9 ms·

The first point is just really wrong. The harmonic series is not integer multiples, and you don't have to get very far in the harmonic series before you encount

The first point is just really wrong. The harmonic series is not integer multiples, and you don't have to get very far in the harmonic series before you encounter radical microtones that no European musician would have ever considered in-tune for the last thousand years. The order of intervals in the harmonic series is: Octave Perfect Fifth Perfect Fourth Major Third Minor Third Subminor Third Diminished Third Supermajor Second Major Second Neutral Second Minor Second And that's just the leading edge of the series. Between distant notes in the series you will find the neutral third, subfourth, superfourth, superaugmented fourth, subfifth, superaugmented fifth, supermajor sixth, and neutral seventh. If you want to claim that our intervals all come from the harmonic series, then in order to include the Minor Second, which everyone would agree is pretty dang important for our music, you must also include all the intervals listed above that nobody's ever heard of. It's simply wrong. The major scale has nothing to do with the harmonic series except by coincidence. The Tritone for instance, never appears in the harmonic series, and any music theorist will tell you that's way more important to our music than the Minor Second.

whiddershins 5y ago
It does somewhat though. You build it by taking the 3rd harmonic (perfect fifth) cut it in half to get it back into the octave, rinse, repeat.
- hannasanarion 5y ago
  That's not the harmonic series though, that's the first 3 members a bunch of different harmonic serieses smooshed together. It's extremely arbitrary. Like, if you want to say that our musical system revolves aroundt the frequency ratios of the octave (1:2) and fifth (2:3), then that's one thing and it's basically correct. But that's very different from claiming that western music is derived from the universal mathematical properties of the harmonic series, and is entirely irrelevant for making progress towards actual important concepts in music theory, like octave equivalence, let alone scales or dominant resolutions.
  - whiddershins 5y ago
    Sure. Just saying the harmonic series is not completely and utterly irrelevant. Even as you pointed out some of the harmonic series tracks closely to the scale tones, and the relationship between the harmonic series and the scale tones is a useful unit of analysis towards understanding consonance and dissonance and such things.
  - PaulDavisThe1st 5y ago
    > But that's very different from claiming that western music is derived from the universal mathematical properties of the harmonic series, Western (classical) music is marked (one might even say "distinguished") by its attention to harmonicity, which in turns is rooted in ideas about consonance and dissonace. These ideas are absolutely rooted in the mathematical properties of the harmonic series. As noted elsewhere in the comments, the work of Plomp & Levelt and then Sethares has shown how human perception of consonance (with dissonance being its reciprocal) is rooted in the averaged amplitude-weighted sum of the pair-wise disonnances between all the partials.
    - hannasanarion 5y ago
      They are related to the mathematical properties of the harmonic series, but they are definitely not rooted in it. If it was true that western musical ideas of consonance is rooted in simple harmonic ratios, and that our musical scales were based on the harmonic series, then we should be seeing subminor thirds (6:7) and supermajor seconds (7:8) everywhere. They are way more "consonant" than major seconds (9:10), or minor seconds (15:16). But in reality, they are seen as exotic and wrong.
      - PaulDavisThe1st 5y ago
        Unfortunately, nobody said "western musical ideas of consonance is rooted in simple harmonic ratios". The whole point of Plomb & Levelt's work (recommended if you have no read it) is that provides an explanation for consonance that has simple harmonic ratios as an existential but insufficient component. From their work, what matters is not the ratios of the two fundamentals, because there are essentially no natural tones that consist of only the fundamental (1). Instead, consonance/dissonance is a result of the sum of the consonsance relationships between each pair of partials in the two tones. This is a good overview (linked elsewhere in the comments): https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html Sethares extended their conclusions a bit by noting that since the partial spectrum is the definition of timbre, the most consonant/dissonant intervals would vary by timbre, which they claimed is observed in the real world. (1) Indeed, the first step of their work, defining the "human dissonance response curve" relies on being able to use a signal generator to create pure sine tones.
PaulDavisThe1st 5y ago
This is just wrong. You are using western musical culture terms for what are actually physical phenomena. I will quote to you from wikipedia: > The harmonic series is an arithmetic progression (f, 2f, 3f, 4f, 5f, ...). In terms of frequency (measured in cycles per second, or hertz, where f is the fundamental frequency), > The second harmonic, whose frequency is twice the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher). > The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals https://en.wikipedia.org/wiki/Harmonic_series_(music) https://en.wikipedia.org/wiki/Harmonic_series_(music) What you are describing is a musically significant set of intervals that are merely a subset of the harmonic series, created by what the wikipedia page describes as: > If the harmonics are octave displaced and compressed into the span of one octave, some of them are approximated by the notes of what the West has adopted as the chromatic scale based on the fundamental tone
- hannasanarion 5y ago
  > some of the pitches in the harmonic series are approximated by the notes of the chromatic scale is not the same thing as > the chromatic scale is derived from the harmonic series which is what the OP article claims. You can find the chromatic scale in the harmonic series, yeah, if you ignore the majority of the notes in the harmonic series. To find the chromatic scale in the harmonic series, you need to take the 2nd, 3rd, 4th, 5th, 9th, 15th, and 44th harmonics, and ignore of the rest. That's not a mathematically justified derivation, that's a post-hoc rationalization built on coincidence alone.
  - PaulDavisThe1st 5y ago
    day-after update: 12T in the octave, based on the following harmonics: (in pitch/frequency order) 1st, 17th, 9th, 19th, 5th, 21st, 11th, 3rd, 13th, 27th, 7th, 15th In harmonic order: 1,3,7,9,11,13,15,17,19,21,27 So, sure, fair question why these harmonics and not any of the others? Well, powers of 2 are out because they are just higher octaves. Then we have a whole series of harmonics that are equivalent ratios to the fundamental when folded down into the octave range (3 (3:2),6 (6:4), 12 (12:8)), (5 (5:4), 10 (10:8)), (7,14,28), (13,26) and so on. You'll notice the pattern: the harmonics the define the intervals in a 12T system are those that introduce new ratios into the list of intervals, so they lean toward being prime or only having factors not already introduced. By the time you go through the list, it's easy to see that going up to the 31st harmonic really only leaves out a couple of possibilities from a 12T system: 25, 29, 31 and as far as I am aware this is because introducing them into the pitch class produces results extremely close to already existing members. And sure, you could go higher, but the pattern will repeat: harmonics whose ratio folded into the octave range are identical, or which give rise to pitches extremely close to pitches defined already. It's not a post-hoc rationalization at all.
PaulDavisThe1st 5y ago
> The Tritone for instance, never appears in the harmonic series, The tritone is the eleventh harmonic of the series, folded back down into the octave to give a ratio of ... "well, it depends". Again, from wikipedia: > The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as Fhalf sharp4 above C1), known as the lesser undecimal tritone or undecimal semi-augmented fourth, is found in some just tunings and on many instruments. For example, very long alphorns may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F♯ above C, for example), as in Brahms's First Symphony
- hannasanarion 5y ago
  The eleventh harmonic is a neutral second. Even according to the wikipedia section you quoted, you have to round up from F half-sharp to F sharp to get it, which doesn't make sense if you're trying to derive a musical scale from the harmonic series, as you would have already passed over the neutral third, diminished third, supermajor second, and all the rest of the intervals listed above which require no rounding at all. The actual 11th harmonic is the neutral second against the 10th harmonic, which corresponds to the super-fourth compared to the root.
  - PaulDavisThe1st 5y ago
    You are confusing the physical harmonic series with the series of "harmonics" typically named in western music.
    - hannasanarion 5y ago
      You're the one saying that F half-sharp (the actual 11th harmonic of C) is the same thing as F-sharp.
      - PaulDavisThe1st 5y ago
        The problem here is that "tritone" has varying definitions, including at least: half an octave, three whole tones, two minor thirds, and that particular in JI, that leaves things a bit wooly to say the least. It appears fairly conventional to say at least that the lesser undecimal tritone is the 11th harmonic, which corresponds to a ratio (folded into the octave) of 11/8, but other defintions (64/45 or 45/32 for example) also exist.
Aidevah 5y ago
Thank you. The insistence on using the harmonic series to explain the source of the notes diatonic scale is anachronistic, post hoc and never really made any sense even on its own terms. There is nothing inherently "natural" about the seven basic notes that make up the diatonic scale, and there are even more problems than the major omissions that have been made in order to fit the members of the harmonic series into a diatonic mold that you have mentioned. The harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator, unless you invoke reciprocals. Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one.
- PaulDavisThe1st 5y ago
  > Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one. Again, Plomp & Levelt (1965!). It's not about "simple ratios".
  - quadrangle 5y ago
    No, but it is partly about beats and matching partials. See https://sethares.engr.wisc.edu/ttss.html https://sethares.engr.wisc.edu/ttss.html which describes how the spectra of timbres affects which harmonies blend. You can synthesize inharmonic timbres to get arbitrary scales and tunings to blend more. You can try this on a guitar. If you pluck a string right in the middle to kill all the even harmonics, major 7ths and minor 9ths don't sound so dissonant as they did when the original string had its 2nd harmonic octave present.
- PaulDavisThe1st 5y ago
  > The harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator Let's consider the major sixth. It is the 27th harmonic of the fundamental. Expressing that in the conventional octave range (1:1 .. 2:1) requires us to write it as 27:16. This is Pythagorean major sixth. JI can use the same ratio, but it is common there to use 5-limit tuning, which builds all intervals from ratios built from powers of 2, 3 and 5. This is conceptually equivalent to the sort of "ratio distortion" that occurs with ET, though for an entirely different purpose. The Pythagorean major sixth (27:16) is expressed in decimal form as 1.6875. The closest 5-limit tuning ratio to that is 5:3, or 1.6666.... By contrast, the ET major sixth is 2^(9⁄12) or 1.681793. The description of the 5-limit tuning major sixth as 5:3 is no different in its deviation from the Pythagorean version (27:16) than the reason why the ET version also does not match the ratio given by the harmonic series: musical/compositional preference. Neither version precisely matches the harmonic series, but has its own musically-rooted reasons to use a nearby ratio (that will sometimes be expressed with denominators that are not powers of two). I would also note that in some other musical cultures, they also express the same musical ambivalence. In some Indian scales for example the pitch denoted "Dha" can be either 27:16 or 5:3 with respect to "Sa", the fundamental. So sure, in one sense, "the harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator" is true. But the full explanation is that "the real ratio is fully explained by the harmonic series, but for performance/instrument/compositional reasons, many music cultures use a ratio that deviates from the real value, just as is the case for much of ET tuning and other ratios derived from the harmonic series".
quadrangle 5y ago
Meta: this whole conversation amounts to a bunch of people who don't really understand tuning talking over-confidently about it. There are communities of experts on the subject, and if you bother to engage with any of them, you can learn. But you don't learn by starting out sure that you already know everything. > The Tritone for instance, never appears in the harmonic series The first of multiple tritones that show up is the one from harmonic 5 to harmonic 7. We're not talking weird abstract, esoteric harmonics here. The major scale has SOMETHING to do with harmonics of course. It's true that Just Intonation purists can be dogmatic and make claims that don't hold up to scrutiny, but jumping to the conclusion that their claims have NOTHING to them is even more stupid a position. The major scale became the reference scale in chordal music specifically because of the way it facilitates blended harmonies in closely related chords. It's because people discovered the blend of the major chord which is a thing because of how harmonics work. And you put three overlapping major chords together and you get the major scale. Before chordal-harmony, other modes and scales were just as or more prominent.
- PaulDavisThe1st 5y ago
  The harmonic series does not define the major scale, which is a heptatonic (7 tone) scale, at least not any more (or less) than it defines any other scale. The harmonic series helps to explain the relationships between the twelve tones that we (people composing and performing in "western" musical traditions) use to divide the octave. It's likely not a coincidence that many other cultures, despite their use of additional tones within the octave, also seem to "stumbled" as we did onto the properties we get from 12T-per-octave (regardless of the precise tuning of those 12 tones). I'm not sure where you are getting your information about tritones from. Everything I've learnt about tritones would say that you're wrong to claim they occur between the 5th and 7th harmonic. Got any references on that? Also, this thing about communities of experts ... one of the problems with this is that at least in the USA and Europe, the majority of the community of experts appears to have deliberately constrained their expertise to a somewhat narrow view of tuning. There's a reason why Bohlen-Pierce and the tunings of Harry Partch are considered "out there": it's not just that they are quite challenging for 12T(ET) ears to listen to, but also that our musical academies have shrunk the possibilities for tuning down to a sadly constrained range. Consider for example, the bulk of musical set theory, which is incredibly interesting at the scale (mode) level, but still appears from the literature to overwhelming assume only a handful of possible tuning systems (and a big chunk of it considers only one possible tuning system).

whiddershins 5y ago

It does somewhat though. You build it by taking the 3rd harmonic (perfect fifth) cut it in half to get it back into the octave, rinse, repeat.

hannasanarion 5y ago
That's not the harmonic series though, that's the first 3 members a bunch of different harmonic serieses smooshed together. It's extremely arbitrary. Like, if you want to say that our musical system revolves aroundt the frequency ratios of the octave (1:2) and fifth (2:3), then that's one thing and it's basically correct. But that's very different from claiming that western music is derived from the universal mathematical properties of the harmonic series, and is entirely irrelevant for making progress towards actual important concepts in music theory, like octave equivalence, let alone scales or dominant resolutions.
- whiddershins 5y ago
  Sure. Just saying the harmonic series is not completely and utterly irrelevant. Even as you pointed out some of the harmonic series tracks closely to the scale tones, and the relationship between the harmonic series and the scale tones is a useful unit of analysis towards understanding consonance and dissonance and such things.
- PaulDavisThe1st 5y ago
  > But that's very different from claiming that western music is derived from the universal mathematical properties of the harmonic series, Western (classical) music is marked (one might even say "distinguished") by its attention to harmonicity, which in turns is rooted in ideas about consonance and dissonace. These ideas are absolutely rooted in the mathematical properties of the harmonic series. As noted elsewhere in the comments, the work of Plomp & Levelt and then Sethares has shown how human perception of consonance (with dissonance being its reciprocal) is rooted in the averaged amplitude-weighted sum of the pair-wise disonnances between all the partials.
  - hannasanarion 5y ago
    They are related to the mathematical properties of the harmonic series, but they are definitely not rooted in it. If it was true that western musical ideas of consonance is rooted in simple harmonic ratios, and that our musical scales were based on the harmonic series, then we should be seeing subminor thirds (6:7) and supermajor seconds (7:8) everywhere. They are way more "consonant" than major seconds (9:10), or minor seconds (15:16). But in reality, they are seen as exotic and wrong.
    - PaulDavisThe1st 5y ago
      Unfortunately, nobody said "western musical ideas of consonance is rooted in simple harmonic ratios". The whole point of Plomb & Levelt's work (recommended if you have no read it) is that provides an explanation for consonance that has simple harmonic ratios as an existential but insufficient component. From their work, what matters is not the ratios of the two fundamentals, because there are essentially no natural tones that consist of only the fundamental (1). Instead, consonance/dissonance is a result of the sum of the consonsance relationships between each pair of partials in the two tones. This is a good overview (linked elsewhere in the comments): https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html Sethares extended their conclusions a bit by noting that since the partial spectrum is the definition of timbre, the most consonant/dissonant intervals would vary by timbre, which they claimed is observed in the real world. (1) Indeed, the first step of their work, defining the "human dissonance response curve" relies on being able to use a signal generator to create pure sine tones.

PaulDavisThe1st 5y ago

This is just wrong. You are using western musical culture terms for what are actually physical phenomena. I will quote to you from wikipedia: > The harmonic series is an arithmetic progression (f, 2f, 3f, 4f, 5f, ...). In terms of frequency (measured in cycles per second, or hertz, where f is the fundamental frequency), > The second harmonic, whose frequency is twice the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher). > The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals https://en.wikipedia.org/wiki/Harmonic_series_(music) https://en.wikipedia.org/wiki/Harmonic_series_(music) What you are describing is a musically significant set of intervals that are merely a subset of the harmonic series, created by what the wikipedia page describes as: > If the harmonics are octave displaced and compressed into the span of one octave, some of them are approximated by the notes of what the West has adopted as the chromatic scale based on the fundamental tone

hannasanarion 5y ago
> some of the pitches in the harmonic series are approximated by the notes of the chromatic scale is not the same thing as > the chromatic scale is derived from the harmonic series which is what the OP article claims. You can find the chromatic scale in the harmonic series, yeah, if you ignore the majority of the notes in the harmonic series. To find the chromatic scale in the harmonic series, you need to take the 2nd, 3rd, 4th, 5th, 9th, 15th, and 44th harmonics, and ignore of the rest. That's not a mathematically justified derivation, that's a post-hoc rationalization built on coincidence alone.
- PaulDavisThe1st 5y ago
  day-after update: 12T in the octave, based on the following harmonics: (in pitch/frequency order) 1st, 17th, 9th, 19th, 5th, 21st, 11th, 3rd, 13th, 27th, 7th, 15th In harmonic order: 1,3,7,9,11,13,15,17,19,21,27 So, sure, fair question why these harmonics and not any of the others? Well, powers of 2 are out because they are just higher octaves. Then we have a whole series of harmonics that are equivalent ratios to the fundamental when folded down into the octave range (3 (3:2),6 (6:4), 12 (12:8)), (5 (5:4), 10 (10:8)), (7,14,28), (13,26) and so on. You'll notice the pattern: the harmonics the define the intervals in a 12T system are those that introduce new ratios into the list of intervals, so they lean toward being prime or only having factors not already introduced. By the time you go through the list, it's easy to see that going up to the 31st harmonic really only leaves out a couple of possibilities from a 12T system: 25, 29, 31 and as far as I am aware this is because introducing them into the pitch class produces results extremely close to already existing members. And sure, you could go higher, but the pattern will repeat: harmonics whose ratio folded into the octave range are identical, or which give rise to pitches extremely close to pitches defined already. It's not a post-hoc rationalization at all.

PaulDavisThe1st 5y ago

> The Tritone for instance, never appears in the harmonic series, The tritone is the eleventh harmonic of the series, folded back down into the octave to give a ratio of ... "well, it depends". Again, from wikipedia: > The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as Fhalf sharp4 above C1), known as the lesser undecimal tritone or undecimal semi-augmented fourth, is found in some just tunings and on many instruments. For example, very long alphorns may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F♯ above C, for example), as in Brahms's First Symphony

hannasanarion 5y ago
The eleventh harmonic is a neutral second. Even according to the wikipedia section you quoted, you have to round up from F half-sharp to F sharp to get it, which doesn't make sense if you're trying to derive a musical scale from the harmonic series, as you would have already passed over the neutral third, diminished third, supermajor second, and all the rest of the intervals listed above which require no rounding at all. The actual 11th harmonic is the neutral second against the 10th harmonic, which corresponds to the super-fourth compared to the root.
- PaulDavisThe1st 5y ago
  You are confusing the physical harmonic series with the series of "harmonics" typically named in western music.
  - hannasanarion 5y ago
    You're the one saying that F half-sharp (the actual 11th harmonic of C) is the same thing as F-sharp.
    - PaulDavisThe1st 5y ago
      The problem here is that "tritone" has varying definitions, including at least: half an octave, three whole tones, two minor thirds, and that particular in JI, that leaves things a bit wooly to say the least. It appears fairly conventional to say at least that the lesser undecimal tritone is the 11th harmonic, which corresponds to a ratio (folded into the octave) of 11/8, but other defintions (64/45 or 45/32 for example) also exist.

Aidevah 5y ago

Thank you. The insistence on using the harmonic series to explain the source of the notes diatonic scale is anachronistic, post hoc and never really made any sense even on its own terms. There is nothing inherently "natural" about the seven basic notes that make up the diatonic scale, and there are even more problems than the major omissions that have been made in order to fit the members of the harmonic series into a diatonic mold that you have mentioned. The harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator, unless you invoke reciprocals. Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one.

PaulDavisThe1st 5y ago
> Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one. Again, Plomp & Levelt (1965!). It's not about "simple ratios".
- quadrangle 5y ago
  No, but it is partly about beats and matching partials. See https://sethares.engr.wisc.edu/ttss.html https://sethares.engr.wisc.edu/ttss.html which describes how the spectra of timbres affects which harmonies blend. You can synthesize inharmonic timbres to get arbitrary scales and tunings to blend more. You can try this on a guitar. If you pluck a string right in the middle to kill all the even harmonics, major 7ths and minor 9ths don't sound so dissonant as they did when the original string had its 2nd harmonic octave present.
PaulDavisThe1st 5y ago
> The harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator Let's consider the major sixth. It is the 27th harmonic of the fundamental. Expressing that in the conventional octave range (1:1 .. 2:1) requires us to write it as 27:16. This is Pythagorean major sixth. JI can use the same ratio, but it is common there to use 5-limit tuning, which builds all intervals from ratios built from powers of 2, 3 and 5. This is conceptually equivalent to the sort of "ratio distortion" that occurs with ET, though for an entirely different purpose. The Pythagorean major sixth (27:16) is expressed in decimal form as 1.6875. The closest 5-limit tuning ratio to that is 5:3, or 1.6666.... By contrast, the ET major sixth is 2^(9⁄12) or 1.681793. The description of the 5-limit tuning major sixth as 5:3 is no different in its deviation from the Pythagorean version (27:16) than the reason why the ET version also does not match the ratio given by the harmonic series: musical/compositional preference. Neither version precisely matches the harmonic series, but has its own musically-rooted reasons to use a nearby ratio (that will sometimes be expressed with denominators that are not powers of two). I would also note that in some other musical cultures, they also express the same musical ambivalence. In some Indian scales for example the pitch denoted "Dha" can be either 27:16 or 5:3 with respect to "Sa", the fundamental. So sure, in one sense, "the harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator" is true. But the full explanation is that "the real ratio is fully explained by the harmonic series, but for performance/instrument/compositional reasons, many music cultures use a ratio that deviates from the real value, just as is the case for much of ET tuning and other ratios derived from the harmonic series".

quadrangle 5y ago

Meta: this whole conversation amounts to a bunch of people who don't really understand tuning talking over-confidently about it. There are communities of experts on the subject, and if you bother to engage with any of them, you can learn. But you don't learn by starting out sure that you already know everything. > The Tritone for instance, never appears in the harmonic series The first of multiple tritones that show up is the one from harmonic 5 to harmonic 7. We're not talking weird abstract, esoteric harmonics here. The major scale has SOMETHING to do with harmonics of course. It's true that Just Intonation purists can be dogmatic and make claims that don't hold up to scrutiny, but jumping to the conclusion that their claims have NOTHING to them is even more stupid a position. The major scale became the reference scale in chordal music specifically because of the way it facilitates blended harmonies in closely related chords. It's because people discovered the blend of the major chord which is a thing because of how harmonics work. And you put three overlapping major chords together and you get the major scale. Before chordal-harmony, other modes and scales were just as or more prominent.

PaulDavisThe1st 5y ago
The harmonic series does not define the major scale, which is a heptatonic (7 tone) scale, at least not any more (or less) than it defines any other scale. The harmonic series helps to explain the relationships between the twelve tones that we (people composing and performing in "western" musical traditions) use to divide the octave. It's likely not a coincidence that many other cultures, despite their use of additional tones within the octave, also seem to "stumbled" as we did onto the properties we get from 12T-per-octave (regardless of the precise tuning of those 12 tones). I'm not sure where you are getting your information about tritones from. Everything I've learnt about tritones would say that you're wrong to claim they occur between the 5th and 7th harmonic. Got any references on that? Also, this thing about communities of experts ... one of the problems with this is that at least in the USA and Europe, the majority of the community of experts appears to have deliberately constrained their expertise to a somewhat narrow view of tuning. There's a reason why Bohlen-Pierce and the tunings of Harry Partch are considered "out there": it's not just that they are quite challenging for 12T(ET) ears to listen to, but also that our musical academies have shrunk the possibilities for tuning down to a sadly constrained range. Consider for example, the bulk of musical set theory, which is incredibly interesting at the scale (mode) level, but still appears from the literature to overwhelming assume only a handful of possible tuning systems (and a big chunk of it considers only one possible tuning system).