Can the harmonic series explain the origin of the major scale?The major scale - why and how?What are the actual notes of each natural harmonic?Is it possible to create the illusion of a sub-harmonic?Where is a natural harmonic for the note C on guitar?Why Is Just Intonation Impractical?How often does each interval appear in the harmonic series (relative to the fundamental)?How does the harmonic series affect consonance?How to understand a minor chord using the harmonic series?In a musical note (A for an example) are all the other frequencies harmonic?Why do we scale/shift up an octave in the overtone series?Why is the Major-Minor Scale unused?

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Can the harmonic series explain the origin of the major scale?


The major scale - why and how?What are the actual notes of each natural harmonic?Is it possible to create the illusion of a sub-harmonic?Where is a natural harmonic for the note C on guitar?Why Is Just Intonation Impractical?How often does each interval appear in the harmonic series (relative to the fundamental)?How does the harmonic series affect consonance?How to understand a minor chord using the harmonic series?In a musical note (A for an example) are all the other frequencies harmonic?Why do we scale/shift up an octave in the overtone series?Why is the Major-Minor Scale unused?













4















This Wikipedia article is the source for two overtone series charts...



enter image description here




The numbers above the harmonic indicate the number of cents' difference from equal temperament (rounded to the nearest cent).




enter image description here



  • Regarding harmonic 11, is it a 'out of tune' Gb or an F?

  • Am I reading the charts correctly that there is no A natural in the series?

If I understood this article about cents, -49 on harmonic 11 is about 1/4 step - enough to be a horribly out of tune P4 - and -12 on harmonic 15 may drift past the threshold of perceptible depending on the person and the exactly sound context (it may be close enough.) Roughly the same for harmonic 5 as 15.



So the resulting major scale (based on these very weak harmonics) will have FA horribly out of tune, and MI and TI questionable.



Why would anyone use the harmonic series to explain the origin of the major scale with such bad tuning?



Based on the strength of the overtones the major triad is supported by the series. But the next strongest thing is a diminished triad! Past that everything is weak and out of tune!










share|improve this question

















  • 1





    @topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question

    – Michael Curtis
    3 hours ago






  • 1





    Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground.

    – Michael Curtis
    3 hours ago






  • 1





    now it's only +18

    – Michael Curtis
    3 hours ago






  • 1





    THE PEOPLE HAVE SPOKEN. WE MUST RESPECT THEIR VOTE. FAILING TO DO SO WOULD BE A BETRAYAL OF DEMOCRACY. (sorry if this humor doesn't work outside the UK ;). Anyway +1 from me :)

    – topo morto
    2 hours ago






  • 1





    @ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen.

    – Albrecht Hügli
    57 mins ago















4















This Wikipedia article is the source for two overtone series charts...



enter image description here




The numbers above the harmonic indicate the number of cents' difference from equal temperament (rounded to the nearest cent).




enter image description here



  • Regarding harmonic 11, is it a 'out of tune' Gb or an F?

  • Am I reading the charts correctly that there is no A natural in the series?

If I understood this article about cents, -49 on harmonic 11 is about 1/4 step - enough to be a horribly out of tune P4 - and -12 on harmonic 15 may drift past the threshold of perceptible depending on the person and the exactly sound context (it may be close enough.) Roughly the same for harmonic 5 as 15.



So the resulting major scale (based on these very weak harmonics) will have FA horribly out of tune, and MI and TI questionable.



Why would anyone use the harmonic series to explain the origin of the major scale with such bad tuning?



Based on the strength of the overtones the major triad is supported by the series. But the next strongest thing is a diminished triad! Past that everything is weak and out of tune!










share|improve this question

















  • 1





    @topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question

    – Michael Curtis
    3 hours ago






  • 1





    Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground.

    – Michael Curtis
    3 hours ago






  • 1





    now it's only +18

    – Michael Curtis
    3 hours ago






  • 1





    THE PEOPLE HAVE SPOKEN. WE MUST RESPECT THEIR VOTE. FAILING TO DO SO WOULD BE A BETRAYAL OF DEMOCRACY. (sorry if this humor doesn't work outside the UK ;). Anyway +1 from me :)

    – topo morto
    2 hours ago






  • 1





    @ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen.

    – Albrecht Hügli
    57 mins ago













4












4








4








This Wikipedia article is the source for two overtone series charts...



enter image description here




The numbers above the harmonic indicate the number of cents' difference from equal temperament (rounded to the nearest cent).




enter image description here



  • Regarding harmonic 11, is it a 'out of tune' Gb or an F?

  • Am I reading the charts correctly that there is no A natural in the series?

If I understood this article about cents, -49 on harmonic 11 is about 1/4 step - enough to be a horribly out of tune P4 - and -12 on harmonic 15 may drift past the threshold of perceptible depending on the person and the exactly sound context (it may be close enough.) Roughly the same for harmonic 5 as 15.



So the resulting major scale (based on these very weak harmonics) will have FA horribly out of tune, and MI and TI questionable.



Why would anyone use the harmonic series to explain the origin of the major scale with such bad tuning?



Based on the strength of the overtones the major triad is supported by the series. But the next strongest thing is a diminished triad! Past that everything is weak and out of tune!










share|improve this question














This Wikipedia article is the source for two overtone series charts...



enter image description here




The numbers above the harmonic indicate the number of cents' difference from equal temperament (rounded to the nearest cent).




enter image description here



  • Regarding harmonic 11, is it a 'out of tune' Gb or an F?

  • Am I reading the charts correctly that there is no A natural in the series?

If I understood this article about cents, -49 on harmonic 11 is about 1/4 step - enough to be a horribly out of tune P4 - and -12 on harmonic 15 may drift past the threshold of perceptible depending on the person and the exactly sound context (it may be close enough.) Roughly the same for harmonic 5 as 15.



So the resulting major scale (based on these very weak harmonics) will have FA horribly out of tune, and MI and TI questionable.



Why would anyone use the harmonic series to explain the origin of the major scale with such bad tuning?



Based on the strength of the overtones the major triad is supported by the series. But the next strongest thing is a diminished triad! Past that everything is weak and out of tune!







harmonics






share|improve this question













share|improve this question











share|improve this question




share|improve this question










asked 4 hours ago









Michael CurtisMichael Curtis

10.3k637




10.3k637







  • 1





    @topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question

    – Michael Curtis
    3 hours ago






  • 1





    Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground.

    – Michael Curtis
    3 hours ago






  • 1





    now it's only +18

    – Michael Curtis
    3 hours ago






  • 1





    THE PEOPLE HAVE SPOKEN. WE MUST RESPECT THEIR VOTE. FAILING TO DO SO WOULD BE A BETRAYAL OF DEMOCRACY. (sorry if this humor doesn't work outside the UK ;). Anyway +1 from me :)

    – topo morto
    2 hours ago






  • 1





    @ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen.

    – Albrecht Hügli
    57 mins ago












  • 1





    @topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question

    – Michael Curtis
    3 hours ago






  • 1





    Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground.

    – Michael Curtis
    3 hours ago






  • 1





    now it's only +18

    – Michael Curtis
    3 hours ago






  • 1





    THE PEOPLE HAVE SPOKEN. WE MUST RESPECT THEIR VOTE. FAILING TO DO SO WOULD BE A BETRAYAL OF DEMOCRACY. (sorry if this humor doesn't work outside the UK ;). Anyway +1 from me :)

    – topo morto
    2 hours ago






  • 1





    @ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen.

    – Albrecht Hügli
    57 mins ago







1




1





@topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question

– Michael Curtis
3 hours ago





@topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question

– Michael Curtis
3 hours ago




1




1





Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground.

– Michael Curtis
3 hours ago





Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground.

– Michael Curtis
3 hours ago




1




1





now it's only +18

– Michael Curtis
3 hours ago





now it's only +18

– Michael Curtis
3 hours ago




1




1





THE PEOPLE HAVE SPOKEN. WE MUST RESPECT THEIR VOTE. FAILING TO DO SO WOULD BE A BETRAYAL OF DEMOCRACY. (sorry if this humor doesn't work outside the UK ;). Anyway +1 from me :)

– topo morto
2 hours ago





THE PEOPLE HAVE SPOKEN. WE MUST RESPECT THEIR VOTE. FAILING TO DO SO WOULD BE A BETRAYAL OF DEMOCRACY. (sorry if this humor doesn't work outside the UK ;). Anyway +1 from me :)

– topo morto
2 hours ago




1




1





@ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen.

– Albrecht Hügli
57 mins ago





@ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen.

– Albrecht Hügli
57 mins ago










1 Answer
1






active

oldest

votes


















4














There are several numerological methods to derive a major scale. Historically, several sets of six notes were used (called "hexachords") from which by various roundabout means, the various modes were derived. (Not every theorist used the same modes nor the same derivations.) The overtone series doesn't quite generate a nice scale (nor mode); neither do any others.



One method is to take combinations of overtones based on the first overtone (octave) and second (fifth). One gets "Pythagorean Tuning" using ratios containing only twos and threes. Tones an octave apart are considered "equivalent" for these constructions and the rations all become between 1.0 and 2.0. Thus the octave has a ratio of 2:1 (I prefer fractions 2/1) and the fifth (not called that yet) with a ration of 3/2. The inverse of the octave is 4/3 (the fourth). One can get twelve tones by noticing that 12 fifths is about the same as four octaves: (3/2)^12 (reduced) is 531441/252144 or 2.027.... which is close to an octave. Nothing ever comes out exact because no power of two is a power of three (except the 0th power which is 1). (Actually they are never too close except 8 and 9). One can take the twelve powers of 3/2 (0 through 11), reduce them to between 1 and 2, then sort. One gets an approximation to the 12-note chromatic scale. The first 7 powers are similar to the 7-notes of the major scale (or any of its modes).



One problem is that the "major third" (interval from C to E in modern terminology) has a ratio of 81/64 which is a bit out of tune (whatever that means) for lots of people. The next step is to expand the multipliers to using 2s, 3s, and 5s. This yield "just tuning" which sounds pretty good for a few chords. (Check in the Wiki: just tuning, temperament, Pythagorean tuning, mean-tone tuning, hexachord theory, etc. Iterate the bibliography operator.)



The fact that (except for the zeroth power) no prime powers are equal means that all system fail. One must make some type of compromise; either some keys are not in tune or all keys are out of tune. But this is all a bit of a digression (people do get unreasonably passionate about tunings though).



The Greeks defined several tetrachords (four-note collections). Medieval theorists used various tetrachord combinations to derive scales. The Greeks used three descentind tetrachords (later theorists used ascending tetrachords). The "diatonic" tetrachord was (in modern notation) A-G-F-E (two tones and a semitone), the "chromatic" tetrachord A-Gb-F-E (a minor third and two semitones), and the "enharmonic" tetrachord A-Gbb-Fq-E (a major third and two quarter-tones). The major scale (or the Greek version) was made of two (descending again) tetrachords joined by a whole tone: E-D-C-B+A-G-F-E (the pattern of modern major and minor scales by permutation, and modes for that matter). That's how it was done harmonically. The Wiki on "tetrachords" gives a nice description.



My conclusion is that the major (and other) scales were not really derived from overtones. Overtones were later used to "explain" (actually, "almost explain") these scales.






share|improve this answer























  • If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

    – Michael Curtis
    3 hours ago











  • “The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

    – leftaroundabout
    58 mins ago







  • 1





    And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

    – leftaroundabout
    49 mins ago











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









4














There are several numerological methods to derive a major scale. Historically, several sets of six notes were used (called "hexachords") from which by various roundabout means, the various modes were derived. (Not every theorist used the same modes nor the same derivations.) The overtone series doesn't quite generate a nice scale (nor mode); neither do any others.



One method is to take combinations of overtones based on the first overtone (octave) and second (fifth). One gets "Pythagorean Tuning" using ratios containing only twos and threes. Tones an octave apart are considered "equivalent" for these constructions and the rations all become between 1.0 and 2.0. Thus the octave has a ratio of 2:1 (I prefer fractions 2/1) and the fifth (not called that yet) with a ration of 3/2. The inverse of the octave is 4/3 (the fourth). One can get twelve tones by noticing that 12 fifths is about the same as four octaves: (3/2)^12 (reduced) is 531441/252144 or 2.027.... which is close to an octave. Nothing ever comes out exact because no power of two is a power of three (except the 0th power which is 1). (Actually they are never too close except 8 and 9). One can take the twelve powers of 3/2 (0 through 11), reduce them to between 1 and 2, then sort. One gets an approximation to the 12-note chromatic scale. The first 7 powers are similar to the 7-notes of the major scale (or any of its modes).



One problem is that the "major third" (interval from C to E in modern terminology) has a ratio of 81/64 which is a bit out of tune (whatever that means) for lots of people. The next step is to expand the multipliers to using 2s, 3s, and 5s. This yield "just tuning" which sounds pretty good for a few chords. (Check in the Wiki: just tuning, temperament, Pythagorean tuning, mean-tone tuning, hexachord theory, etc. Iterate the bibliography operator.)



The fact that (except for the zeroth power) no prime powers are equal means that all system fail. One must make some type of compromise; either some keys are not in tune or all keys are out of tune. But this is all a bit of a digression (people do get unreasonably passionate about tunings though).



The Greeks defined several tetrachords (four-note collections). Medieval theorists used various tetrachord combinations to derive scales. The Greeks used three descentind tetrachords (later theorists used ascending tetrachords). The "diatonic" tetrachord was (in modern notation) A-G-F-E (two tones and a semitone), the "chromatic" tetrachord A-Gb-F-E (a minor third and two semitones), and the "enharmonic" tetrachord A-Gbb-Fq-E (a major third and two quarter-tones). The major scale (or the Greek version) was made of two (descending again) tetrachords joined by a whole tone: E-D-C-B+A-G-F-E (the pattern of modern major and minor scales by permutation, and modes for that matter). That's how it was done harmonically. The Wiki on "tetrachords" gives a nice description.



My conclusion is that the major (and other) scales were not really derived from overtones. Overtones were later used to "explain" (actually, "almost explain") these scales.






share|improve this answer























  • If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

    – Michael Curtis
    3 hours ago











  • “The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

    – leftaroundabout
    58 mins ago







  • 1





    And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

    – leftaroundabout
    49 mins ago
















4














There are several numerological methods to derive a major scale. Historically, several sets of six notes were used (called "hexachords") from which by various roundabout means, the various modes were derived. (Not every theorist used the same modes nor the same derivations.) The overtone series doesn't quite generate a nice scale (nor mode); neither do any others.



One method is to take combinations of overtones based on the first overtone (octave) and second (fifth). One gets "Pythagorean Tuning" using ratios containing only twos and threes. Tones an octave apart are considered "equivalent" for these constructions and the rations all become between 1.0 and 2.0. Thus the octave has a ratio of 2:1 (I prefer fractions 2/1) and the fifth (not called that yet) with a ration of 3/2. The inverse of the octave is 4/3 (the fourth). One can get twelve tones by noticing that 12 fifths is about the same as four octaves: (3/2)^12 (reduced) is 531441/252144 or 2.027.... which is close to an octave. Nothing ever comes out exact because no power of two is a power of three (except the 0th power which is 1). (Actually they are never too close except 8 and 9). One can take the twelve powers of 3/2 (0 through 11), reduce them to between 1 and 2, then sort. One gets an approximation to the 12-note chromatic scale. The first 7 powers are similar to the 7-notes of the major scale (or any of its modes).



One problem is that the "major third" (interval from C to E in modern terminology) has a ratio of 81/64 which is a bit out of tune (whatever that means) for lots of people. The next step is to expand the multipliers to using 2s, 3s, and 5s. This yield "just tuning" which sounds pretty good for a few chords. (Check in the Wiki: just tuning, temperament, Pythagorean tuning, mean-tone tuning, hexachord theory, etc. Iterate the bibliography operator.)



The fact that (except for the zeroth power) no prime powers are equal means that all system fail. One must make some type of compromise; either some keys are not in tune or all keys are out of tune. But this is all a bit of a digression (people do get unreasonably passionate about tunings though).



The Greeks defined several tetrachords (four-note collections). Medieval theorists used various tetrachord combinations to derive scales. The Greeks used three descentind tetrachords (later theorists used ascending tetrachords). The "diatonic" tetrachord was (in modern notation) A-G-F-E (two tones and a semitone), the "chromatic" tetrachord A-Gb-F-E (a minor third and two semitones), and the "enharmonic" tetrachord A-Gbb-Fq-E (a major third and two quarter-tones). The major scale (or the Greek version) was made of two (descending again) tetrachords joined by a whole tone: E-D-C-B+A-G-F-E (the pattern of modern major and minor scales by permutation, and modes for that matter). That's how it was done harmonically. The Wiki on "tetrachords" gives a nice description.



My conclusion is that the major (and other) scales were not really derived from overtones. Overtones were later used to "explain" (actually, "almost explain") these scales.






share|improve this answer























  • If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

    – Michael Curtis
    3 hours ago











  • “The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

    – leftaroundabout
    58 mins ago







  • 1





    And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

    – leftaroundabout
    49 mins ago














4












4








4







There are several numerological methods to derive a major scale. Historically, several sets of six notes were used (called "hexachords") from which by various roundabout means, the various modes were derived. (Not every theorist used the same modes nor the same derivations.) The overtone series doesn't quite generate a nice scale (nor mode); neither do any others.



One method is to take combinations of overtones based on the first overtone (octave) and second (fifth). One gets "Pythagorean Tuning" using ratios containing only twos and threes. Tones an octave apart are considered "equivalent" for these constructions and the rations all become between 1.0 and 2.0. Thus the octave has a ratio of 2:1 (I prefer fractions 2/1) and the fifth (not called that yet) with a ration of 3/2. The inverse of the octave is 4/3 (the fourth). One can get twelve tones by noticing that 12 fifths is about the same as four octaves: (3/2)^12 (reduced) is 531441/252144 or 2.027.... which is close to an octave. Nothing ever comes out exact because no power of two is a power of three (except the 0th power which is 1). (Actually they are never too close except 8 and 9). One can take the twelve powers of 3/2 (0 through 11), reduce them to between 1 and 2, then sort. One gets an approximation to the 12-note chromatic scale. The first 7 powers are similar to the 7-notes of the major scale (or any of its modes).



One problem is that the "major third" (interval from C to E in modern terminology) has a ratio of 81/64 which is a bit out of tune (whatever that means) for lots of people. The next step is to expand the multipliers to using 2s, 3s, and 5s. This yield "just tuning" which sounds pretty good for a few chords. (Check in the Wiki: just tuning, temperament, Pythagorean tuning, mean-tone tuning, hexachord theory, etc. Iterate the bibliography operator.)



The fact that (except for the zeroth power) no prime powers are equal means that all system fail. One must make some type of compromise; either some keys are not in tune or all keys are out of tune. But this is all a bit of a digression (people do get unreasonably passionate about tunings though).



The Greeks defined several tetrachords (four-note collections). Medieval theorists used various tetrachord combinations to derive scales. The Greeks used three descentind tetrachords (later theorists used ascending tetrachords). The "diatonic" tetrachord was (in modern notation) A-G-F-E (two tones and a semitone), the "chromatic" tetrachord A-Gb-F-E (a minor third and two semitones), and the "enharmonic" tetrachord A-Gbb-Fq-E (a major third and two quarter-tones). The major scale (or the Greek version) was made of two (descending again) tetrachords joined by a whole tone: E-D-C-B+A-G-F-E (the pattern of modern major and minor scales by permutation, and modes for that matter). That's how it was done harmonically. The Wiki on "tetrachords" gives a nice description.



My conclusion is that the major (and other) scales were not really derived from overtones. Overtones were later used to "explain" (actually, "almost explain") these scales.






share|improve this answer













There are several numerological methods to derive a major scale. Historically, several sets of six notes were used (called "hexachords") from which by various roundabout means, the various modes were derived. (Not every theorist used the same modes nor the same derivations.) The overtone series doesn't quite generate a nice scale (nor mode); neither do any others.



One method is to take combinations of overtones based on the first overtone (octave) and second (fifth). One gets "Pythagorean Tuning" using ratios containing only twos and threes. Tones an octave apart are considered "equivalent" for these constructions and the rations all become between 1.0 and 2.0. Thus the octave has a ratio of 2:1 (I prefer fractions 2/1) and the fifth (not called that yet) with a ration of 3/2. The inverse of the octave is 4/3 (the fourth). One can get twelve tones by noticing that 12 fifths is about the same as four octaves: (3/2)^12 (reduced) is 531441/252144 or 2.027.... which is close to an octave. Nothing ever comes out exact because no power of two is a power of three (except the 0th power which is 1). (Actually they are never too close except 8 and 9). One can take the twelve powers of 3/2 (0 through 11), reduce them to between 1 and 2, then sort. One gets an approximation to the 12-note chromatic scale. The first 7 powers are similar to the 7-notes of the major scale (or any of its modes).



One problem is that the "major third" (interval from C to E in modern terminology) has a ratio of 81/64 which is a bit out of tune (whatever that means) for lots of people. The next step is to expand the multipliers to using 2s, 3s, and 5s. This yield "just tuning" which sounds pretty good for a few chords. (Check in the Wiki: just tuning, temperament, Pythagorean tuning, mean-tone tuning, hexachord theory, etc. Iterate the bibliography operator.)



The fact that (except for the zeroth power) no prime powers are equal means that all system fail. One must make some type of compromise; either some keys are not in tune or all keys are out of tune. But this is all a bit of a digression (people do get unreasonably passionate about tunings though).



The Greeks defined several tetrachords (four-note collections). Medieval theorists used various tetrachord combinations to derive scales. The Greeks used three descentind tetrachords (later theorists used ascending tetrachords). The "diatonic" tetrachord was (in modern notation) A-G-F-E (two tones and a semitone), the "chromatic" tetrachord A-Gb-F-E (a minor third and two semitones), and the "enharmonic" tetrachord A-Gbb-Fq-E (a major third and two quarter-tones). The major scale (or the Greek version) was made of two (descending again) tetrachords joined by a whole tone: E-D-C-B+A-G-F-E (the pattern of modern major and minor scales by permutation, and modes for that matter). That's how it was done harmonically. The Wiki on "tetrachords" gives a nice description.



My conclusion is that the major (and other) scales were not really derived from overtones. Overtones were later used to "explain" (actually, "almost explain") these scales.







share|improve this answer












share|improve this answer



share|improve this answer










answered 3 hours ago









ttwttw

9,001932




9,001932












  • If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

    – Michael Curtis
    3 hours ago











  • “The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

    – leftaroundabout
    58 mins ago







  • 1





    And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

    – leftaroundabout
    49 mins ago


















  • If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

    – Michael Curtis
    3 hours ago











  • “The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

    – leftaroundabout
    58 mins ago







  • 1





    And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

    – leftaroundabout
    49 mins ago

















If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

– Michael Curtis
3 hours ago





If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument.

– Michael Curtis
3 hours ago













“The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

– leftaroundabout
58 mins ago






“The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma.

– leftaroundabout
58 mins ago





1




1





And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

– leftaroundabout
49 mins ago






And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here.

– leftaroundabout
49 mins ago


















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