or at all, depending on the sound) for complex chords and counterpoint and this is because of our cognitive system and hearing apparatus, not because of some ideological limitations).

    Gamelan and certain African musical tribal traditions are good example for original musical cultures, not based on harmonic series (Arabic music can be explained as using higher harmonics like 11th and 13th), but on the timbres of their gongs and similar instruments.
    Chopi people in Africa use the anti-diatonic scale where major and minor are interchanged and we have augmented triad, not diminished on "B" (the scale can be generated by a flat "fifth" - 675 cents) , so any work composed with diatonic scales has a dual equivalent there. Of course, it will sound terrible and out of tune on typical Western instruments with harmonic overtones; we have to use some instruments that work in this tuning - like Chopi marimbas, xylophones (I'm not too sure about what other instruments they use, wikipedia mentions musical bows and pipes/flutes).

  5. #20
    Senior Member millionrainbows's Avatar
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    Quote Originally Posted by BabyGiraffe View Post

    "Are scales and octave divisions "created" and derived from principles and/or procedures, or do scales "just exist?" -

    You are going into philosophical territory. For mathematicians or platonists such constructions probably "exist".
    Howard Hanson's book "Harmonic Materials of Modern Music" describes ways of creating scales by using "interval projection."

    How do scales arise? are they made, or do they form out of relationships of notes?
    "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

  6. #21
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    How do scales arise? -



    Compositional resources are abstractions and this has nothing to do with music theory, it's a question from the field of philosophy of music.


    are they made, or do they form out of relationships of notes? -

    It doesn't matter; they may exist in some kind in some non-physical space or they could be a construction, invented by human mind.
    You can call any pitch collection used as a scale a "scale" - even these without any structural relationships between the tones (as long as it is used in a musical piece). Wikipedia gives us this definition: " sequence of ordered musical notes". Or: "In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale. " - It is interesting that last sentence better describes Indian ragas or Medieval modes - these are something like proto-melodic patterns, not something like our current scales or pitch class sets, which are more abstract. ( I wonder who is the genius behind the ascending/descending minor - certainly it's not hard to find counterexamples, even pieces where Dorian or Harmonic minor are used along natural and melodic minor in all kinds of motion.
    The modern, vague concept of minor tonality (with "flexible" notes or "alterations", and different ascending and descending patterns) is almost as bad in describing actual music as the theory behind "blues" scale in rock music.)

  7. #22
    Senior Member Phil loves classical's Avatar
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    Quote Originally Posted by millionrainbows View Post
    Howard Hanson's book "Harmonic Materials of Modern Music" describes ways of creating scales by using "interval projection."

    How do scales arise? are they made, or do they form out of relationships of notes?
    The basis of musical foundations is all in the math. Scales were constructed from the symmetry of those relationships. (tetrachords). The human brain is like a wave analyser, it can detect consonance and dissonance. So it is possible for us to perceive a posteriori without actually understanding the math.

    Here is a good overview

    https://golem.ph.utexas.edu/category...tical_ori.html

    It's because of this, that tonality is natural, and even if we sent a race of humans to another planet that never heard a note of music, over time (a few thousand years or so) they would still develop the same scales The only thing that is arbitrary or may be different is another mode could be more popular than major/minor
    Last edited by Phil loves classical; Jul-14-2019 at 22:40.
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  8. #23
    Senior Member millionrainbows's Avatar
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    Quote Originally Posted by BabyGiraffe View Post
    How do scales arise? -



    Compositional resources are abstractions and this has nothing to do with music theory, it's a question from the field of philosophy of music.

    You mean all those scales you talk about, like the Pythagoran scale with its 9/8 seconds, are not based on relationships?

    And you're always talking about "just" intonation. Any "just" scale has to have a starting point, from which interval relationships proliferate.

    Take just one interval, the 3:2 fifth. When placed in an octave, we immediately get its reciprocal, the just fourth 3:4.
    Last edited by millionrainbows; Jul-15-2019 at 12:57.
    "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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