Talk:Consonance and dissonance/Archive 1

Archive 1

Archive 2

Notation

In the Symbol Sourcebook (ISBN 0-4712-8872-1), Henry Dreyfuss asserts the existence of a symbol in musical notation for dissonance. He denotes it by two notes with their stems converging at a common point, with each notehead flanked by a different accidental symbol. Can anyone corroborate such a symbol?  Denelson83  07:27, 1 December 2005 (UTC)

Split

Please keep this as a single article. — Omegatron 20:14, 6 July 2006 (UTC)

I agree, oppose split. Also, that template is meant for pages about different topics with the same name, which is why it mentions disambiguation. In this case, a disambiguation page would make no sense, because consonance and dissonance have different names but nevertheless belong together. This is an encyclopedia, not a dictionary. —Keenan Pepper 01:16, 7 July 2006 (UTC)

It seems to me, the interwiki links are totally wrong because the article is not splitted. There is enough room for both of the articles. Agree to splitting. Avjoska 07:53, 21 May 2007 (UTC)

Example dissonance

For the examples of baroque, classical era, and modern dissonance, can someone please add sound files that correspond to each? 66.72.192.130 (talk) 18:50, 15 February 2009 (UTC)

Midis provided. Hyacinth (talk) 03:09, 6 July 2009 (UTC)

The Bach excerpt on the page (Prelude XXI)

I looked at the sheet music for the piece, and the one I've seen doesn't match the excerpt. The first chord should have an A♭ rather than an A♮. Also, this midi file seems much faster than normal (the prelude is mostly 32nd notes, after all). Aurora Illumina 03:24, 16 January 2010 (UTC)

Never mind, changed it myself. Except the image now shows the wrong notes, and I don't have an easy way to edit that. Aurora Illumina 01:53, 1 February 2010 (UTC)

most articles on music theory on wikipedia...

suffer from a serious case of too much classical lore. i know, it's easier to write about it but it should be confined to subsections with names such as "con/dis in medieval music" etc. —Preceding unsigned comment added by 195.66.94.4 (talk) 08:39, 26 August 2010 (UTC)

Diverging tones graphic

Why is the diverging tones so split off from the text by the table/outline? Hyacinth (talk) 01:14, 7 July 2008 (UTC)

Speaking of which... re/ the graphic under "Diverging tones"... despite its poor image quality, it is very interesting to inspect closely. How was this generated? In the name of verifiability, I think the reader deserves an explanation of Blackman Spectral Analysis or a reliable source for this diagram or something very close to it. --Ds13 (talk) 01:19, 7 July 2008 (UTC)

While I have only begun to look through the music theory articles, my impression so far is that they tend to dabble superficially in highly theoretical matters (like this) that can not possibly be made clear in articles of this size and generality, while treating the fundamentals in a way that is at best foggy and at worst simply erroneous. Fenneck (talk) 19:04, 26 September 2008 (UTC)

I've restored the deleted description on Commons:File:Dissonance-a220-a440-notated.jpg The captions and quality of this image are pretty poor, and I think the subject matter is rather misleading. If there are overtones, then these are not (quite) sinusoids. The fact that you've got lines going up and down at the same time means that there is significant aliasing, too. This is because the tones were generated in Cool Edit, which does a "naive" waveform generation instead of a bandlimited waveform. This seems to be the reason for the pronounced effect at consonances, not the consonance itself. I'm not sure if these bands are actually even due to consonance, or if they're due to the waveform frequency being a submultiple of the sampling frequency at those points.

I tried to recreate this image with sawtooth waves and no aliasing, and the "bands" of consonance are not visible. Vertical lines could be drawn to show the coincidence of overtones at each consonance, but I'm not sure if that would be helpful.

Then you've got the lossy nature of both the ogg vorbis and the .jpg... — Omegatron (talk) 02:34, 31 July 2009 (UTC)

It has nothing to do with overtones in the original signal. As you can see from the original sine wave, there are only even-order harmonics, which suggests that the sine wave in question is more of a square wave. However, I doubt that he simply used square waves to do this experiment, due to all of the sum and difference tones present (which forms the basis of this guy's argument).

What is really going on, most likely, is that this guy performed the experiment with two sine waves that were clipping. He probably had each sine wave at full volume, so that when the two were added together, the signal clipped a bit. This is sort of like running the signal through a guitar distortion pedal, which explains the combination tones. So this graph doesn't really explain consonance or dissonance at all. It is simply a graph of what intervals will wound maximally "coherent" when run through what we DSP types call a "nonlinearity" -- think some kind of distortion or clipping here -- and which ones will sound maximally incoherent.

As anyone who has ever tried to play a tritone on a distorted guitar will attest, it sounds like a bunch of noise and generally sounds awful. The graph reflects this. On the other hand, fifths sound nice, and so do octaves. This is what we are seeing, but the metaphor doesn't hold true for actual consonance and dissonance - a maj7 chord, for example, will sound spectacular with a clean guitar tone, but when distortion is added it will sound awful. Same with a minor triad.

In short, this entire image and explanation and everything about it is completely inaccurate and just plain wrong. It's a nice guess, but it is, unfortunately, not what is happening. The theory that nonlinearities are responsible for consonance and dissonance has been considered out-of-date for years now (see [Missing Fundamental] for more info on that). It sounds nice and will probably pass for truth among many musicians, but it is, at its core, inaccurate.

I say we get rid of it, but I don't know what to replace it with. Battaglia01 (talk) 01:41, 14 January 2010 (UTC)

Just noticed that the "sinusoids" are actually supposed to be filtered triangle waves. Nonetheless, some nonlinearity is clearly taking place here. I don't think it's aliasing, rather I think some kind of harmonic distortion is present. Most likely the OGG compression either added some combination tones, or the guy screwed up when rendering the file and there was clipping. Battaglia01 (talk) 04:15, 14 January 2010 (UTC)

I second your suggestion for just getting rid of it. When you have to wonder about what the picture really shows, it simply isn't contributing to the informativeness of the article. You may be right that the waves are supposed to be triangle waves, but the description still says sinusoids and that is clearly just wrong. We may not have anything better at the moment to replace the picture with, but I think we are better off with nothing at all here than with confusing and misleading information. Kevin Nelson (talk) 07:59, 10 February 2010 (UTC)

The graphic text, "sinusoid waves" should at least be modified. It conflicts with the graphic text, "Overtones". Sinusoid waves don't have overtones. The graphic doesn't seem to go with the article text either. Maybe it should be sent to the cornfield after all. I third. There are some nice illustrations in Helmholtz (page 193) of "roughness" (dissonance) curves across interval space--but those are copyrighted aren't they? What's the statute of limitations?. Maybe someone can recreate them for wikipedia. I think making x the frequency of the moving tone and y the beating frequency between the moving and non-moving tones would do it. Helmholtz' illustrations spanned two octaves. Another Stickler (talk) 10:35, 13 February 2010 (UTC)

I fourth that the Blackman spectral analysis graphic be removed, because it is original research. The methods are not well described and the observed effect is surely some sort of artifact. In a computer generated audio signal, the 'dissonance' frequencies are there because they were (likely unintentionally) programmed to be there. The presence of the overtone at about 660 Hz (and the higher one) is disturbing and not explained.75.13.130.16 (talk) —Preceding undated comment added 14:57, 29 December 2010 (UTC).

I see it's one year later and this is still around - I'm going to be bold and get rid of it. What a misleading graphic. Battaglia01 (talk) 22:38, 12 January 2011 (UTC)

consonance as zero value

consonance is to dissonance as zero is to infinity. chords can not be but can move towards. an octave is dissonant. a unison is as close as you can get to consonance. I am new to both wikipedia editing and music theory. hopefully some smart people will read this and tell me if this is already understood as common knowledge, or another way of using the words(symbols) in our simple Sweetgum75 (talk) 21:01, 16 February 2011 (UTC)language.

I'm new too, so my apologies for any guidelines I may violate. I think the "consonance is to dissonance as zero is to infinity" statement should say something like "consonance is to dissonance as complementaty colors are to clashing colors". Or perhaps, simply remove it. Thanks. Jon 23:43, 24 February 2011 (UTC) —Preceding unsigned comment added by 74.192.140.32 (talk)

I kind of agree. The statement is too poetic, and not necessarily correct. It would probably be more correct to say something like, "Consonance is to dissonance, as pi is to chaos," or "... as circles are to irregular shapes." But then, proteins are irregular shapes, and no biologist would say proteins are dissonant. And, of course, chaos can't be said to be truly dissonant either (ex, white noise isn't dissonant).

Zero would be absolute silence, and absolute silence is neither consonant nor dissonant (unless you hold the view that silence isn't truly silent - then zero would actually equal infinity).

As I understand dissonance, it's not the absence of everything (zero); it's just the presence of something that doesn't belong - like an off-key singer in a choir or a toxin in a biological system. It's a bit like throwing something into an equation that simply doesn't belong: e=mc^2(linoleum). See? Linoleum doesn't fit the equation. It makes the equation dissonant and throws the whole universe off.

Since there is no citation, the statement should probably be removed. Most likely, it's just someone trying to sound clever. Or, you could say the author of the statement was just trying to be dissonant, in which case it's very clever. Mpilting (talk) 17:48, 5 March 2011 (UTC)mpilting

fun!! glad to hear your opinions. let me first explain why i added it. i was trying to understand the meaning of these words with respect to their use as being subjective or objective. it took awhile of reading for me to come to the conclusion that they might be objective. to cite my source is the paragraph above it. (Latin com-, "with" + sonare, "to sound") and (Latin dis-, "apart" + sonare, "to sound"). my reasoning is: how close is "with"? how far is "apart"? hence zero and infinity. really just trying to make this more accessable to everyone.

second let me respond to your comments:

"complimentary colors are to clashing colors" good example. color and sound might even be the same forces, just that color is a much higher wavelength. color theory gets confusing, we can talk additive or subtractive (its about light and/or pigments) as far as light goes its a continuum (a spectrum,zero to infinity) in the practice of mixing pigments we use the model of a color "wheel". in the wheel model, the farthest "away" you can be is directly across a.k.a. complimentary color, the closest "together" would be considered analogous.

"Pi is to chaos" after first reading i was like "no way". then it hit me yes you are right(i think) shit dog,i am tripping out on it! (: seriously, its making writing very difficult for me. if it's what i think, yours is a three(or more) dimensional model, mine is two. you might even be talking about god. i think i just shat myself.

what is the difference between sound and music? we make a distinction between the two, yet undefinable. i thinks its simply called sound when speaking objectively, and music when speaking subjectively. if i was to cite specific work, it would be that of Elaine Walker and her work on the Bohlen-Pierce scale: an interesting 12tet alternative discovered independently by two different microwave engineers (sound-color-microwaves as to lions-tigers-bears?????)

how can i cite these properly?

maybe it should read:

Objectively speaking, consonance is to dissonance, as zero is to infinity. Subjectively speaking, consonance is to dissonance, as yes is to no

too confusing!!! "black-white-chicano-hell-if-i-know" -lyrics born

(g7 was diss but now is con) g7 never changed our opinions did.

we are talking about intervals. so we really need to know the chord before the one we are talking about. on which side of the spectrum, which direction are we heading? towards or away?

p.s. yes i am suggesting that a unison is as close as you can get to consonance, and an octave could be considered dissonant - subjectively speaking of course. 1/1 does not equal 1/2 which is an octave,Sweetgum75 (talk) 03:13, 17 March 2011 (UTC)

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Typo/Mix-up

• Consonance has been defined variously through: With...

The article currently reads as above. Hyacinth (talk) 00:34, 31 March 2012 (UTC)

Found its origin in the edit history [1] and corrected. Hyacinth (talk) 00:51, 31 March 2012 (UTC)

"Pure tones cause neural firing exactly with the period or some multiple of the pure tone."

This sentence seems to imply that T_neurons=k*T_sinusoid, with k an integer (if not an integer, or let alone a constant, I would probably agree: our neurons gotta fire sometime if our brain is to interpret the world its input senses sense).

I thought our auditory sensors primarily measured intensity in spectral domain, further I thought that at the lower cognitive levels frequency is communicated by connectivity not by signal... I could be wrong.

This sentence gives me the impression that what is insinuated is that at every say peak of the sinusoid, the neuron fires. Like a Ruth Goldberg Machine. —Preceding unsigned comment added by 157.193.10.29 (talk) 21:40, 2 May 2011 (UTC)

Ruth appears to be a film scholar, but I bet you meant Rube. Hyacinth (talk) 00:55, 31 March 2012 (UTC)

Minor second is a bad example

While this example works in equal temperament because all minor seconds are dissonant, I think it's confusing to cite minor seconds as being dissonant. In other tunings the interval C-Db is perfectly fine, it's the interval C-C#, augmented unison that is dissonant. Hogdotmac (talk) 18:07, 18 August 2014 (UTC)

This is fairly sensational news to me. Do you have a reliable source to verify this?—Jerome Kohl (talk) 22:27, 18 August 2014 (UTC)

Well, I admit there is a lot of subjectivity to it and equal temperament is perhaps not really that dissonant as I say, and there is some grey area between the boundaries of consonance and dissonance. But if you compare the minor second to intervals that are clearly dissonant, those intervals even sound awful when you play the notes in consecutive order, such as the augmented unison does in meantone temperament in a trill/mordent (and hence you will almost never find such an occurrence in renaissance music), while these are common with minor seconds. Hogdotmac (talk) 14:28, 19 August 2014 (UTC)

I take it this means you have no source for this claim. Still, how do you square this claim with the more traditional classification of tritones, major and minor seconds and sevenths, and unsupported perfect fourths as dissonances (I presume these are the "clearly dissonant" ones), as against the consonances of perfect unisons, octave, fifths, and supported fourths, along with major and minor thirds and sixths? It sounds to me like you may be positing a continuum from most consonant to most dissonant, and are then splitting the finest of hairs at one end of this spectrum. When you re-cast this in black-and-white terms ("any interval is either consonant, or it is dissonant"), you end up with one dissonant interval and all the rest consonant. A little too original for my tastes, if this is really what you are claiming, and certainly not in line with conventions of musical theory or practice.—Jerome Kohl (talk) 16:17, 19 August 2014 (UTC)

Perfect fourth as dissonance

In the current article: "The perfect fourth is considered a dissonance in most classical music when its function is contrapuntal."

This is vague. What is a "contrapuntal function" for an interval? Quite simply, a perfect fourth is generally considered a dissonance in classical music when it is the interval above the bass. I can only assume that "contrapuntal function" is somehow trying to hint at the fact that the fourth is not part of the chord (as in a sus4 chord in jazz or whatever), but even that is inaccurate for "classical" music. Why? Because a fourth between two upper voices, even in a "contrapuntal" context (e.g., Bach fugue) is a perfectly acceptable consonance. It is only the interval with the bass that is generally considered dissonant. 65.96.161.79 (talk) 02:17, 7 February 2011 (UTC)

Hindemith showed that (tuning specifics aside) the P4 is innately consonant. In some counterpoints, the matrix (math) mechanics sometimes require the 4ths to **resolve according to rules as dissonances** to prevent bad harmonic progressions or final sonorities, and so P4 is treated as dissonance in that sense. Just which 4ths are subject to the restriction varies with author and his rules, as do the related injunctions about parallel motion and 6/4 inversion. In some texts on species CP, a terminal 4th in the upper voices is bad in spots; in some texts on ornamented counterpoint and fugue, concentrations of 4ths and 11ths in the submetric notes or the hypermetric motif layout (entrances and strettos) are still problematic. But remember: said rules apply ONLY when resolution is required (translated above as: "its function is contrapuntal"), and that being left undefined. In counterpoints using parallel 4ths planing, quartal chords, Gothic consonances, or those avoiding dominant-tonic tonal orientation ("the 7th" i.e.: Gregorian chant accompaniment), these resolution requirements are not present and so the 4th can be treated exactly as the consonance it is. Our music texts, oriented towards making the student resolve his resolutions rightly in exercises, neglect to specify the whole classes of musics where 4ths are never pretentious dissonances. 70.115.48.136 (talk) 23:45, 19 January 2015 (UTC)

Quite so. "Our music texts", of course, are designed to teach relative beginners first the rules of European music of the 18th and 19th centuries and, after that, the contrapuntal practice of the 16th century. This is just as well because, if we tried to explain the music of all times and places all at once, it would drive our students mad. The problem with an encyclopedia entry like this one is that it must respect this didactic approach to a degree, but at the same time must not pretend that such explanations can be extrapolated to all music. It was of course a widespread belief in the 18th and 19th centuries that European music of those times was the summation and goal toward which all music was striving, and therefore the defects of such "primitive" or "inferior" music could be explained as an inadequate understanding of how music really worked. This belief is no any longer accepted by most scholars, but it does make the universe much less tidy.—Jerome Kohl (talk) 01:06, 20 January 2015 (UTC)

A questionable article?

This article strikes me as quite questionable, certainly not worth a "B-class" rating, in its present state at least (it may have been better in earlier versions). Aside from the many requests for clarification in the text, my questions concern the following, among others:

• The lead opens "In music, a consonance...". The definition that follows may be acceptable, but shouldn't something be said about Consonance and dissonance in acoustics? The very common confusion between music and acoustics in this respect makes things quite bad. The article includes a section on "Physiological basis of consonance", which abandons the musical context for the physical/physiological one, without trying to explain how this can be reconciled with the cultural variations described elsewhere. But this section ends with ... a perfect cadence ii–V⁷–I, one of the most conventional and of the least dissonant usages of dissonance.
• The lead continues with the statement that "musics other than those from the western art music tradition, e.g., Balkan, Arabic, Chinese, do not follow this definition", and supports this statement by quoting a book on Music in Bulgaria. What is one supposed to conclude? That these countries follow another definition? Or have none?
• The section on Consonance gives several definitions, among which 'fusion' is somewhat redundantly mentioned twice:

– "Blend/fusion: perception of unity or tonal fusion between two notes" [The first reference, to Stumpf, is unclear; the second, to Butler and Green, has an uncorrect link.]

– "Fusion or pattern matching [...] Harmonics may be perceptually fused into one entity." [The harmonics are either those of a single note and the definition does not belong in this article, or those of two (or more) notes, and the two definitions are the same.]

• I defy any normal person to understand what is meant by "fundamentals may be perceived through pattern matching of the separately analyzed partials to a best-fit exact-harmonic template or the best-fit subharmonic"; and those having understood the sentence to recognize in what sense the perception of fundamentals concerns that of consonance/dissonance (the phrase probably belongs to the Missing fundamental article).
• The definition of consonance as continuous and dissonance as intermittent is a grossly oversimplified quotation of Helmholtz. Intermittent tones are discussed in the Ellis translation pp. 168 ff., where Helmholtz explains that beating around 20-30 Hz produces the same type of jarring of rattling as intermittent tones. This is not exactly the same thing as saying that "dissonances are intermittent".
• "In Western music, dissonance is the quality of sounds that seems unstable and has an aural need to resolve to a stable consonance." It seems to me that the need to resolve is conventional, rather than "aural"; many dissonances resolve on another dissonance (as Rameau already had made clear).
• The section on Dissonance should include concrete definitions, starting from Helmholtz' definition of dissonance resulting from beats.
• I think that the last chord in the first staff of the Krenek examble should include an Eb (C–Eb–Ab).
• In the section 'Dissonance and musical style', the quotation attributed to Schenker is most probably not by him and certainly has nothing to do with matters of musical style. The word [guidelines] appended to "voice leading" is utterly puzzling.
• "This interval [the neutral second] is known to create the maximum sharpness and is known in German ethnomusicology under the term Schwebungsdiaphonie". This is a misunderstanding of what Messner (1976) defines as Schwebungsdiaphonie: the interval could be much less than a neutral or even than a minor second, and the effect is not that of a dissonance, but of a beating (schwebende) sound.

Etc. etc. Correcting all this will require a thorough rewriting of the whole, and a lot of research to find the necessary references. — Hucbald.SaintAmand (talk) 21:05, 24 January 2015 (UTC)

Following the above, I started writing a revised version of the article, which can be found on my Sandbox page, where for a while it may grow. Comments and suggestions will be most welcome (on the Talk page of my Sandbox), as this is a complex enterprise. — Hucbald.SaintAmand (talk) 14:04, 25 January 2015 (UTC)

Consonance vs dissonance

The sections "Dissonance and musical style" "Dissonance throughout the history of western music" and "objective basis of dissonance" could be retitled by replacing dissonance with consonance without changing the content. Hyacinth 15:35, 16 October 2006 (UTC)

Is it correct to say that dissonance in music is considered transitional, or temporary? Does this not presume that all sounds "ought to" or "should" resolve into consonance? — Preceding unsigned comment added by Grelluz (talk • contribs) 05:24, 9 January 2015 (UTC)

Good point. To make such a claim assumes a value judgment, which in turn requires the establishment of an ethical system giving priority to consonance over dissonance.—Jerome Kohl (talk) 06:34, 9 January 2015 (UTC)

The Middle Ages section has been fleshed out, and the "George Russell" Neo-classicism section fixed, too. That ought to give a better basis, deciding for or against an historical structuring of the article. 70.115.48.136 (talk) 03:46, 20 January 2015 (UTC)

Perhaps "Consonance and Dissonance" should become a trio to include Resonance? That's the magic binding together stacks of thirds and seconds such that { C Db } hurts, but { C E G Bb Db } can be the fundamental structural basis of much of Romantic music.70.115.48.136 (talk) 22:53, 8 February 2015 (UTC)

If you have got reliable references for this theory then, why not? I like your recent addition of inharmonic instruments, by the way. The issue of consonance and dissonance with bells and such is a vexed problem, which I should think might interact significantly with such a theory of "resonance".—Jerome Kohl (talk) 23:29, 8 February 2015 (UTC)

Comments on the "simple ratio" hypothesis

The section above ("The objective (physical/physiological) basis of dissonance") assumes the use of harmonic timbres, which have dominanted Western music since ancient times. This assumption is highlighted by the example, which uses harmonic timbres. Had it used pure sine waves, then there would have been no discernable difference between "simple ratios" and non-simple ratios, once outside of the critical band (ibid, Sethares, 1992). If the timbres used in the experiment were complex (i.e., included many partials) but did not fall in a strictly harmonic series, the widths of the consonant intervals would change accordingly (ibid., Sethares, 1992). Some cultures whose dominant instruments produce inharmonic spectra use tunings that may maximize consonance by aligning the pitches of the notes with the partials of the instruments (ibid., Sethares, 2004) -- tunings that do not follow Pythagoras' "simple ratios." Hence, the "simple ratios" hypothesis does not generalize beyond the special case of harmonic timbres. Since this special case is common in Western musical culture, it may have led some Western writers into an unintended ethnocentrism. The "coincidence of partials" hypothesis, also described above, avoids this by embracing harmonic and inharmonic timbres equally. - Horseshit, who cares about some hideous wog noise, we're talking about music.

Furthermore, the "simple ratio" hypothesis explains only what has already been done in Western' music's past rather than what could be done in music's future. For example, the "simple ratio" hypothesis cannot explain the consonance of the intervals in the piece C To Shining C, because the widths of the intervals in that piece are constantly changing as the piece's tuning progresses from quarter-comma meantone to 5-tone equal temperament and back (as described here). Nor can the "simple ratio" hypothesis explain the tension and release imparted by the same piece's progression from tuning-aligned timbres to fully-harmonic timbres (and back). These progressions are both examples of dynamic tonality. The "simple ratios" hypothesis cannot explain their consonance, but the "coincidence of partials" hypothesis can and does (ibid., Milne et al., 2007, 2008, 2009). - Horseshit, this poovery has no future.

I removed the above since comments do not belong in articles but rather on talk pages. Hyacinth (talk) 03:16, 6 July 2009 (UTC)

I'm glad you did, because furthermore, it's wrong. The "simple ratios hypothesis", as this guy decided to put it, applies even for sinusoids that have no overtones. Even with sinusoids, a tritone will sound more dissonant than a perfect fifth. This is due to the virtual fundamental phenomenon. Paul Erlich has done a lot of really groundbreaking work on this with his Harmonic Entropy model, which Sethares has also written about: http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf

So the above, if it ever were posted, is really a guy who is missing the forest for the trees; there is more than one type of dissonance, critical band dissonance being just one type. Using specialized timbres such as those of 5-equal will make 5-equal fifths a lot more pleasant. However, the fact that 5-equal fifths still sound like fifths at all, and the fact that they still sound consonant in the sense that fifths are consonant to begin with, is due more to temporal periodicity process and is more what Erlich is getting at with his Harmonic Entropy model. Battaglia01 (talk) 13:14, 23 January 2011 (UTC)

The worst parts are the two sentences that were obviously added by a vandal, not the original writer--those two sentences that start with "Horshit". The original writer made some good points. It's good they were moved here for discussion instead of being deleted.

Erlich's "Harmonic Entropy" is derivative, not ground-breaking. It's basically just a digital re-construction of a graph originally from Helmholtz (see page 193 of "On the Sensations of Tone", in Chapter X, under the "Degree of Harmoniousness of Consonances" section), and it actually supports the idea that complex timbres give the ear more information with which to discern consonance and dissonance than simple timbres do. You can do your own audible experiments with pairs of sine waves (as I have and as anyone who writes about it should) to discover that the perception of consonance and dissonance is much less, or even missing, without upper partials. 108.60.216.202 (talk) 01:47, 4 May 2015 (UTC)

Non-harmonic consonance

Although a reference to Sethares' work is made in the "see also" section, it is not mentioned in the body of the article. This seems like a very significant omission. His work in establishing the relationship between the timbres of non-Western instruments and the tunings in which they are played would seem to validate the idea that consonance arises from the alignment of tuning & timbre, thus providing a physical, culture-independent basis for the perception of sonance.

Recently, he co-authored a paper which introduced the previously-unrecgnized property of tuning invariance in isomorphic keyboards, which shows that some two-dimensional note-patterns present a given musical interval with the same shape wherever they occur, independent of tuning, across a wide tuning continuum that includes the tunings of many non-Western cultures (in addition to those of the West). This raises the possibility -- still speculative, and probably not yet suited to discussion in Wikipedia -- that the human brain uses a tuning invariant note-layout to classify interval relationships.

The point being that Sethares' previously-published work is worthy of mention in the main body of this article not only because it illuminates the subject at hand, but also because it illuminates new areas of research that are only now opening up.

I am reluctant to edit this article to reflect this recent (published, peer-reviewed) scholarship without first mentioning it to this article's editors.

Your thoughts?

JimPlamondon (talk) 22:17, 25 February 2008 (UTC)

I don't think the layout of controllers has any bearing on the way we hear. It's more the other way around. Instrument design is always a compromise between the ideal goal of unencumbered expression and construction practicalities. The human voice and whistling are two instruments that get pretty close to unencumbered expression, and they don't have a "layout" at all, because they don't need one--you can immediately produce any note you want by ear. Unfortunately, they're monophonic. 38.86.48.38 (talk) 03:25, 4 May 2015 (UTC)

References

The first figure, titled "Triads consisting of three consonances," is inaccurate. The third chord in the first system is augmented (C-E-Aflat). The author's intention was probably C-Eflat-Aflat. — Preceding unsigned comment added by 70.168.62.52 (talk) 19:55, 6 January 2012 (UTC)

I don't have the book to confirm the illustration, but doesn't (C-E-Aflat) have three consonances: two major thirds and a minor sixth? 38.86.48.38 (talk) 03:25, 4 May 2015 (UTC)

Unless I am mistaken, E-Ab is a diminished fourth, not a major third. That is that in any temperament other than equal, it probably won't sound as a major third. Besides, a consonant triad should be consonant in all its inversions: Ab-C-E includes an augmented fifth. The main problem is that Krenek's example is presented as expressing a fact, while it merely gives an opinion (Krenek's); the caption fails to present the context of Krenek's book, namely atonal counterpoint. Hucbald.SaintAmand (talk) 08:30, 4 May 2015 (UTC)

You're absolutely right, E to Ab is a diminished fourth, but I think those examples implicitly assume 12 EDO, which makes it enharmonically equivalent to a major third, which is supposedly "consonant" in the rest of that chart. Notice that the chart doesn't say that an augmented triad (first inversion) is consonant. It merely says it contains three consonant intervals. That means the dissonance of an augmented triad emerges not from its component intervals, but from the total combination, which is interesting too. An augmented fifth is enharmonically equivalent to a minor sixth in 12 EDO, so if one is "consonant" the other must be. About C-Ab, if you look at line 4, second triad (C, Db, Ab), it supposedly has two consonances and a "sharp dissonance". I assume that means C to Db is the sharp dissonance and Db to Ab is a consonant, which leaves C to Ab as the remaining consonant, according to Krenek. However, I agree, the chart is an opinion. Also, if it really is supposed to be a collection of all the atonal triad shapes in all inversions (in close position), organized by dissonant and consonant component intervals, then it should have 55 examples, not 29, so something's off. If you allow Just Intonation, then there are a bunch of possible interpretations of C-E-Ab (and of every triad shown), each with its own consonant or dissonant component intervals, which would require a larger chart (understatement). Basically, this just reinforces the problem that consonance and dissonance are difficult to nail down and that different people who try to catalog specimens are going to come up with different arrangements. Will you add the context of 12-tone atonal counterpoint to the caption? 38.86.48.38 (talk) 23:12, 4 May 2015 (UTC)

Another user had suggested that Krenek had intended to write C Eb Ab example: this is another possibility. It seems farfetched to choose 12 EDO to illustrate consonance and dissonance, and the figure, as you say, is quite incomplete: one important triad that seems missing is C Eb Gb. I cannot imagine what the figure really shows, and I would suggest removing it alltogether; but such a suggestion will not be well received, I am afraid.

I had begun rewriting the article, opening the page User:Hucbald.SaintAmand/Consonance_and_dissonance for the purpose. But my invitation to work there did not meet much reaction and the existing article has been modifided meanwhile, to an extent that frightens me. Also, some of us wanting to work here have been called to work elsewhere: see User:Hucbald.SaintAmand/Tonality. — Hucbald.SaintAmand (talk) 06:16, 5 May 2015 (UTC)

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Monteverdi one of the firsts to use dissonance?

The article contains the following paragraph:

One of the early composers known for use of dissonants was Monteverdi (Kempers and Bakker 1949, p. 14: "His use of dissonants was in contradiction to all known rules; his melodies showed unheard-of intervals; he even neglected the rules of counterpoint, secure in his conviction that the new subjective art form needed a new means of expression"). The use of unresolved dissonance was also practiced by Moscheles (Anon. 1826, p. 349: "It however appears, that Mr. Moscheles, according to the most esteemed writers and the rules of composition, is perfectly justified in writing it D flat. The D flat is here a diminished 7th, which, though a dissonant, is not necessarily resolved in all cases") and taught by Chopin (Eigeldinger, Howat, and Shohet 1988, p. 42: "Musical prosody and declamation; phrasing—Here are the chief practical directions as to expression which Chopin often repeated to his pupils: A long note is stronger, as is also a high note. A dissonant is likewise stronger, and equally so a syncopated note. The ending of a phrase, before a comma, or a stop, is always weak. If the melody ascends, one plays crescendo, if it descends, decrescendo. Moreover, notice must be taken of natural accents.)"

None of the quotations included in this paragraph say what they are supposed to say.

• Kempers and Bakker say that Monteverdi's "use of dissonants was in contradiction to all known rules" (which I don't think is true, but they wrote 60 years ago), not that he was among the firsts to use them. If there were "known rules", it probably means that dissonances usually were practiced along these rules. And this whole affair of Monteverdi and the dissonances is well known to originate in Fétis; it concerned unprepared dissonances, not unresolved ones.
• The text about Moscheles says that a diminished 7th "is not resolved in all cases", not that Moscheles practiced the use of unresolved dissonance. The problem of the diminished 7th mainly is that any of its notes could be taken as the dissonance (usally involving enharmony, which is the probable reason why D flat is mentioned); but I trust that at least one of the notes is resolved in all cases.
• The quotation about Chopin never concerns unresolved dissonances.

I think that something must be done about this, but I see no other solution than to delete the whole paragraph – I'll do wo, unless somebody wants to argue otherwise. — Hucbald.SaintAmand (talk) 08:57, 16 September 2018 (UTC)

Reorganizing the article?

I don't think that reorganizing the article, moving this or that section before or after this or that other one, may save the article. There may be a problem in the organization, but the problem is mainly about the content.

I fully sympathize with the attempts at explaining consonance and dissonance outside Western music, but I believe that there is a fundamental problem here. The reason why these two concepts are dealt with in a single article is that it may be impossible to define one without mentioning its opposition with the other. As the lede says (I think I am responsible for this statement), consonance and dissonance form "a structural dichotomy in which they define each other by mutual exclusion". One may discuss whether this or that interval or this or that tuning in non Western music is a consonance or a dissonance, but this hardly makes sense unless by comparison with Western music. To say that "Vocal polyphonic traditions from Bulgaria, Serbia, Bosnia-Herzegovina, Albania, Latvia, Georgia, Nuristan, some Vietnamese and Chinese minority singing traditions, Lithuanian sutartinės, some polyphonic traditions from Flores and Melanesia are predominantly based on the use of sharp dissonant intervals and chords" makes little sense unless one opposes these intervals and chords with consonant ones. These may or may not exist in the cultures considered, but in any case I am afraid that the intervals and chords described as "sharply dissonant" are such for Western ears only. Not that the Bulgarians and others ear differently, but that they don't mind: whether their music is dissonant or not is not an important category for them. In other terms, to call their music "sharply dissonant" is to view it (or hear it) from a Western bias.

To state (with Gouwens 2009) that "The overtones of the inharmonic series produced by such instruments may differ greatly from that of the rest of the orchestra, and the consonance or dissonance of the harmonic intervals as well" boils down to the same problem: it is not that the "consonance" or "dissonance" differs from that of the rest of the orchestra, but rather that these categories (consonance and dissonance) do not apply for these inharmonic instruments. Otherwise, once again, one starts from a preconception of what a consonance or dissonance is for "the rest of the orchestra".

The section on "Physiological basis" first states that "Musical styles such as traditional European classical music consider [beating] objectionable ... Other musical styles such as Indonesian gamelan consider this sound an attractive part of the musical timbre and go to equally great lengths to create instruments that produce this slight roughness". Nothing, up to there, is said about consonance or dissonance. The text then continues: "Sensory dissonance and its two perceptual manifestations (beating and roughness) are both closely related to a sound signal's amplitude fluctuations". Now, if these statements are to be taken seriously, they appear to indicate that European classical music considers dissonance "objectionable", while Indonesian gamelan considers it "attractive". Aside from the fact that, having played the Indonesian gamelan for years, I never noticed any of this, there is a contradiction in the terms, namely that a "dissonance" should be considered "attractive". I would rather say that what is considered dissonant in European classical music is considered consonant in Indonesian gamelan...

The section further discusses at length the production of the sensations of "beating" and "roughness", in terms that even I, who may consider myself rather competent in these matters, hardly can understand (by which I mean that it could be explained –and has been– in much simpler terms). And it then comes to this highly questionable statement: "Dissonance is more generally defined by the amount of beating between partials (called harmonics or overtones when occurring in harmonic timbres)". But that is the Western definition of dissonance, and it should by no means be considered universal!

Once again, one cannot merely consider that exotic musics are dissonant, and built the article on such basis, because one couldn't construct any coherent universal definition of dissonance on such premises. The fact is that some cultures (e.g. the Western one) do associate dissonance with beating and roughness (and, more important perhaps, consonance with the absence of these phenomena) while others do not make such associations, merely because they hardly can make the distinction between beating and non beating.

Consonance and dissonance are Western concepts. One should not believe that Western music developed to adapt to the ideas of consonance and dissonance, it is the other way around: Western music developed in such a way that it was able to construct the concepts of consonance and dissonance and to make them an essential part of the musical game.

As long as the article does not understand that, it will remain a mess. But this is only my opinion, and I am willing to hear that of others. — Hucbald.SaintAmand (talk) 21:28, 25 July 2017 (UTC)

Dear Hucbald.SaintAmand,

This is the second note that I have addressed to you on this page, so I am taking the liberty of saluting you as "dear." Please advise me if you find this to offend your dignity.

I beg to differ with you, Dear Hucbald.SaintAmand, regarding the unique Western-ness of consonance, for many reasons.

1. Having lived in Southeast Asia for many years, I can assure you that most of the music I hear in the streets here is as consonant as that I hear in the West. Listen, for example, to the Molam music of north-east Thailand: [Molam: Thai Country Groove From Isan Vol. 2]. It has been strongly influenced by Western music, but it is not "Western." After all, such influence is a two-way street. I suspect that the same is true of much of other non-Western regions' musics. One can reasonably disagree about how much Western influence makes music more or less Western, but...do we really want to open that can of worms here?
2. Thai classical music, played on the traditional renat, is consonant in its traditional 7-tet scale. Of course, you would not know that, because you are "averse" to reading the works of "Sethares and others." Let me make it easy for you: Here's the relevant chapter, for free, on line. It is missing some pages; that's the price you pay for "free." Most university libraries have it.
3. Likewise, here's a link to the chapter on the Indonesian gamelan. The gamelan's slendro scale is, essentially, 5-tet, and the gongs of the gamelan orchestra are carefully tuned to be consonant with that tuning, within the bounds of local variation. I would expect that you, "having played the gamelanfor years," would be particularly interested in a theory of consonance that worked exactly the same for the gamelan (and renat) as for the human voice and piano. However, to understand that theory -- and to stop stripping it out of Wikipedia articles! -- you would need to overcome your "aversion" to reading the works of "Sethares and others."
4. You will please note that the two tunings just described -- that of the Thai and Indonesian classical music -- fall at the endpoints of the syntonic temperament's valid tuning continuum. "Hmmmm...," you might say to yourself, upon realizing this. "Hmmmm..., that's interesting." Or maybe not, due to your "aversion" to reading the recent works of "Sethares and the others." Really, Dear Hucbald.SaintAmand, I promise that you have nothing to fear. Despite what your mother may have told you, the self-gratification of keeping up with the scientific literature won't make you go blind. Indeed, you might find the experience to be...stimulating. Once you read Sethares' "Tuning, Timbre, Spectrum, Scale" (see links below) you may find your understanding of consonance to be remarkably expanded.

Once you understand the theory of consonance and dissonance described by "Sethares and the others," I trust that you will see this article with new eyes, and will see the structure and nature of the needed revisions much more clearly. As long as you do not understand that, this article will remain a mess. But this is only my opinion, and I am willing to hear that of others...AFTER they have done the minimum expected of any article editor, and kept up with the relevant literature.

Respectfully,

--JimPlamondon (talk) 12:15, 3 May 2020 (UTC)

Work to be done on this article

Isn't it time to rekindle the work on this article – especially now that several of us have little else to do? (I hope that all readers are well, and wish you the best for the time coming.)

I must first apologize for the unsigned comment in the section above this one. I wrote it more than a year ago (25 March 2019), if I trust the revision history of this page. I am surprized that it was left unsigned since then, I thought that WP was patrolled for such cases.

The lead of the article seems acceptable for the time being and we would probably do best to work on the following sections. For today, I'd like to question some statements in the section devoted to the definition of consonance. My comments are inserted in the text:

The definition of consonance has been variously based on experience, frequency, and both physical and psychological considerations (Myers 1904, p. 315). These include:

• Frequency ratios: with ratios of lower simple numbers being more consonant than those that are higher (Pythagoras[full citation needed]). Many of these definitions do not require exact integer tunings, only approximation.[vague][citation needed]

Pythagoras obviously cannot have considered frequency ratios, as the concept of frequency did not become available before more than a thousand years later. String length ratios should be mentioned, but that would require some additional explanation. The second sentence refers to "many of these definitions", but none is given! Also, it does not seem to be the definitions that don't require exact integers, but our perception. There are texts about that, that should be quoted.

• Coincidence of partials: with consonance being a greater coincidence of partials (Helmholtz & 1954 [1877],[page needed]). By this definition, consonance is dependent not only on the width of the interval between two notes (i.e., the musical tuning), but also on the combined spectral distribution and thus sound quality (i.e., the timbre) of the notes (see the entry under critical band). Thus, a note and the note one octave higher are highly consonant because the partials of the higher note are also partials of the lower note (Roederer 1995, p. 165). Although Helmholtz's work focused almost exclusively on harmonic timbres and also the tunings, subsequent work has generalized his findings to embrace non-harmonic tunings and timbres (Sethares 1992; Sethares 2005[failed verification][page needed]; Milne, Sethares, and Plamondon 2007,[page needed]; Milne, Sethares, and Plamondon 2008,[page needed]; Sethares et al. 2009,[page needed]).
• Fusion: perception of unity or tonal fusion between two notes (Stumpf 1890, pp. 127–219; Butler and Green 2002, p. 264).

If Helmholtz' theory is defined in terms of the coincidence of partials, the difference with Stumpf's theory of fusion is not clear (neither theory is explained, in addition). The important point about Helmholtz is that he linked consonance with the absence of beats – or, better said, his theory is one of dissonance, associated with beats.

It is true that these theories explain why consonance is not dependent only on the width of the interval, but the mention of "the musical tuning" in this context puzzles me: shouldn't it rather be "the intonation", or something like that? When a violinist plays a major third, she does not "tune" it...

The matter of the spectral distribution also seems problematic to me. It would seem almost impossible to expect a coincidence of partials if the partials were not more or less harmonic. I am somewhat averse to reading Sethares and the others, but I don't see how Helmholtz' ideas could be "generalized to embrace non-harmonic tunings and timbres". It is not that Helmholtz "focused on harmonic timbres", rather that his ideas implied harmonic timbres.

And none of this says much about how music defines consonance. I am somewhat at loss to see what could be done about the quotations above. I'd like to hear the opinion of others about all this. — Hucbald.SaintAmand (talk) 10:54, 1 May 2020 (UTC)

All of this is truly appalling, Hucbald, I couldn't agree more. Checking the edit history I see that most of my own sustained work on this article was done in 2014. I remember placing that demand for publication details of Pythagoras' book, article, pamphlet, YouTube video, or blog, naturally with satyrical intent. I should not have left it for so long. I'm not sure I ever tried tracing the citations (in particular Myers 1904) to see if they really supported the ridiculous or vague claims, though as you know, on Wikipedia what matters is verifiably, not truth or even plausibility. I suspect there may be a lot of misunderstanding of what some of the sources actually say, and it is high time that these passages (to start with) were more critically scrutinised.—Jerome Kohl (talk) 18:01, 1 May 2020 (UTC)

As a matter of fact, it seems hardly feasible to treat consonance and dissonance separately, e.g. to provide a definition of the one without (or before) a definition of the other. I reorganized the first sections of the article with this in mind, under the heading "Definitions". I didn't yet change much to the content, but this already seems more satisfying. Whaddya think? — Hucbald.SaintAmand (talk) 08:38, 3 May 2020 (UTC)

Hucbald.SaintAmand,

Thank you for writing these two sentences: “It would seem almost impossible to expect a coincidence of partials if the partials were not more or less harmonic. I am somewhat averse to reading Sethares and the others, but I don't see how Helmholtz' ideas could be ‘generalized to embrace non-harmonic tunings and timbres’.”

For many years now, I have been searching for a concise, modern example of the Semmelweis reflex. With those two sentences, my search is over. Thank you!  :-)

I cannot help but be reminded of the philosophers who were “averse” to peering through Galileo's telescope, lest they see that the moons of Jupiter orbited around -- gasp! -- JUPITER, rather than the Earth, thus throwing their Aristotelian model of the universe into chaos. [Chaos! Dogs and cats living together! MASS HYSTERIA!]

Hucbald.SaintAmand, I promise that the Pope will not excommunicate you if you peer through the metaphorical telescope created by “Sethares and others” and see -- gasp! -- pseudo-Just temperaments and pseudo-Harmonic timbres living together in consonance.

C’mon, buddy, try our first paper: The X_System. The first one is free! I will warn you, though, that it’s a gateway paper, not even peer-reviewed (which is one reason that it is simple and clear). The next thing you know, you might want to read our peer-reviewed papers! Their heavy mathematics proves that our approach is valid -- that’s the “hard stuff,” right there. Or maybe you might move up to experimenting with some of our Dynamic Tonality synths, which moves our ideas from being merely "ideas" to the realm of sounds. They’re addictive, I’ll admit.

In the meantime, Hucbald.SaintAmand, you might want to refrain from editing out content that describes recent advances in the state of the art which, for whatever reason, you are averse to seeing for yourself.

Respectfully,

--JimPlamondon (talk) 11:01, 3 May 2020 (UTC)

JimPlamondon, I am happy to see that at least this all amuses you. Keep in mind that English is not my mother tongue. I would not be "averse" (it may not be the right word) to reading you if I had never done so – on the contrary, it is because I read you that I became "averse". You write that "their heavy mathematics proves that our approach is valid", which in my turn I find amusing: life would be too good if "heavy" mathematics were sufficient to prove anything valid.

I don't think I ever "edited out" any of your contents, but since you mention "editing out", you might perhaps help me understand the concept of "tempering out", or "tempering to unison" (especially applied to commas) that I often find in your writings. In your X_System paper, for instance, you write that "in the meantone temperament, the syntonic comma (81/80 = 2^-43⁴5^-1) is tempered to unison, which means that every integral power of the syntonic comma (e.g. (81/80)² or (81/80)^-7) is also tempered to unity, so any of these infinite number of different commas could be used to define the temperament." I sometimes tuned harpsichords in ¼–comma meantone, but I never succeeded in "tempering" its syntonic commas to anything, nor to make all of them vanish. [I tuned by tempering fifths, not commas, but never mind.] It is true that when my tuning was successful, the major thirds were ... in unison with themselves (is that what you mean by "comma tempered to unison"?), but they were a syntonic comma narrower than Pythagorean ones – which had been the purpose of the whole operation. As to the minor thirds, they were better than in Pythagorean tuning, but still about 1/4 of a syntonic comma away from their "just intonation" value. You seem to indicate that "any of the infinite number of different commas" resulting from integral powers of the "syntonic comma tempered to unity" (i.e. to unison?) could define the temperament, but do you mean that any power of the unison (1^x – isn't that heavy mathematics?) not only could define the temperament, but also should be called a comma?

As you can see, the probable reason of my aversion seems to be that I don't understand. It may be that I don't understand your English. I'll be glad to hear and, hopefully, to learn. — Hucbald.SaintAmand (talk) 13:31, 3 May 2020 (UTC)

Woops! I obviously have problems with heavy mathematics: when I wrote 1^x, I obviously meant 0^x. It doesn't change much, yet ... — Hucbald.SaintAmand (talk) 09:03, 4 May 2020 (UTC)

Tempered timbres

Dear Hucbald.SaintAmand, et al.,

The problem with mathematics is not that it explains so little, but that is explains an unreasonably large percentage of natural phenomena so accurately. That said, mathematics is not my mother tongue, so I defer to my collaborators Dr. Andrew Milne and Prof. William Sethares to explain the mathematics of Dynamic Tonality.

I think visually, not algebraically. Let me share a visualization that helped me "get it." Maybe it will help you, too.  :-)

Today, I uploaded to Wikipedia an animated GIF (shown as Figure 1 below) that should, perhaps, more-visually explain what we're doing.

Figure 1: Tempering a Harmonic timbre to be a pseudo-Harmonic timbre, as per the syntonic "temperament-as-defined-by-a-comma-sequence," such that its partials align with the notes of similarly-tempered tuning.

The animation in Figure 1 starts by showing the partials of the Harmonic Series and the 5-limit Just Intonation intervals derived from them in the syntonic "temperament-as-defined-by-a-comma-sequence" (hereinafter, "temperament"). Then, it shows those same partials (except the 7th partial and its octaves) tempered to a 5-limit pseudo-Harmonic timbre according to the 5-limit syntonic temperament at various different widths of the tempered P5 (the generator): 700¢ (12-tet), 686¢ (7-tet), and 720¢ (5-tet). The result of each tempering is a pseudo-Harmonic 5-limit syntonic-tempered timbre, in which the partials are aligned perfectly with the notes of a similarly-tempered pseudo-Just 5-limit syntonic-tempered tuning. Using an identically-tempered pseudo-Harmonic timbre and pseudo-Just tuning together, one can play with maximal consonance for any width of the tempered P5 that is within their valid tuning range.

Figure 2: Valid tuning ranges of the syntonic temperament. The 5-limit range (pale blue) is very wide; the 7-limit range (purple) is much narrower; the 11-limit range (orange) is narrower still. The 13-limit, 17-limit, and each progressively-higher-prime-limit ranges (not shown) would be progressively narrower.

BTW, if I recall correctly, I first saw the kind of diagram shown in Figure 1 in the book Hearing and Writing Music by the late Ron Gorow, although I could be mistaken about that. — Preceding unsigned comment added by JimPlamondon (talk • contribs) 13:34, 5 May 2020 (UTC)

You asked about "tempering out" and "tempering to zero." Here's my explanation of why we're using that term.

• All of the notes of the syntonic temperament are derived from an octave-reduced stack of tempered P5's. If you start the stack on Do, then Re is two P5's up and an octave down. Mi is, from Do, four P5's up and two octaves down. Hence, the width of Do-Re (the tempered major second) is half the width of Do-Mi (the tempered major third).
• One definition of "the syntonic comma" is "the difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third." Importantly, that this definition is based on the Stack of Fifth's, not on ratios-of-small-whole-numbers. For our purposes, we generalize that definition slightly to "the difference between four justly tuned tempered perfect fifths, and two octaves plus a justly tuned tempered major third."
• In the syntonic temperament, the difference between four tempered P5's and two octaves plus a tempered major third is 0¢ for every width of the tempered P5 within the syntonic temperament's valid tuning range. Hence, the syntonic comma -- by our generalized definition above -- has been tempered to zero ("tempered out") of both the timbre and the tuning all across the valid tuning range of the syntonic temperament. That's what having the syntonic comma as the first comma in a temperament's comma sequence MEANS. It also means that the 5th partial is mapped to the major third (as opposed to, say, the diminished fourth, as it would be if the first comma in the comma sequence were the schisma).
• One of the most problematic occurrences of the syntonic comma is in the chord progression I-vi-ii-V-I. In Just Intonation, the initial I and the final I differ in pitch by a syntonic comma. Using syntonic pseudo-Just tuning & pseudo-Harmonic timbres, this drift is eliminated... while retaining consonance.

In Figure 1, the 7th partial (and its octaves) is not shown as moving around at different widths of the tempered P5. The next comma in the temperament's comma sequence, after the syntonic comma, would tell the temperament to which note to map the 7th partial (and its octaves): augmented sixth? Flat seventh? Similarly, the next comma in the comma sequence, after that, would indicate the note to which the 11th partial (and its octaves) would be mapped, and so on up the prime-numbered partials (and their octaves).

In Figure 2 above, the whole tuning range is labeled "5-limit" (pale blue, on the right-hand side). The valid tuning range labeled "7-limit" (purple) is narrower than the 5-limit tuning range. The valid tuning range labeled "11-limit" (orange) is narrower still, each a subrange of the previous range. Each successive comma in the temperament-defining comma sequence is an additional constraint that can only narrow, not expand, the temperament's valid tuning range.

Figure 3 below shows one possible mapping of the higher-prime partials to the note-controlling buttons on the Wicki-Hayden note-layout, as used in the instrument generically known as a "jammer."

Figure 3: One possible mapping of the higher prime partials (7th, 11th, 13th, etc.) to the note-controlling buttons of a Wicki-Hayden note-layout as used on the keyboard of the instrument known generically as a "jammer."

You mentioned not seeing how Sethares et al. could claim to have generalized Helmholtz's definition of consonance. Helmholtz defined the source of consonance as arising specifically from the alignment of a Just Intonation tuning's notes with the partials of a timbre that followed the Harmonic Series. Sethares et al. have generalized that definition of consonance to include pseudo-Just tunings and their related pseudo-Harmonic timbres -- all the way to their endpoints, at which we find notable non-Western tunings, thus showing that our generalization is a very useful generalization indeed!  :-)

I understand that the idea of tempering one's timbres to align with the notes of one's tuning is relatively novel. It wasn't documented until 2006, and not published in a peer-reviewed publication until 2007. It isn't possible with acoustic instruments or the human voice unless their sound is digitally processed in real time (which we can do, BTW). It is particularly easy to do with digital synthesis. More importantly, one must use an note-layout that is isomorphic with the temperament -- an isomorphic keyboard -- to control it dynamically, and (sigh) I failed to bring mine to market (sigh).

I care most about the syntonic temperament, for a variety of reasons, but one can similarly temper the tuning and timbre of any rank-2 temperament, and -- using a 2D keyboard whose note-mapping is isomorphic with that rank-2 temperament -- play with consistent fingering all across that temperament's valid tuning range.

Aligning a tuning's notes and a timbre's partials maximizes consonance all across the valid tuning range of any given temperament. However, one can also systematically mis-align the notes and timbres to increase dissonance. One can also re-distribute the energy among a timbre's partials systematically, or do any of a number of other musically-interesting things.

Together, the novel musical effects enabled by this systematic [mis]alignment of a tuning's notes and a timbre's partials, across the valid tuning range of a temperament (including modulations between, and progressions through, such temperaments) we call Dynamic Tonality.

I suspect that those who have been immersed for decades in "ratios of small whole numbers" are going to have massive cognitive dissonance (ahem) with the explanations above. Dynamic Tonality does not give a s*** about "ratios of small whole numbers" except as a starting point for mapping partials to notes. After that, it's all about the Stack of Generators and the width of the generator. Period (ahem).

I hope that these images and explanations have clarified how Sethares et al. are "tempering timbres."

I trust that you now also see that the syntonic temperament includes non-diatonic tunings (at 5-tet and 7-tet, the extreme ends of its valid tuning range). As it includes non-diatonic tunings, it should not be subsumed into Wikipedia's article on "Regular diatonic tunings," and should instead revert to being a stand-alone article.

Next time, perhaps, we can chat about the the Tone Diamond, and the irregular tunings that one can get from it, thus providing another reason why the syntonic temperament should not be so subsumed.

Thank you for raising these questions. Your doing so caused me to create the PowerPoint presentation on from which Figure 1 was derived, which I now realize I should add to Wikipedia's Dynamic Tonality article.

Respectfully,

--JimPlamondon (talk) 11:21, 5 May 2020 (UTC)

Back to consonance and dissonance

Dear JimPlamondon, many thanks for this quite impressive lengthy demonstration. Let me repeat that the reason why I developed some aversion for yours and your colleague's writings is that I read them. Your demonstration therefore does not learn me much that I didn't know, but it may be useful for others. I am not sure however that this kind of in-depth dialogue between you and me has its place in this talk page. And we should keep in mind that the question at stake here is a description of what is called "consonance" and "dissonance".

But let's continue for a short while, as long as nobody stops us. I'll tell you some more of myself than I usually do on WP. After a few other businesses, I became by profession a historian of music theory. My field of competence extends, say, from Antiquity to late tonal music. I very much regret not to have been able to develop the same competence in other types of music that I like, say, jazz or modern "popular" music. Looking back, in these times that remind us that we are mortals, I realize that the musics that I studied are characterized by the fact that they are written (I have worked a lot in the field of musical semiotics). That is to say that the musics that interest me are notable, i.e. can be notated (and read), which entails strong semiotic limitations. Several of my students induced me in also studying the theory of non-Western musics, particularly but not exclusively "Arabic" music (whatever that means). All these musics are "notable", mainly in that they consist of discrete pitch categories (even when the pitches themselves are quite variable). For similar reasons, I am very much attached to the meaning of words, and to the history of the meaning of words.

A first point that I note in your demonstration is that you stress that "tempering one's timbres to align with the notes of one's tuning [...] isn't possible with acoustic instruments or the human voice unless their sound is digitally processed in real time (which we can do, BTW). It is particularly easy to do with digital synthesis." You add that this was not described before about thirteen or fourteen years ago. This appears to me extremely marginal in the context of the 2500 or 3000 years of music and music theory that I tried to cover in my own researches. Also, compared to the practice of the musicians (dead or alive) that I met during these researches, what you describe appears to me quite "unreal". (Call it once again my "Semmelweis reflex", I don't mind.) The fact is that most musicians throughout history and throughout the world were and are unable to control the partials of their sounds, unless by trying to make them as periodic as possible. I do not doubt that you and your friends (and some others) can, with digital synthesis, but once again I doubt the importance of that in "real" music. By real music, I mean the singing or playing of women and men merely because musiking, as speaking, is a capacity (and a necessity) of mankind.

Your demonstration makes use of terms that have a long history – a history which, I think, creates obligations. You make use of the shorthand "P5", obviously meaning "pure 5th". But this questions both the meaning of "pure" and that of "5th". In your fig. 2, the "P5" generator extends from 686 to 720 cents, and the figure suggests that it could extend farther in both directions. I wonder however how an interval that varies over a span of at least 34 cents, and probably up to a quarter tone and more, can be considered "pure" in all these variations. And, above all, I wonder in what sense it may be a "5th".

Your usage of the word "temper" also is puzzling. This word appeared in its musical sense about five centuries ago, if I am not mistaken, and means "altering" an interval – more specifically, it means altering a "rational" interval (one that can be expressed as a ratio) into an "irrational" one. You suggest tempering the comma. But the comma does not appear as a concrete interval, allowing to be tempered, in any of the tunings that you suggest. The comma is only a difference between intervals, and it is clear that if you somewhere reduce this difference (possibly to 0) it will necessarily increase elsewhere. The "just" major third (that of "just intonation", wich is an early 18th-century term), always is a syntonic comma away from a Pythagorean one, and tuning ¼-comma meantone in no way modifies this difference. A comma "tempered to unison" merely is not a comma anymore – no more than a P5 tempered to 686 or to 720 cents is a P5 anymore.

What I mean is that you should feel free to develop any theory that you want, and to concretize it in instruments and in musical compositions. But you should not allow yourself to divert words from their meaning established since centuries. And this brings us back to the matter of consonance and dissonance. Helmholtz' and Stumpf's theories of the coincidence of partials ultimately tell us that the very concepts of consonance and dissonance only make sense in the case of (more or less) harmonic partials (i.e. more or less periodic sounds). This not only is a matter of the coincidence of partials, but it also involves difference tones, non-linearities, reverberation, room acoustics, and the like. The "artificial" consonances and dissonances that you may be able to construct (I do not doubt your capacity to do so) first do not correspond to the usual definition of these terms, and second may not be possible unless in "artificial" surroundings.

Or, to say it otherwise, your definitions of consonance and dissonance do not merely "generalize" those by Helmholtz or Stumpf, they propose a novel description of the whole matter, with an utterly different meaning. I have nothing against that – as long as it is presented for what it is, not merely a "generalization", but a novel definition. But that, for the layman to which WP adresses, needs much more than your explanation above. — Hucbald.SaintAmand (talk) 20:45, 5 May 2020 (UTC)

Dear Hucbald.SaintAmand,

Your reply confirms what I have suspected might be true: that those who have spent decades thinking of music in terms of ratios of small whole numbers have cognitive dissonance when encountering an alternative perspective.

Consider the piano, and specifically, the issue of stretch in piano tuning, discussed more fully in the article describing the Railsback Curve. The piano is only equally-tempered in the middle of its range. Outside of that, its very tight (at the high end) and very loose strings (at the low end) do not vibrate as ideal strings would, thereby emitting the Harmonic Series, but instead emit a slightly inharmonic timbre. The partials of the extreme notes to not align with the partials of the center notes or with each other, nor with the fundamentals of the notes of a strictly equal temperament. To maximize consonance when faced with the reality of slightly inharmonic timbres, the skilled piano tuner tempers the tuning to match the timbre, by tuning notes such that their fundamentals align with the actual partials of other notes, despite these partials' inharmonicity. Note also that the piano's hammers evolved to dull the piano's tone, stripping it of its 7th and higher-prime partials, to better fit with 12-tet equal temperament. That is, the instrument evolved to maximize the alignment of its partials with its timbre. Hammer placement, wood choices and varnishes, aliquot strings, all evolved to maximize the alignment of 12-tet tuning and timbre in the piano.

A number of useful points can be take from this example. First, there are no ratios of small whole numbers involved. It is purely a matter of aligning the notes of a tuning to match the partials of a timbre.

Second, no pianist would dispute that a piano, thus tuned, was properly "equally tempered," despite many of its notes' tunings deviating from the calculated 12-tet equal values by upwards of 30 cents. Point being that the definition of the word "temper" is not strictly tied to ratios of small whole numbers, as you assert above.

Third, you noted that "what [Sethares et al.] describe appears to me quite 'unreal'." Please note that the piano is often used in real music, suggesting that the alignment of inharmonic tuning and timbre, regardless of ratios of small whole numbers, is of real musical utility. Similarly, the early Hammond organ's tonewheels emitted a timbre whose partials were aligned (more or less) with the notes of equal temperament, and it is used in rather a lot of real music. Furthermore, most Western wind instruments were redesigned in the mid-1800's, partly to enable more-chromatic music and partly to make their partials align better with the emerging 12-tet tuning in which more-chromatic music was often played (to facilitate freer modulation); these instruments, and the compositions performed on them, are certainly "real." Finally, please allow me to point out that the music of Indonesian gamelans (in which the notes of the 5-tet slendro tuning are aligned with the partials of the gamelan gong's inharmonic timbre) and of the Thai renat and Mandinka balafon (in which the notes of the 7-tet tuning are aligned with the partials of the ranat's and balafon's inharmonic partials) are "real," too, despite not being Western. In a previous comment on this Talk page, you commented that you did not find the music of the gamelan to be "dissonant," and cited this as evidence that the notion of consonance and dissonance was purely Western.

Fourth, I note that you wrote that "the 'artificial' consonances and dissonances that you may be able to construct (I do not doubt your capacity to do so) first do not correspond to the usual definition of these terms, and second may not be possible unless in 'artificial' surroundings." Yet the piano, the Hammond organ, the re-designed Western wind instruments, the gamelan, the renat, the balafon, etc., all have such "artifical" (that is, non-Just and non-Harmonic) consonances, and they seem to be accepted just fine as being "real," despite the issues that you correctly note wrt "difference tones, non-linearities, reverberation, room acoustics, and the like."

In summary, Sir, with respect, you are simply wrong in your claim that "ratios" are the ultimate source of consonance. Specifically:

1. Consonance and dissonance are highly valued by every musical culture we have identified. It is not a purely Western concept.
2. In every such culture, considerable effort has been made to align their tuning's notes with their dominant instrument's timbre's partials and/or vice versa, because such alignment is the ultimate source of consonance.
3. The alignment of the notes of a Just Intonation tuning with the partials of a Harmonic timbre is merely a special case of the general fact above.

As Thomas Kuhn pointed out, a new paradigm in a given domain is often incommensurablewith the domain's old paradigm. The old paradigm, in this case, is a misplaced focus on ratios of small whole numbers. Western music theorists, since Pythagoras, have focused on a symptom (ratios of small whole numbers) that was specific to Just Intonation tunings and Harmonic timbres, rather than on the general cause (alignment of a tuning's notes with a timbre's partials). Hence they were unable to generalize their music theory -- or its nomenclature, such as "consonance" -- to embrace the "real world" occurrences such as stretch in a piano's strings, the tone wheels of a early Hammond organ, the gamelan, the renat, etc. These, they viewed -- as you have stated above -- as something "unreal," unnatural, beyond their theory. Metaphorically, they were and are still viewed as were Marine iguanas, deviations in the orbit of Mercury, and magnetic banding along mid-ocean ridges. This is no surprise; it is entirely typical of how paradigm shifts happen: by questioning the fundamental assumptions of a theory -- often starting with a "thought experiment" -- and then seeing if the new theory explains both (a) everything explained by the old theory, and also (b) the phenomena that the old theory could not explain. "What if parallel lines DO cross?" => Non-Euclidean geometry. "What if time is NOT constant?" => relativity. "What is the continents are careening around and bouncing off of each other, on a geologic time scale?" => Plate Tectonics.

Sethares was the first to ask, "What if consonance arises from the alignment of a vibrating object's partials with the notes of a tuning?" => Dynamic Tonality. As you can see from the above evidence of instruments as familiar to you as the piano and the gamelan, the result is a new paradigm that explains both (a) all of the consonance explained by the old paradigm (because the alignment of a Just tuning's notes with a Harmonic timbre's partials is a special case of the general theory) and (b) all of the consonances that were unexplained aberrations under the old theory of "ratios."

Transitions between paradigms always involve new perspectives on old words. Consider the definition of "mass" in Newtonian physics vs. relativistic physics, as discussed here. In a paradigm shift, some words get redefined; some new words are introduced; and some old words are deprecated. That is normal, and while it needs to be done thoughtfully, it is not anathema. Shift happens.

We have redefined "temperament" to mean "temperament-as-defined-by-a-comma-sequence," as per Gene Ward Smith. We propose that "consonance" and "dissonance" need NOT be redefined, but rather, that their source needs to be properly identified, now that its general source is understood -- which is diametrically opposed to the Euro-centric changes that you propose to make to this Consonance and Dissonance page, by claiming that consonance is a strictly Western concept. (Note that we recognize two different sources of dissonance: one arising from misalignment of partials, and one arising from two notes being played within each others' Critical band. We have not discussed the latter source of dissonance in this Talk page.)

We are not proposing these re-definitions lightly. We have good reason.

Finally, please allow me to note that the fact that some idea has been widely believed for hundreds or thousands of years does not mean that it is correct. The Earth goes around the Sun, not vice versa. The blood circulates. Communicable disease is caused by germs, not miasmas. Emotion is seated in the brain, not the heart. Bleeding with leeches is not medically effective. Etc. Quite the contrary! The fact that we're still using musical instruments made of wood and brass in an otherwise digital age suggests that there is fundamentally wrong with the West's underlying theory of music that is holding it back.

Sethares et al. have cracked that problem, so that music, too, can finally move into the Digital Age. With our innovations, people can choose to play purely-Harmonic timbres in purely-Just tunings (using a very wide range of tunings, in many different temperaments, up to arbitrary p-limits, including tunings that are regular or irregular, major-biased or minor-biased, using an arbitrary number of notes per octave, etc.) -- options that they did not have conveniently before. But also, our innovations give them the option of playing consonantly with pseudo-Harmonic timbres in pseudo-Just tunings, and changing those tunings on the fly, and playing around with the structure of their timbres dynamically. We give you your cake, and encourage you to eat it, too.

And to this, you are objecting?

It may be, Dear Hucbald.SaintAmand, that you remain unconvinced, and will ALWAYS remain unconvinced, no matter what evidence is presented to you. That is not uncommon in paradigm shifts. As Kuhn pointed out, sometimes one must simply wait for the Old Guard to die out.

Wishing you a long and healthy life, I remain

Respectfully,

--JimPlamondon (talk) 02:05, 6 May 2020 (UTC) P.S.: "P5" is an abbreviation for "Perfect Fifth," which is why I consistently preface it with "tempered."

JimPlamondon, Sir, this discussion is pointless. Let me stress a few points:

1. I am not "thinking of music in terms of ratios of small whole numbers". When I wrote that "the very concepts of consonance and dissonance only make sense in the case of (more or less) harmonic partials", more or less was meant to allow for (slight) inharmonicity.
2. A sound with widely inharmonic partials has no clearly identifiable pitch. The perceived pitch often is around partial 1, but oscillating both in pitch and intensity at rates that can be estimated close to that of the true (GCD) fundamental, which depends on the inharmonicity. The phenomenon is of the order of frequency modulation, which as is well known makes fussy mathematics: I don't think that one has at this point anything better than estimates of the oscillation frequencies.
3. Western music, once it had chosen for polyphony, favored harmonic timbres, periodic sounds, stable emissions, etc., to an extent unknown in many musics of the world: instruments of sustained sound (bowed instruments, winds, organ, etc.) took a central role in the orchestra. The distinction between consonance and dissonance became one of the main issues of the musical game. We lost other aspects of the game, e.g. rhythmic complexity.
4. In other types of musics, where the distinction consonance/dissonance does not play the same role, or not to the same extent, fluctuating sounds are less problematic. You mention the case of the gamelan. It happens that I played in both a slendro and a pelog gamelan ensemble for several years. Our instruments were certainly tuned neither in 5TET, nor in just intonation (I wonder whether you figure out how one tunes bonang gongs). We played heterophony, i.e. the same melodies more or less freely varied. This often resulted in simultaneities of adjacent intervals (i.e., roughly, fifths of octave, tones, or semitones), which we never resented either as particularly dissonant or particularly consonant: consonance and dissonance merely were unimportant and not really differenciable.
5. You seem to believe that Western music (and music in general) should abandon wood and brass instruments in favor of electronic ones. (You write: "The fact that we're still using musical instruments made of wood and brass in an otherwise digital age suggests that there is fundamentally wrong with the West's underlying theory of music that is holding it back.") Well, at times I still use an ink pen, even although I do have a computer, and I still read books printed on paper – some are not in English, some even are in Latin!: do you consider this disease incurable, doctor?

But once again, I don't see the point in further discussing all this. — Hucbald.SaintAmand (talk) 14:43, 6 May 2020 (UTC)