Talk:Consonance and dissonance

Jonas on dissonance and voice leading

After removing the paragraph discussed in the section above, I also removed the quotation from Jonas that opened the section on Musical style in the article. This statement of Jonas indeed has nothing to do with style. The quotation read as follows:

The concept of dissonance does not belong to the domain of harmony as it is presented to us by Nature, but is derived from voice leading, which is an essential constituent of Art.

And what it meant is this: in Schenker's conception, the model presented by Nature is the (major) triad, on the imitation of which all harmony is based. Dissonances do not belong to this model and can only result from a voice leading that adds different kinds of non harmonic notes (passing or neighbor notes) filling in the distances between the notes of the triad. This is a quite strict conception of harmony, and Jonas' own view of the matter was even more restricted than Schenker's. Jonas writes:

The triad is the simultaneity given by Nature, and the intervals that it comprises are the consonances given by Nature. The concepts of consonance and dissonance are more disputed today than ever before (p. 20 of the 2d edition, 2005).

And he continues with a criticism of Schoenberg's idea that "the distance between [consonance and dissonance] is but a matter of degree, not of kind" (Jonas, 2d edition, p. 20, quoting Schoenberg, Harmonielehre, 3d edition, 1922, p. 18). Jonas clearly associates dissonances (and their treatment) with counterpoint, and accepts as (harmonic) consonances only the intervals of the triad and their inversions: third, fourth, fifth, sixth and octave. These indeed are the intervals described as consonant in the section Consonance of the article. But this has nothing to do with style, nor even with a definition of consonance and dissonance: a list of consonances is not a definion of consonance.

Open discussion of Sethares claims

I've found a lot of very useful information in the "Consonance and Dissonance" wikipedia page on many levels. I like the way the article is laid out overall. I also like the information I have found on William Sethares and his efforts, but I want to challenge a cornerstone of Sethares's propositions (as a trained musician - I tune pianos, play piano, sing, teach, and I have played piano for over 45 years), not in order to change this Wikipedia article in any way, but in order to help give perspective to Sethares's work as well as a reasonable amount of weight.

In the first chapter of his book most associated with this topic, Tuning Timbre Spectrum Scale, Sethares offers the reader the "octave challenge" to open the book, two specially constructed notes ostensibly placed an octave apart. He creates what he is calling "notes" that do not adhere to the harmonic series of overtones. From my perspective as a musician, Sethares pulls a little sleight-of-hand to open the entire text with that "octave challenge," though I doubt it is an intentional sleight-of-hand. I think Sethares has spent a lifetime exploring the edges of music rather than becoming more learned in music theory. He's a physics/computer guy, and music is a research area. Let me be clear - Sethares asks a lot of very worthwhile and interesting questions in the book as well. The slight of hand is this: Sethares provides a recorded example with two notes an octave apart and then played as a dyad. The assumption is the notes are "an octave" because the fundamental frequency is one octave, but then the overtones do not conform to a typical musical note.

Here is the problem: Sethares flings at the feet of many novices a problem that even trained musicians might not quickly pick up on unless given more warning. Large intervals are typically the most difficult for singers to accurately sing. Listening similarly, to a *SEQUENTIALLY* played large interval and properly identifying it...much less whether it is well tuned or not? That's a tall order. It's not a fair question. I would have much preferred first Sethares approach a thousand accomplished musicians and ask the question, "Does this sound like: (a) An EXACT octave? (b) NARROW of an octave? or (c) WIDE of an octave? I'd lay a c-note on the table says 90%+ would say it is off, and say it is off in the same direction. It's been a few months since I looked at it. I forget whether it was wide or narrow. I just know it did not sound like an octave to me. So when I hear Sethares's two notes sequentially? ***NO*** I definitively do NOT hear an octave. Fwiw, when I'm in heavy "piano tuning" mode? (days on end...) Then my ear flirts with perfect pitch. My daughter played marimba in a high school band, and one time I weirdly caught myself asking, "remember when you played that high A?" I stopped. I was hearing the note in my head. Went to a piano. It was A. I can't always do that, though.

The moment Sethares played the second note, it sounded "off" to me...and the "off" helps explain it as a dissonant dyad. In effect Sethares says, "don't these sound like and octave played one after the other?" It's a leading question. My ear says, "Nope. Sorry Sethares, but that's a hard non-starter for me." ...And that is the opening of his book, the basis of the thesis.

The alteration of overtones may have altered the perception of distance between notes. All I know is those first few pages in the book are supposed to throw into question the hegemony of the octave and the book fails to accomplish that. I would also observe at this point that it is possible, within our understanding of "notes" to produce sounds that possess internal dissonance. I also think Sethares does this a bit in that very first example with his book. Sethares writes, "Individually, they [the notes] sound normal enough, although perhaps somewhat electronic or 'bell-like' in nature." Not to me they don't. The specially constructed sounds do not contain the expression of the singular note. They are close, and they are certainly interesting and "musical." But I do not hear a crisp musical "note," as might be played by a flute, stringed instrument, or sung. I hear a more nebulous sound, something far less "scalar" in possibility.

We constructed the first flutes in the neighborhood of 40,000 years ago. Certainly we have produced a wide variety of instruments with a sense of pitch that do not adhere to the standard overtone series. (Timpani springs to mind.) From my perspective, the overtone series buried right inside our own voices (and in the voices of many living things) bears a *strong* likelihood for explaining the mathematical perceptual distance between notes even more than consonance/dissonance.

At a precise octave, the overtone series becomes exactly like a Russian nested doll: Every single overtone in the higher note is effectively subsumed in the overtones of the lower note. (This shouldn't happen with Sethares's notes, in part because of the math involved) The upper note of an octave has NO OVERTONES that do not already exist in the lower note. The lower note is like a vessel containing the entirety of overtones in the higher note. There is an incredibly precise sameness, "This is a 'Mini-Me,' a littler version of me."

The "voice" is extra special in the musical world, the lone flute, or lone cello note - It possesses a quality that beckons. The voice is special in a way not because of what we DO hear, but because of what we DON'T hear. We don't hear overtones automatically. Our hearing synthesizes them into the singular thing, an extra special sound that says, "whatever made this might be alive." Sethares's tones contain an "edginess" that our voices do not. He purports to explore consonance. He's really exploring the attractiveness of dissonance, and these days edgy sounds are certainly all the rage. That's a worthwhile quest, too. Just don't label it something it is not.

To this extent, there is a "resting home" feel of the singular note. I like to call it the "Call of the Wild," alluding to Jack London's novel.

My own work, to be released soon, explores the math/physiology of consonance/dissonance.

I like Sethares's work a lot. I really do! But Sethares does not effectively "lay claim" to consonance and dissonance. What he does successfully achieve, I think, is to explore features of a different "type" of consonance/dissonance, but he does not override or effectively undermine tonal consonance (or sometimes perhaps "harmonic" consonance)... He simply does not. In my own work, I will explain why. Sethares's "octave challenge" is a very interesting question, even if an unintentional "sleight of hand." The real point perhaps reinforced is that there may be different *kinds* of consonance/dissonance, but Sethares really doesn't throw into question older questions/understandings of consonance/dissonance. He just doesn't. He adds a new interesting layer, but he does not rewrite older understandings at all, not a bit. KWSager (talk) 20:40, 11 November 2020 (UTC)

KW: As someone who has collaborated with Prof. Sethares on a small portion of his work -- specifically the creation of Dynamic Tonality -- please allow me to take up the gauntlet in his defense. 🙂

First, let me point out that my perspective on this matter has shifted over time due largely to discussions on Wikipedia's talk pages, notably with Hucbald.SaintAmand (Hi, Huc! 🙂). With that in mind, my current perspective is this: you are correct, that Sethares/Milne/my view of consonance is not the universal, overwhelming, all-conquering definition of consonance. It is a layer.

It is a very useful definition, however, in that it enables Dynamic Tonality, which has the potential to expand the frontiers of tonality in very rich and fruitful ways. (If only I'd shipped the d**ned Thummer, I would have been able to point to actual music that made this point for me. My greatest regret in life...so far!..was not managing the Thummer company's finances better. Ah, well. As Huc would say, C'est la vie.) Even lacking the Thummer, Milne has leveraged tuning-aligned timbres to do some very interesting work in the cognition of musical structures. He has used tuning-aligned timbres to isolate "that which is learned" from "that which is innate" (in either the sound itself or in the ear/brain's processing thereof). Others are joining the research program enabled by Dynamic Tonality, slowly but surely, despite the absence of the Thummer.

As to the stretched octave example that starts Sethares' book (which I have not read for a while)... is it not true that the upper octaves of a piano are stretched? If so, then if Sethares' example had used piano sound samples from the piano's upper octaves, would that have made the example more or less effective, from your perspective as an experienced piano tuner? Why (or why not)?

--JimPlamondon (talk) 11:57, 20 July 2021 (UTC)

Hi, Jim 🙂. Allow me to continue our (facetious) discussion, which is more serious than it may seem. I attentively followed your video and I would have a few questions.

A. "The syntonic temperament's period is the Just Perfect Octave". I think that we should distinguish the theoretical description of temperaments from their practical realization. So far as I know, all temperaments are theoretically based on the octave, even if in practice piano temperament (based on the coincidence of partials) has to take in account that octaves may be somewhat stretched because of the inharmonicity of piano strings. This can be described only in very general terms because the inharmonicity may depend not only on the size of the piano, but also on the strings used, on their age, etc.

B. "The syntonic temperament's generator in the Just Perfect Fifth". I begin to loose foot, because up to there the "syntonic temperament" (I am not sure to understand what that is) has untempered intervals both as its period and its generator: is it a temperament?

C. "We map this to the tempered perfect fifth, P5 (~700c)". I don't see what "to map" means here, but at least we seem now to be turning to 12-TET, although I fail to see why that puts the 3d partial on G3 (etc.).

D. "In the syntonic temperament [...] the syntonic comma is 'tempered out' (to zero)". We often discussed that before. From the description given, it is the syntonic comma between the Pythagorean and the perfect major thirds that is "tempered out", but not the one between the Pythagorean and the perfect minor third. Up to there, the syntonic temperament ressembles just intonation (just perfect fifths and perfect major thirds), but only for the C triad.

E. "The 9th partial falls between the 8th and the 10th partials" – well, is that not a property of whole numbers, that 9 falls between 8 and 10? I thought that the "5-limit" system was defined by using partials up to 5 (and their duplicates) — or, as Zarlino and others described it, up to 6 (the senario), with 6 obviously duplicating 3. The difference that you seem to introduce is that 9/8 is not described as (3/2)², but as corresponding to the 9th partial...

F. "The 5-limit syntonic temperament has a tuning range of 34 cents". Here, I am completely at loss: how is this range determined? It appears to be a range of values for the perfect fifth, from 686 to 720 cents, i.e. from -16 to +18 cents compared with the just perfect fifth, but where do these figures come from, and what is the purpose of widening the just perfect fift?

I stop here, because what follows, about adding a second comma, becomes unintelligible to me: where? how? why? That the resulting "temperament" is isomorphic merely seems to mean that it is a temperament of fixed intervals (and fixed notes). But with tempered fifths (686 to 720 cents, or 695 to 700 cents), it cannot be fully isomorphic: unless for fifths of 700 cents, there must be a wolf somewhere.

I'd very much like to understand, Jim, and I'll be grateful for any explanation. — Hucbald.SaintAmand (talk) 17:49, 22 July 2021 (UTC)

Venerable Huc,

I have taken the liberty of editing your questions above ONLY to identify them with unique letters A-F, so that I could refer to them as I (attempt to) answer them.

These are all excellent questions, which were all answered in our peer-reviewed publications on Dynamic Tonality (which had not yet been named as such), notably "Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum" (Milne et al., 2007), which is available in full, for free, at the given link, to which I refer you for details and mathematical proofs. I will refer to that article henceforth as "ICDT." I will attempt to summarize the key points of ICDT here VERY briefly.

ICDT starts by describing the two research questions that we, its authors, set out answer in our research:

1. Is it possible to create a keyboard surface that is capable of supporting many possible tunings?
2. Is it possible to do so in away that analogous musical intervals are fingered the same throughout the various tunings, so that(for example) the 12-TET fifth is fingered the same as the just fifth and the 17-TET fifth?

Many of your questions above arise, it seems to me, from the issue of "analogous musical intervals." This is, in fact, the first of four issues that we had to resolve in order to answer our above-listed research questions. We don't claim that these resolutions are universal; they are the resolutions that were necessary to make Dynamic Tonality possible. They are not the only possible resolutions.

(Similarly, Euclidean geometry is not the only geometry. Negate its parallel postulate, and one gets a variety of Non-Euclidean geometries that enable practical solutions to real-world problems that Euclidean geometry could not solve—for example, the use of spherical geometry in oceanic navigation.) Dynamic Tonality is to the Static Timbre paradigm what Non-Euclidian geometry is to Euclidean geometry.)

ICDT lists four issues that must be resolved to answer the research questions above, which I will summarize here as follows:

1. Intervals: There must be a number of distinguishable intervals by which the invariance can be measured. For example, it must be possible to say that the P5 in 12-tet, 17-tet, quarter-comma meantone, etc. are all analogous in some musically-useful way to Just Perfect Fifth (3/2). (If one insists that these are all unique intervals, not "the same" in any musically-meaningful way, then one cannot gain the musical benefits of Dynamic Tonality. To get Non-Euclidean geometry, parallel lines must cross, so to speak.)
2. Tuning System and Temperament: There must be a tuning system that is itself transpositionally invariant with regard to both rational and ordinal modes of interval identification (the latter terms being discussed in detail in the article).
3. Scales: There must be musically-useful sets of scales in said tuning system.
4. Keyboard: There must be mapping of said tuning system to the note-controlling devices of a keyboard that maintains transpositional and tuning invariance.

Each of these issues is addressed, sequentially, in sections of ICDT. I am not going to recapitulate those sections here. Rather, with the information above, I can address your questions by reference to that paper's sections.

A1. You state, "I think that we should distinguish the theoretical description of temperaments from their practical realization."

When executed using a synthesizer compatible with Dynamic Tonality—meaning that its timbres' partials can be aligned with the tuning's notes—then there is no difference between the theoretical temperament and its practical realization. The difference that you seek to recognize is a weakness of the Static Timbre paradigm, arising from the non-alignment of its timbre's partials with its tuning's notes in any tuning except rank-∞ Just Intonation.

A2. You state, "So far as I know, all temperaments are theoretically based on the octave."

Please allow me to draw your attention to the Bohlen–Pierce scale, which is not based on the octave.

B. You state, "I begin to loose foot, because up to there the "syntonic temperament" (I am not sure to understand what that is) has untempered intervals both as its period and its generator: is it a temperament?"

Syntonic temperament: Please allow me to draw your attention to the definition of the syntonic temperament, which appears on page 21 of ICDT. The section of ICDT titled "Tuning Systems and Temperaments" lays the groundwork for that definition.

Just and tempered intervals: Addressed in the sections of ICDT titled "Intervals" and "Tuning Systems and Temperaments." The JI intervals are the Platonic ideals, signified by the tempered versions thereof.

Is it a 'temperament'?: Dynamic Tonality's definition of a "tuning system" and a "temperament" are discussed in the section of ICDT titled "Tuning Systems and Temperaments." I will admit that what is defined, re "temperament," is the more-specific term "regular temperament," as follows:

The embodiment of such a temperament-mapping in a suitable tuning system is called a regular temperament, and it can be characterized by the small JI intervals called commas that are tempered to unison (Smith 2006).

I will plead guilty to consistently using the term "temperament" instead of the more-specific "regular temperament." Please let me know if using "regular temperament" consistently instead of just "temperament" would address some of your concerns.

C1: You state, "I don't see what "to map" means here."

The mapping of rank-∞ Just Intonation to a lower-rank [regular] temperament is discussed in the section of ICDT titled "Tuning Systems and Temperaments."

Please allow me to quote from page 20 of ICDT: "This is analogous to the way a projection of the three-dimensional surface of a globe to a two-dimensional map inevitably distorts distance, area, and angle. However, so long as the countries have identifiable shapes, the projection can be considered valid. Different map projections result in different distortions, and some map projections are more or less suitable to specific purposes. Some projections (such as the Mercator Projection) have the virtue of wide familiarity; so it is also with temperament-mappings (such as those that lead to 12-TET)."

C2: You state, "I fail to see why that puts the 3d partial on G3."

See page 21 of ICDT, searching for the text "A second example is in 5-limit JI". The quick answer, in the least-mathematical language possible, is that the series of generators (period, generator, commas in the comma sequence) define the notes to which the successive prime partials are mapped. In the syntonic temperament, given a fundamental (1st partial):

• the first generator (the period) determines the note to which the 2nd partial is mapped (the octave);
• the second generator (the generator) determines the note to which the 3rd partial is mapped (the fifth); and
• the third generator (the first comma in the comma sequence, in this case the syntonic comma) determines to which note the 5th partial is mapped (the major third). If the third generator had been the schisma instead of the syntonic comma, then the result would have been the Schismatic temperament, in which the 5th partial is mapped to the diminished fourth.
• the fourth generator (the second comma of the comma sequence) determines the note to which the 7th partial is mapped (the augmented sixth, if the septimal semicomma, also known as the starling comma, giving Septimal meantone temperament (if the previous comma was the syntonic comma).

I can't explain the math behind the mapping of the comma sequence's commas to notes, because I don't understand it myself. I will refer you instead to the "Tempering by commas" section of Tuning Continua and Keyboard Layouts, here. Personally, I suffer from "math blindness," which is akin to color blindness: all mathematical formulae more complex than "1+1=2" look essentially the same to me: impenetrable. I am overstating for effect, but you get my meaning. My co-authors did all of the heavy mathematical lifting.

D: I do not understand your question, sorry; could you please rephrase it?

E: I do not understand your question, sorry; could you please rephrase it?

F: You state, "how is this range [the valid tuning range of the syntonic temperament] determined?"

Please see the section of ICDT titled "Scales" (p. 21), and also "Tuning Continua and Keyboard Layouts," section "Valid tuning range for consistent fingering" (page 4). In brief, for the syntonic temperament, the endpoints of the range are the tunings at which a diatonic interval shrinks or grows to become indistinguishable from another diatonic interval. In the syntonic temperament, the endpoints are the tunings at which the minor second shrinks to unison (5-tet) and (at the other end) the minor second grows to equal the major second (7-tet).

Please let me know, Huc, what further questions these answers spark.

Respectfully,

--JimPlamondon (talk) 05:32, 23 July 2021 (UTC)

@JimPlamondon: Venerable Jim ;–)), I should make it clear that I am not particularly interested by Dynamic Tonality. I had read your (and your colleagues') books and I have some idea of what Dynamic Tonality can achieve. However, I had choices to make in my career as a music theorician, I have for 20 years worked in the domain of the history of music instruments, and for 20 other years in that of the history of theory, and music for synthetizers merely does not figure in the choices that I had to make. I have been interested on the other hand by our discussion and by the challenge of explaining such complex things in simple terms for WP. Let's therefore forget the idea of convincing each other, that's not the point, and merely enjoy the discussion. Let's review my questions and your answers.

• A1 "When executed using a synthesizer compatible with Dynamic Tonality—meaning that its timbres' partials can be aligned with the tuning's notes—then there is no difference between the theoretical temperament and its practical realization". Of course, unless that in that case you don't really struggle with the problem of making the best of turning pegs reluctant to turn, of distributing tension on the length of strings reluctant to slide on their bridge, of comprimising inharmonic partials that fail to form any regularity, etc. Adjusting a synthetizer in the way implied by Dynamic Tonality has little in common with the task of a piano tuner. You reject this task together with the "Static Timbre Paradigm", which seems to assume that musical notes produce harmonic partials. If you believe that seriously (I mean, harmonic partials), then there is little that remains to be discussed. Harmonic partials suppose strictly periodic vibrations, which merely are a hypothetical view of the mind.
• A2 Bohlen–Pierce scale. Well, any truth has its exceptions.
• B "The JI intervals are the Platonic ideals, signified by the tempered versions thereof." Do you mean that untempered intervals are "signified" by their tempered versions? I don't think that this can be explained by the restriction to "regular temperaments". As we already discussed before, I don't understand what tempering commas to unison could possibly mean. 1/4-comma meantone tempers the fifths by 1/4 comma, with the result that 1/4 comma fifths are a syntonic comma lower than their Pythagorean equivalent (this difference, by the way, being a definition of the syntonic comma. I don't see how all that could be described as "tempering commas to unison", I fail to see how tempering fifths by 25% of a comma (which in this case is a measure) could make the comma vanish – or what would be the dimension of 25% of a comma "tempered to unison".
• C1 "Mapping partials to notes." Well, let's leave that one aside for the time being.
• C2 I can understand that the 2nd partial of C2 is C3, that its 3d partial is G4, its 5th partial is E5, etc., and that this may remain true for so long as the partials are not way too inharmonic. What I don't understand is why there should be a particular justification for this and, if that is what is meant, while this would be particularly true in any specific tuning or temperament.
• D Once again, "tempering to zero". One never tempers the syntonic comma itself. For instance, one tempers the 5ths, say, by 1/4 comma, which results in lowering the major 3rds by a full comma, and in raising the minor thirds by 3/4 comma. I utterly fail to see how the comma is "tempered to zero" in all this: the comma should be considered as the measure of a distance rather than as an interval properly speaking. Your video shows how the 2nd, 3d, 5th (and 9th) partial of C2 can be "mapped". But what is it that allows to say that the same is true for all fundamentals other than C2? It seems to me that either you are speaking of some ET (possibly with more than 12 notes), then indeed this is true, or the "mapping" is "dynamic", i.e. may depend on the particular notes considered, then it is not true.
• E What has the 9th partial to do in a 5-limit tuning or temperament?

Mathematics is a formal language, and equations may in the end appear to have a meaning of their own. What keeps me puzzled is the relation to "real life", to real instruments, with real strings, really inharmonic partials, etc. – and, perhaps also, to real sythetizers, heard through real loudspeakers, in real acoustic environments, etc. Best, Hucbald.SaintAmand (talk) 21:05, 24 July 2021 (UTC)

Venerable Hucbald,

Many of our disagreements/misunderstandings come down this one or both of two things:

1. You assert that it is impossible to "temper a comma to zero," where I assert that it both possible and useful.
2. You assert that Dynamic Tonality is nonsensical unless it limits itself to the constraints imposed by traditional instruments; I assert that compatibility with traditional instruments is the least of Dynamic Tonality's concerns, so long as the SOUNDS of those instruments (that is, their partials) can be reproduced and manipulated digitally (e.g., via TransFormSynth).

TEMPERING A COMMA TO ZERO

You define a comma as a ratio of small whole numbers. Such a ratio can never be "tempered to zero," I agree. So, don't use that definition. Instead, define a comma as the difference between two other intervals. That "difference" can be tempered to zero by expanding one or both of the two comma-defining intervals. End of problem. This latter approach is entirely in line with Wikipedia's definition and discussion of commas, and with long-standing usage in the microtonal community (see here, here, and here).

I agree that I need to find a way to explain this in a clear, concise, simple way, and that I have thus far failed to do so, for which I apologize sincerely. It's a fundamentally mathematical issue, and math has never been my strong suit. To be able to explain it in non-mathematical terms, I would need to understand the mathematics completely, and I don't. That's my fault, not any reader's fault.

COMPATIBILITY WITH TRADITIONAL INSTRUMENTS

This one is easy: I do care one whit -- not the slightest little bit -- about making Dynamic Tonality compatible with traditional musical instruments. Digital synthesis is now so good (because computers are now so powerful) that only the most expert listener can distinguish the recording of an acoustic instrument from the recording of a modern-digitally-synthesized instrument. Even acoustic instruments are, more often than not, amplified (and hence, inevitably, distorted, more or less) to reach the ears of the listener in a live setting. I care only about making Dynamic Tonality compatible with the recorded SOUNDS of traditional instruments, not with the instruments themselves. The sounds are what matter; the instruments themselves are just a physical means of producing sounds, easily replaced by digital means.

I understand that you, as an expert in traditional musical instruments, find this line of thinking to be horrifying, just as dairymen are horrified by the thought of precision fermentation disrupting traditional dairying and the engineers of internal combustion engines are horrified by the thought of electric vehicles disrupting traditional cars. And yet it moves. I appreciate your helping me understand your position, because I will surely encounter frequently in the years ahead, and I am not sure how to turn it around. Thomas Kuhn said that old & new paradigms are 'incommensurable' "if they are embedded in starkly contrasting conceptual frameworks whose languages do not overlap sufficiently to permit scientists to directly compare the theories or to cite empirical evidence favoring one theory over the other." That sounds like what is happening here. However, even if our paradigms are incommensurable, it is incumbent on me as the challenger of the status quo to attempt to overcome the incommensurability, or at least to explain the differences is a clear and concise manner. I suspect that I am failing to do that, and I appreciate your patience in helping me work it out.

As to your specific questions:

• A1 I wrote, "When executed using a synthesizer compatible with Dynamic Tonality—meaning that its timbres' partials can be aligned with the tuning's notes—then there is no difference between the theoretical temperament and its practical realization". You replied, "Of course, unless that in that case you don't really struggle with the problem of making the best of turning pegs reluctant to turn, of distributing tension on the length of strings reluctant to slide on their bridge, of comprimising inharmonic partials that fail to form any regularity, etc." Exactly! Ain't Dynamic Tonality grand? As Bach said, "All one has to do is hit the right keys at the right time and the instrument plays itself." Admittedly, any DT-compatible synthesizer that sought to credibly simulate the sounds of a piano would have to model the piano's weaknesses, such as the deviation of its highest and lowest strings from proper harmonicity as per the Railsback curve. However, you seem to be claiming that these weaknesses are strengths. I do not understand how that could be the case. Please explain.

• B I wrote, "The JI intervals are the Platonic ideals, signified by the tempered versions thereof." You replied, "Do you mean that untempered intervals are "signified" by their tempered versions?" Yes, as described in the reference provided.

• C2 You wrote, "I can understand that the 2nd partial of C2 is C3, that its 3d partial is G4, its 5th partial is E5, etc., and that this may remain true for so long as the partials are not way too inharmonic. What I don't understand is why there should be a particular justification for this and, if that is what is meant, while this would be particularly true in any specific tuning or temperament." The justification is that said alignment enables dynamic changes in tuning while maintaining consonance, which expands the frontiers of tonality, thereby liberating musical creativity to explore those frontiers. The association of a given partial with a given note is defined by the temperament, which is defined by the comma sequence, which is the list of commas to be "tempered to zero" (see above).

• D You wrote, "Once again, 'tempering to zero'." See Tempering commas to zero, above.

• E You wrote, "What has the 9th partial to do in a 5-limit tuning or temperament?" The prime factorization of 9 is 3*3, and 3 is lower than 5, so the mapping of the 9th partial to a note of the temperament is defined by a 5-limit temperament (unless I am missing something).

Respectfully,

--JimPlamondon (talk) 07:06, 25 July 2021 (UTC)

Venerable @JimPlamondon:,

• The matter of "tempering the comma to 0" merely is one of logic. I am ready to define the comma as the difference between two other intervals. Let's agree that the syntonic comma is the difference between the Pythagorean major third and the just major third. As soon as you modify one or the other of these thirds (or both), the modified third(s) cease to correspond to their definition (the Pythagorean third ceases to be Pythagoran, the just major third ceases to be just) and the comma itself ceases to be the syntonic one. In addition, this is not achieved by tempering the comma itself, but by tempering the fifths – there exists no historical temperament that is not a temperament of the fifths (or of at least some of them). And when the interval in question is reduced to zero, it ceases to be a comma, it becomes a unison. To define the unison as one type of comma among others really stretches the logic beyond what seems acceptable.
  • An additional problem is that it is often said that, say, 1/4-comma meantone "tempers the comma to 0" (or "makes it vanish"). But the major third reached by 1/4-comma meantone is still a syntonic comma lower than the Pythagorean one: the definition of the syntonic comma as the difference between those is not affected by the temperament. It is you who wanted to define the comma as the difference between two other intervals (and I agree with that definition), but you cannot logically maintain it to the end.
  • Also there are at least two syntonic commas involved in meantone temperaments, one between the Pythagorean and just major thirds, and one between the Pythagorean and just minor thirds (plus those beween sixths, but let's admit that these are the same). I won't repeat that any meantone temperament modifies each of these two differently. If 1/4 comma-meantone fully diminishes the major third from Pythagorean to just, it cannot at the same time widen the minor third from Pythagorean to just. "Tempering" one of these commas "to 0", as you say, cannot achieve the same for the other.
• We agree that Dynamic Tonality has and can have nothing to do with traditional (acoustical) instruments. I merely think that it should be said more clearly. When a WP reader writes us that he is a piano tuner and has problems with Dynamic Tonality's tempered octaves, he should be told mor clearly that the inharmonicity of piano strings is not of the same order as what Dynamic Tonality considers. You propose that Dynamic Tonality represents a "new paradigm" that will eventually replace the "old paradigm" of acoustic instruments. Yet, you recently added a section on Dynamic Tonality to the Classical music article: how would you define "Classical", in Classical music? Don't you think that Classical music as such is an "old paradigm" (an obsolete paragigm perhaps? Not yet, I think).
  • It seems to me also that Dynamic Tonality may put too much confidence in digital synthesis. It is true that all what truly is digital in this is now in full control and that the time when the high frequency vibrators that were divided to produce the frequency of, say, ET, where not high enough to produce a perfect digital approximation (I have known that time, I did solder quartz oscillators and frequency dividers that were meant to approximate ET). The problem is that our hearing isn't digital yet, and that synthetizers have to feed acoustic interfaces in an acoustic environment. Loudspeakers, if they are not of outstanding quality, do add partials to the sounds that they emit. They also have directional characteristics that may alter the heard spectra. And they usually diffuse in acoustic environments that are not anechoic chambers. Dynamic Tonality is theoretically satisfying, but in practice?

It seems to me that all the rest ensues from this. Let's however try to understand each other. — Hucbald.SaintAmand (talk) 17:53, 25 July 2021 (UTC)

Hi Jim -

Sorry I was so slow in responding. I do not check back here frequently.

I'm going to move straight to your last question regarding Sethares's book opening and piano tones. You asked if Sethares could have used piano tones in his example.

Sethares could not have used piano tones, because he deliberately manipulates overtones, and that then impacts both timbre and pitch.

Manipulating overtones immediately impacts pitch due to the well-researched "missing fundamental" phenomenon: We do not merely derive our sense of pitch from the fundamental. We derive pitch from the "whole package," which Sethares manipulates more aggressively than his language would suggest.

From my perspective, he degrades not only the feeling of an "integrated tone," but pitch as well in an opening gambit regarding pitch for an entire textbook. The "sleight of hand" which I do suspect unintentional is two-layered:

(1) Choosing a large interval which even trained singers can sometimes struggle at delivering well and,

(2) Manipulating pitch vis-a-vis overtones and then pointing down at the fundamental, "See! The pitch is..." And when I listen carefully, his octave challenge fails out of the gates, and neither a different timbre nor register would remedy that.

Wow.

As before, there's a lot of stuff Sethares does that is fun, cool, even thought-provoking. The reason I like Sethares: He's pretty fearless. And I do like people who fearlessly search for answers. I also, as a musician, simply disagree vigorously with a lot of his basic conclusions.

Regarding other tuning systems: I think alternate tuning systems are wizard cool. I like avant-garde jazz. I think Indonesia's gamelan music is beautiful. I play a jazz-blues blend myself littered with minor seconds wherever I can squeeze them in.

...but I do not perceive a lot of room for manipulating/bending the basic language of "consonance vs. dissonance." I do not ever see Sethares's concepts dominating or even substantially altering this corner of the musical lexicon.

My own rather simple work will attempt to show, in stark physics terms (including intermodulation distortion), why the simple basic fractions that dominate western music (3/2, 4/3, 5/3, 6/5, 7/5, 8/5, 9/5) in combination with the overtone series seize our attention and our sense of consonance and lead us directly to a 12-tone system. Doesn't mean other systems aren't not very interesting and even beautiful.

• Does* mean our biology lead us straight where we already went: The small, simple fractions are where consonance arises for reasons of both physics *and* biology.

We are very creative as a species, so we experiment with art/forms and meaning. That's *really* what Sethares is doing, experimenting more than analyzing, unless reviewing music from other cultures.

But suspecting that humanity might be drawn to substantially more pitches or other equal temperament systems besides 12 TET, is nearly perfectly analogous to suggesting that cubism or abstract expressionism will one day dominate visual art because humanities more traditional forms have grown stale. The truth is, we are drawn to classical living forms in every aspect, because living things exchange meaning. We could not escape a sense of pitch wound up in our biology more than 200 thousand years ago even if we wanted.

We came to what we were biologically drawn to as broadest musical form: 12-TET.

From established Duke and Stanford researchers - Another exploration (mine angle be different, conclusions very similar) suggesting that small fractions are "baked in" to the way we hear pitch: We may even "talk" in small fractions" in everyday speech ==>>

https://www.pnas.org/doi/10.1073/pnas.1505768112 KWSager (talk) 20:47, 16 May 2022 (UTC)

PLEASE FORGIVE TYPOS -- WORKING TOO FAST... KWSager (talk) 20:51, 16 May 2022 (UTC)

400 page views a day vs 5

https://pageviews.toolforge.org/?project=en.wikipedia.org&platform=all-access&agent=user&redirects=0&range=latest-20&pages=Consonance_and_dissonance%7CDissonants .. really dissonants should redirect here. In ictu oculi (talk) 22:45, 1 December 2020 (UTC)

Feel free to raise this matter at WP:RFD. Toccata quarta (talk) 06:37, 2 December 2020 (UTC)

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