interval quality question

[shugs]

great info on the guitar.....My original question had to do with textbook definitions, nothing more, and how and where this discussion has evolved has been useful to me.

As a practical matter, I don't care if two notes are called augmented , diminished, perfect, consonant, dissonant, whatever. I just want to know the definitions and to be able to workj with them when we move , next into chords

I was originally perplexed how two identically sounding intervals could be either classified as dissonant or consonant and I think we all fleshed that out pretty good.

As far as what are three half steps I think someone mentioned ? I suppose that would be classified as an augmented second or a minor third depending about the letter of each note ( and by my book would either be dissonant or consonant respectively)

[cryophonik]

As far as what are three half steps I think someone mentioned ? I suppose that would be classified as an augmented second or a minor third depending about the letter of each note ( and by my book would either be dissonant or consonant respectively)

Yes, that would be correct but and you could even call it a double-diminished fourth if you want. As you've been reading here, it's important to separate the textbook definitions/nomenclature of dissonance/consonance (objective) from how they may sound in use (subjective).

[shugs]

triple augmented unison

[vurt]

triple augmented unison

is that a medical condition?

[cryophonik]

triple augmented unison

is that a medical condition?

If it lasts for more than 30 hours, call your doctor

[jancivil]

an augmented second or a minor third depending about the letter of each note ( and by my book would either be dissonant or consonant respectively)

taking the name for a thing over the thing itself. This is just poor pedagogy.

[N__K]

I was originally perplexed how two identically sounding intervals could be either classified as dissonant or consonant and I think we all fleshed that out pretty good.

Yep, some excellent posts in this thread, I think. Not sure if I can add much, but I'll try.


A lot of nomenclature of music theories comes from traditional practices. One of them is naming intervals in context of how pitches in different keys and scales would be marked on the staff of 7 "natural" pitches, with various indications for deviating from them. That makes perfect sense in some cases, but may seem perplexing if one is thinking within context of 12-TET.

Also, from perspective of simple math, the mismatch of number ranges - having to represent 12 or more pitches in a system which was apparently originally intended for 7 - may cause a bunch of "highly illogical, captain" exclamations


I could go on about such things, like why aren't white piano keys represented on the staff by white spaces and black keys by black lines, and why are note lengths represented by symbols instead of actual visual length. One possible answer to those is that visually correlating systems (such as "klavarskribo" or 12-TET piano roll) take more space on paper, and thus the dominant staff system allows for more information to fit in smaller space. It functions effectively as a "visual compression format", at the cost of more cognitive processing required in encoding and decoding the data. But that's a subject for another thread, perhaps...


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Nowadays we have ways to measure and visualize intervals and their partials via high-frequency-resolution spectrum analysis, which may help to understand concepts of consonance and dissonance in relation to actual physical phenomena.
For REAPER users, here's a link to a spectrum analyzer which shows a piano-like semitone grid as a reference: https://forum.cockos.com/showthread.php?t=258602
This is what a major third above Middle C looks like on it, played on a piano:

spectrum - Middle C + E above it on a piano (1600x601).png

Personally I think that there are indeed "consonances" and "dissonances", related to how pitched sounds (and their partials) are perceived. I originally learned them as a mix of mathematics (ratios, decimals etc.) and traditional textbook definitions, while listening to how they sound. Nowadays I also like to look at spectra of sounds, to see how what I'm hearing relates to physics. Speaking of which, it must be noted that FFT spectrum analysis has some compromises that one has to be aware of, such as smearing of time resolution when high frequency resolution is used.


Also, again in terms of simple math, I once came up with this visualization of intervals by simple number ratios (much like in just intonation):

interval ratios (1600x848).png

Those are grids made from interval ratio numbers, all scaled to same width. They suggest a correlation between visual and aural complexity - consonance (simplicity) and dissonance (complexity).

I'm not sure what practical use it may have, and I have no idea whether it correlates in any way to what's actually happening in human neural circuitry related to decoding sound. Probably not But perhaps it's something fun to think about, nevertheless.
Curiously, despite the idea being simple, I have not seen this kind of visualization anywhere else, but I suspect that someone somewhere has made it before...



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an augmented second or a minor third depending about the letter of each note ( and by my book would either be dissonant or consonant respectively)

taking the name for a thing over the thing itself. This is just poor pedagogy.

Indeed. That is why I try to include some aspects of "hard science" in my personal musical practice - it provides at least some amount of objectivity and certainty, no matter how phenomena may be named in any textbook. In a way, it helps me to dare go outside norms, both in theory and practice.

[BertKoor]

Also, again in terms of simple math, I once came up with this visualization of intervals by simple number ratios (much like in just intonation):

[interval ratios (1600x848).png]

Those are grids made from interval ratio numbers, all scaled to same width. They suggest a correlation between visual and aural complexity - consonance (simplicity) and dissonance (complexity).

When looking at size of the squares alone, you might conclude the minor third is just as consonant/dissonant as the tritone. Hmmmmm...

These ratios fit the just intonation systems we used before we settled on the 12-tone equal temperament compromise.
I'd like to take the opportunity to compare them.

Code: Select all

PRIME        1:1    C
minor 2nd   16:15   Db +12 cents
Major 2nd    9:8    D   +4 cents
minor 3rd    6:5    Eb +15
Major 3rd    5:4    E  -14
FOURTH       4:3    F   -2
tritone      7:5    F# -17
FIFTH        3:2    G   +2
minor 6th    8:5    Ab +14
Major 6th    5:3    A  -16
minor 7th    7:4    Bb -31
Major 7th   15:8    B  -12
OCTAVE       2:1    C

[N__K]

Yep - when thinking about qualities of intervals, I see 12-TET and just intonation as sort of "macro and micro" approaches.

In a sense, 12-TET treats the "doubling of frequency" aka "octave" phenomenon as a container of intervals, and divides it into 12 equal pitch slices.
Thinking pedantically, it should be called dodecave, but whatever And, in case there are readers previously unfamiliar with the subject, it's worth mentioning that pitch is logarithmic, while frequency measured in Hz is linear.

Just intonation, on the other hand, seems to look at it mostly from perspective of smaller-than-octave intervals. The results of those approaches come close enough to eachother for some interesting conclusions.



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As another way to look at it, here's a visualization of finding intervals in harmonic series, contrasted with a 12-TET grid in the background:

harmonic series and intervals (2021-11-09, 1600x844).png

The chord name contrivances on the left there may not make much sense - apologies for that.

Most important in that visualization is perhaps that "simplest" or "most consonant" intervals are found earliest in the series - and the order of finding the intervals seems to correspond to "order of complexity" of ratio grids in the previously-posted picture.

[jancivil]

I don't see any necessary correlation between the density of squares and the sound of the intervals.
"They suggest a correlation between visual and aural complexity" to you they do, but maybe you're that infatuated with your premise. Let's deconstruct.

That has a major second as "complex" as a major seventh, for instance. Let's start from the beginning. Dividing a vibrating string into segments: divide in two is the simplest, 2:1 or 1:2 is our result. Divide into three parts should be our next simplest exercise: 3:2 or 2:3 (aka 4:3). Is 3:2 simpler than 2:3? 1 and a half, 1.5, vs .66. The second number looks more complex on the surface, one may suppose.
Is this acoustically the case, though. Apparently the natural consonance of a 1:1 agreement has been disturbed to the same degree. Divide that 3rd of a string some more. No, 15:8 has more going on than 9:8. Few will tell you a major seventh is the same degree of dissonance as a major 9th or major 2nd.

So as to your picture of squares in a square, acoustically this is not going to show as true. If one wishes a science upon music, there is one ready to go without having to invent: acoustics. We can make things more complex by adding harmonics. The analysis of spectrum is not needed for these simpler matters; these do not need that novel a way to look at them. It strikes me as an unnecessary reduction, to pictures, frankly.

"whether it correlates in any way to what's actually happening in human neural circuitry related to decoding sound" that's neuroscience, not music, an actual whole different field of endeavor and thought.

Human perception changes acoustical reality in no way, except that we'll find some people will have a better functioning mechanism for it than others. For me the study of perception of intervals is interesting (for one thing it's shown me beyond doubt that the perception of "octave equivalence" is_not universal); however as interesting as all that may be, it changes nothing about an octave.

[vurt]

I don't see any necessary correlation between the density of squares and the sound of the intervals.
"They suggest a correlation between visual and aural complexity" to you they do, but maybe you're that infatuated with your premise. Let's deconstruct.

i don't know if I'd go that far, some people are just more visually oriented.
for me personally, not so much the above, can't make it out on the phone, but as an example, mixing fundamentals, they really clicked for me after seeing dave gibsons stuff, where he uses colours and shapes, to show the mix elements.

the rest, i can't comment on, you've both gone over my head

[jancivil]

having to represent 12 or more pitches in a system which was apparently originally intended for 7 - may cause a bunch of "highly illogical, captain" exclamations

May cause whom to exclaim? Why? Why not show this logical problem.

First of all, the 12 we still take as essential was established in antiquity (an epoch prior to the modern paradigms) by dividing string lengths, and further calculations in order to make something geometrically impossible possible; in order to make our two simplest products (2:1 and 3:2) cohere, the comma (there are more than one; the first necessary one appears to have been 531441: 524288, called the Pythagorean comma).

So, 3:2, 9:8, 27:16, 81:64, 243:128, 729:512, 256:243 are our first 7 via multiplying the first result that will meaningfully multiply, 3:2. G D A E B F# C# is our first gamut; it extends to G#, D#, A#, E#, B#, Fx*. Our deciding to name certain of these otherwise is strictly from convention. We can order it to spell the exact same things differently. C G D A E B F#; F C G D A E B; Bb F C G D E...
(*: that last one needs that comma to take us back home, or this just keeps going out.)

Which bit is illogical? Sure, our first seven form a scale of seven tones. It's shown to be rather musical by a number of cultures, for a long time (at this juncture I want to point out that primary acoustical heptatonic scale is what we call Lydian).
I don't know if they were all about 7 during the era these establishing experiments were done or not. But ok, seven is a definite thing, but the other five of twelve arrive by simple multiplication originally; and the ways to make 7 letter names into 12 is purely conventional. There is no logical problem just owing to a visual, this should not be reduced to depictions/pictures over what's real. The reasoning behind all sharps and flats is simple and unassailable.

Apparently the real statement there is 'accidentals confuse people'. This one is a real eye-roller. I strongly recommend applying deeper thought to problems, or perhaps draw from music theory knowledge rather than seek to reinvent it to obtain novelty. This picture of smaller and smaller squares is not only of no use, it's a distraction.
That's all in your head, it's not real.

[jancivil]

I don't see any necessary correlation between the density of squares and the sound of the intervals.
"They suggest a correlation between visual and aural complexity" to you they do, but maybe you're that infatuated with your premise. Let's deconstruct.

i don't know if I'd go that far, some people are just more visually oriented.

Not our problem here, since visuals are not sound: can the picture be demonstrated to work. I've demonstrated it doesn't. 9:8 looks as dense as 243:128. Now, we may correct 243:128 to 15:8 for the simpler expression, but the argument the latter is simpler in terms of dissonance {vibration} is not implicit in that depiction either. The difference between those two products of rational intonation is 81:80 ("syntonic comma"). In plain physical terms a major 2nd is not as dissonant as a major 7th (15:8 or what-have-you), not even very close. The picture misleads kind of dramatically.

[vurt]

ok, like i said, that bits over my head!

that said, i wouldn't know of any way this could be visualised so "simply".
without reverting to the old "dm is the saddest key" type thing where an interval rather than its context defines the thing.

[jancivil]

Well, it's rather the same quality of problem as taking the name for a thing over the thing itself, taking a picture over acoustical properties. The picture does not cohere with known properties and its basis is suspect. The requirement to make a simple picture like that is that individual's need, I don't know of another requirement to look at it that way, myself.