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Comment by Aidevah | Hacker News Reader

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.

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> 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".

>Let's consider the major sixth.

>It is the 27th harmonic of the fundamental.

This is the problem I have with discussions concerning the harmonic series, it is presented as equivalent to the notes of the scale, even though no musician before the 19th century was even aware of it. Even in the heyday of just intonation (in theory not in practice), no theorist used the Pythagorean ratio for it, the very simple reason being that the intervals were obtained via dividing the octave and subsequent ratios. This is how it was done in the most important treatises such as Zarlino, Galilei etc, and subsequent theorist followed in their footsteps. The 5:3 major sixth differs from the ET and Pythagorean one for one very good reason: it was the one derivation that theorists actually used. Though how the major sixth was used in practice is a much more difficult question, and that was the reason for Galilei's break with Zarlino showed.

I just want to re-iterate that all these musical concepts have existed long before the discovery of the harmonic series, and were justified differently. The harmonic series definitely does not "explain fully" any ratio as this comment thread has amply shown. It is an invention of the 19th century that people have tried to bolt on to existing concepts which nevertheless requires significant contortions and even so come up deficient.

For the audience, I highly recommend reading through the first chapter of Galilei's Dialogo which presents a historical derivation of the intervals of the scale and the many difficulties attending to just intonation.

https://digital.library.unt.edu/ark:/67531/metadc500437/

It seems to me that you are confusing two claims.

1. a claim about the history of how common western pitches were selected over time.

2. a claim about acoustics and physics shaping the overall process of how humans (and some other animals) select pitches.

I am not suggesting for one second that the actual history of the 12T scale has been explicitly rooted in an understanding of the actual physical harmonic series.

I am suggesting that many of the choices made throughout time, both in western europe and other cultures, and also by some other species that sing (e.g. some birds) have been shaped by the physics of the harmonic series.

> 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.

From: https://musicscience.net/2021/09/12/separating-the-cultural-...

> "The preference for consonance varies across cultures but the aversion to harsh dissonance is universal"

The cited paper uses chord clusters containing four notes, and say nothing about dissonance as a dyadic concept which is what the discussion of ratios have been focusing on. The results are not what the PR piece states since Figure 1 of the cited paper[1] which sums up the results show that for the non-Westerners (Just Pakistani tribes of Kalash and Khow. If that is enough to claim "universality" in the social sciences then God help us.) low valence ratings for chromatic clusters are similar to their responses to major chords, so there is uncertainty about interpreting this result as their aversion to harsh dissonance or simply something they are unfamiliar with.

Just browsing the paper and I come up with some shocking inaccuracies, consider

>Notably, low preference for the major triad among the Northwest Pakistani tribes is not corroborating the theory according to which the attractiveness of consonance is due to harmonic similarity to human vocalizations and is in line with the historical observation according to which the major third became consonant only over time in the framework of Western music as well.

The preference of the Pakistani tribes for the minor triad over the major triad (which is shown in the paper) is most certainly not in line with the gradual acceptance of the major third in Western music for the very simple reason that in the west the two types of thirds were accepted together. And if we consider the acceptance of thirds in the final chord of the piece, the western experience is exactly the mirror of the Pakistani tribes: the minor triad was deemed less 'stable' than the major triad and it took much longer for the minor triad to be regarded as acceptable to the ending of pieces.

Shoddy research like this which betray a lack of historical awareness is why I have very low opinion of most these types of social research. They are not good writers on music, and some, I assume, are good people.

[1] https://nyaspubs.onlinelibrary.wiley.com/doi/full/10.1111/ny...

> 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".

No, but it is partly about beats and matching partials. See 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.

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