The Harmonic Series and Harmony

Here's another idea that may not be new...

In the late 90's I came across the idea of an interval being consonant if it has coincident harmonics. 3's fifth harmonic (15) and 5's third harmonic (15) are the same. For around ten years I believed that both note's harmonic series' had to be considered when working out the harmony value of an interval.

Now I think that this is unnecessary. For me the harmony of an interval depends on "repeating points". If you plot the fundamentals of the 3 and the 5 (in phase) then the waveforms "repeat themselves" every, say, 4 inches. Now plot the first 16 harmonics of both the 3 and the 5 and the "repeating points" are still the same, every 4 inches.

This is assuming of course that the harmonic series of a note whose fundamental is 'x' is exactly x, 2x, 3x, 4x, 5x etc. This is an "ideal" note. In practice however the second and higher harmonics are slightly higher (to a lesser or greater degree, depending on the instrument). I think that this is what timbre means. So in this case the repeating points are blurred.

Someone pointed out that my formula doesn't take timbre into account. Well, I think my formula should serve as a 'rough guide' to consonance, no matter the timbre of the notes. Once the fundamentals are perfectly in tune with each other I wouldn't worry too much about timbre.

John.

>"Now I think that this is unnecessary. For me the harmony of an interval
depends on "repeating points". If you plot the fundamentals of the 3 and the 5 (in phase) then the waveforms "repeat themselves" every, say, 4
inches. Now plot the first 16 harmonics of both the 3 and the 5 and the
"repeating points" are still the same, every 4 inches."

You know, this is exactly why I like to rate scales using things like x/3 (IE 4/3,5/3, 6/3) and x/12 (13/12,14/12,15/12)...those o-tonalities give you a clear idea exactly how long it will take (so far as periodicity for a "worst case scenario" chord within the scale to, yes, repeat. For x/12 the "worst possible interval" is 24/23 (12*2)/(12*2-1). Then you can worry about trying to make the least periodic interval more periodic by, say, knocking the 23rd harmonic out of that series to help make a more periodic scale.

>"Someone pointed out that my formula doesn't take timbre into account."
Ideally (at least for electronically processed music) something like using an FFT-based phase-vocoder to process an existing song to align overtones with notes in the tuning could eliminate that issue entirely. For acoustic instruments though, you'd probably want to build one version of the scale for mostly even timbre instruments (IE guitars) and another for mostly odd timbres (IE flutes). There are only a handful of tunings, such as BP, that I'v seen which are designed for odd-harmonic timbre harmony.

Are (if I were you) I'd just worry about the periodicity between all possible root tones (which JI does a pretty good job with any how) before bothering with anything else.
After that, well...I'd worry a bit about critical band consonance...and I'm still sticking with the idea that intervals closer than 12/11 or so (such as the 15/14 half step) are fairly unusable in chords...unless you have very wide intervals around them such as exists in the C E F G and C E F A B chords (and there aren't many of such types of chords possible).

> Someone pointed out that my formula doesn't take timbre into account. Well, I think my formula should serve as a 'rough guide' to consonance, no matter the timbre of the notes. Once the fundamentals are perfectly in tune with each other I wouldn't worry too much about timbre.
>
> John.

And what if the fundamentals are slightly mistuned?

>

Thanks Michael,

I'm not sure I got all that, I've been working on my own for 14 years and only got in touch with other JI people in January of this year so I'm not up to speed on the terminology.

Critical Band Consonance...for me the narrowest usable interval is 5/6. 6/7 sounds dissonant to me, either on it's own or as part of a chord. No matter what notes are in a chord containing the 6/7 interval I can still hear the dissonance of the 6/7 if I listen carefully.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Now I think that this is unnecessary. For me the harmony of an interval
> depends on "repeating points". If you plot the fundamentals of the 3 and the 5 (in phase) then the waveforms "repeat themselves" every, say, 4
> inches. Now plot the first 16 harmonics of both the 3 and the 5 and the
> "repeating points" are still the same, every 4 inches."
>
> You know, this is exactly why I like to rate scales using things like x/3 (IE 4/3,5/3, 6/3) and x/12 (13/12,14/12,15/12)...those o-tonalities give you a clear idea exactly how long it will take (so far as periodicity for a "worst case scenario" chord within the scale to, yes, repeat. For x/12 the "worst possible interval" is 24/23 (12*2)/(12*2-1). Then you can worry about trying to make the least periodic interval more periodic by, say, knocking the 23rd harmonic out of that series to help make a more periodic scale.
>
>
> >"Someone pointed out that my formula doesn't take timbre into account."
> Ideally (at least for electronically processed music) something like using an FFT-based phase-vocoder to process an existing song to align overtones with notes in the tuning could eliminate that issue entirely. For acoustic instruments though, you'd probably want to build one version of the scale for mostly even timbre instruments (IE guitars) and another for mostly odd timbres (IE flutes). There are only a handful of tunings, such as BP, that I'v seen which are designed for odd-harmonic timbre harmony.
>
> Are (if I were you) I'd just worry about the periodicity between all possible root tones (which JI does a pretty good job with any how) before bothering with anything else.
> After that, well...I'd worry a bit about critical band consonance...and I'm still sticking with the idea that intervals closer than 12/11 or so (such as the 15/14 half step) are fairly unusable in chords...unless you have very wide intervals around them such as exists in the C E F G and C E F A B chords (and there aren't many of such types of chords possible).
>

Mike,

if the fundamentals are slightly mistuned then they don't figure in my system. A case could be made for stretch tuning intervals (as piano tuners do), making them slightly wider but I think that this only applies, over a two octave range, to the 1/2, 1/3 and 1/4 intervals. It seems to me (I have investigated stretch tuning in the past) that all other intervals (unison excepted), over the two octave range would benefit from "squeezing" instead of stretching.

If you "stretch" a small number of strong Major chords will sound better but all other chords will sound worse. If you "squeeze" the opposite is true. For me the best compromise and simplest approach is to neither stretch nor squeeze intervals. In other words the fundamentals should correspond exactly to the integer ratio.

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Someone pointed out that my formula doesn't take timbre into account. Well, I think my formula should serve as a 'rough guide' to consonance, no matter the timbre of the notes. Once the fundamentals are perfectly in tune with each other I wouldn't worry too much about timbre.
> >
> > John.
>
> And what if the fundamentals are slightly mistuned?
>
> >
>

John,

- When are two notes ever going to be so perfectly in tune that they
are perfectly periodic?
- Are you saying that there is no point to temperament?
- Do you deny that there are psychoacoustic phenomena that exist that
correspond to the tempering of intervals, and that these phenomena can
be used to musical effect?

-Mike

On Fri, Apr 9, 2010 at 5:46 PM, jfos777 <jfos777@...> wrote:
>
>
>
> Mike,
>
> if the fundamentals are slightly mistuned then they don't figure in my system. A case could be made for stretch tuning intervals (as piano tuners do), making them slightly wider but I think that this only applies, over a two octave range, to the 1/2, 1/3 and 1/4 intervals. It seems to me (I have investigated stretch tuning in the past) that all other intervals (unison excepted), over the two octave range would benefit from "squeezing" instead of stretching.
>
> If you "stretch" a small number of strong Major chords will sound better but all other chords will sound worse. If you "squeeze" the opposite is true. For me the best compromise and simplest approach is to neither stretch nor squeeze intervals. In other words the fundamentals should correspond exactly to the integer ratio.
>
> John.
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > > Someone pointed out that my formula doesn't take timbre into account. Well, I think my formula should serve as a 'rough guide' to consonance, no matter the timbre of the notes. Once the fundamentals are perfectly in tune with each other I wouldn't worry too much about timbre.
> > >
> > > John.
> >
> > And what if the fundamentals are slightly mistuned?
> >
> > >
> >
>
>

On 10 April 2010 00:19, jfos777 <jfos777@...> wrote:
> Here's another idea that may not be new...
>
> In the late 90's I came across the idea of an interval being
> consonant if it has coincident harmonics. 3's fifth harmonic (15)
> and 5's third harmonic (15) are the same. For around ten years I
> believed that both note's harmonic series' had to be considered
> when working out the harmony value of an interval.
>
> Now I think that this is unnecessary. For me the harmony of an
> interval depends on "repeating points". If you plot the fundamentals
> of the 3 and the 5 (in phase) then the waveforms "repeat themselves"
> every, say, 4 inches. Now plot the first 16 harmonics of both the 3
> and the 5 and the "repeating points" are still the same, every 4 inches.

Right, not a new idea. If you have periodic (harmonic) timbres, the
period of repetition of the chord is as you say. But there are also
radically inharmonic timbres out there, and sensory consonance
theories have to take them into account. William Sethares' "Tuning,
Timbre, Spectrum, Scale" is the standard reference for this.

A locally grown consonance theory that doesn't consider harmonics
explicitly is Paul Erlich's harmonic entropy.

Graham

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Are (if I were you) I'd just worry about the periodicity between all possible root tones (which JI does a pretty good job with any how) before bothering with anything else.
> After that, well...I'd worry a bit about critical band consonance...and I'm still sticking with the idea that intervals closer than 12/11 or so (such as the 15/14 half step) are fairly unusable in chords...unless you have very wide intervals around them such as exists in the C E F G and C E F A B chords (and there aren't many of such types of chords possible).
>

Don't you think that the 15:16:20:24 BCEG chord is a nice voicing of the major 7th chord?

Billy

--- In tuning@yahoogroups.com, "jfos777" <jfos777@...> wrote:
>
> Thanks Michael,
>
> I'm not sure I got all that, I've been working on my own for 14 years and only got in touch with other JI people in January of this year so I'm not up to speed on the terminology.
>
> Critical Band Consonance...for me the narrowest usable interval is 5/6. 6/7 sounds dissonant to me, either on it's own or as part of a chord. No matter what notes are in a chord containing the 6/7 interval I can still hear the dissonance of the 6/7 if I listen carefully.

The most clean and tuneful of the true quadads is the 4:5:6:7 chord. The alternate tunings using a Pythagorean or 5-limit minor 3rd on the top interval sound messy when played alternately alongside the 4:5:6:7. Do you think tuneful music should stick with 5 as the highest prime limit?

Billy

The same example that I always use :)

I love chords like

G E-F B-C G-A

for example. Where the dashes represent a second, so you can see how the
chord is laid out.

Interesting "crunchy" voicings like that basically form half of my
vocabulary, so I dunno why people over here hate them so much sometimes :D

-Mike

On Sun, Apr 11, 2010 at 2:09 AM, Billy <billygard@comcast.net> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Michael
> <djtrancendance@...> wrote:
> > Are (if I were you) I'd just worry about the periodicity between all
> possible root tones (which JI does a pretty good job with any how) before
> bothering with anything else.
> > After that, well...I'd worry a bit about critical band consonance...and
> I'm still sticking with the idea that intervals closer than 12/11 or so
> (such as the 15/14 half step) are fairly unusable in chords...unless you
> have very wide intervals around them such as exists in the C E F G and C E F
> A B chords (and there aren't many of such types of chords possible).
> >
>
> Don't you think that the 15:16:20:24 BCEG chord is a nice voicing of the
> major 7th chord?
>
> Billy
>
>
>

Billy,

for me the n prime limit is irrelevant when constructing chords. If I'm right, all intervals (in harmony) that are wider than an octave are good (e.g. 23/47).

In my system a chord is good only if every interval that occurs in the chord has a value of 0.75 or higher according to my formula for consonance (see earlier posts).

John.

--- In tuning@yahoogroups.com, "Billy" <billygard@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "jfos777" <jfos777@> wrote:
> >
> > Thanks Michael,
> >
> > I'm not sure I got all that, I've been working on my own for 14 years and only got in touch with other JI people in January of this year so I'm not up to speed on the terminology.
> >
> > Critical Band Consonance...for me the narrowest usable interval is 5/6. 6/7 sounds dissonant to me, either on it's own or as part of a chord. No matter what notes are in a chord containing the 6/7 interval I can still hear the dissonance of the 6/7 if I listen carefully.
>
> The most clean and tuneful of the true quadads is the 4:5:6:7 chord. The alternate tunings using a Pythagorean or 5-limit minor 3rd on the top interval sound messy when played alternately alongside the 4:5:6:7. Do you think tuneful music should stick with 5 as the highest prime limit?
>
> Billy
>

>"Don't you think that the 15:16:20:24 BCEG chord is a nice voicing of the major 7th chord?"
How is BCEG a major 7th chord...unless (perhaps) you are thinking of it as C E G B inverted?

But yes, I think it's a good voicing, but only works IMVHO because you have the large gap between C and E to make up for the tiny gap between B and C so far as critical band dissonance is concerned.

As I said before
Me>"intervals closer than 12/11 or so (such as the 15/14 half step) are
fairly unusable in chords...unless you have very wide intervals around
them such as exists in the C E F G and C E F A B chords"
Here your "very wide interval" is (again) that gap between C and E in (BCEG)...it isn't an exception to the "rule" I quote above. Granted though, "even" with the limitations of using "standard" half steps that way there are a few very interesting chords like that available under the scope of "common theory"...just not that many.

Far as rephrasing that "inverted" major 7th chord, I would try something along the lines of 12:13:16:19 as a voicing for that chord (here, yet again, I am using the "forbidden" 13/12 interval instead of the usual JI "major or minor" half steps of 15/14 or 16/15).

>"If I'm right, all intervals (in harmony) that are wider than an octave are good (e.g. 23/47)."
Even then (much as I champion the idea of avoiding critical band dissonance)...I think even with "near perfect" critical band dissonance formed by far-apart tones you can go wrong by being too non-periodic.

Try the interval 33/16 (IE even with just playing two tones)and you'll hear anti-periodic beating fierce enough to ruin any gains you got from critical band consonance (even with sine waves as the notes and no overtones...the beating period takes so long to repeat it makes it sound dissonant).
Note: the example 47/23, despite having a very high "prime" limit, DOES work (at least to my ears), but your brain pretty much rounds the clashing overtone to the 2/1 harmonic and it "close enough" to being periodic that your brain doesn't notice as the high prime 47/23 rounds well to the nearest low-prime of 2/1 (kind of works a bit like temperament does in that way: like comparing 160kb/s mp3 quality to CD quality it's "just close enough to trick you").

Sadly, I don't think there's a way to nail consonance just by either concentrating on pure periodicity theory (IE JI and "harmonic entropy") or the concept of critical band dissonance...to get it right, IMVHO, you have to take both into account.

Thanks Michael,

I clearly have a lot more work to do. I played the 33/16 interval using a piano voice and it clearly sounded "jarring". With a sine lead voice however the beating was obvious but (to my ear and my ear has fooled me before) not too unpleasant.

The 47/23 was a poor choice as an example being so close to the octave but just out of curiosity I played a 47/19 interval and again, with a piano voice it didn't sound great but with the sine lead voice it sounded beautiful.

I need to look into this but I haven't ruled out my formulas just yet. Perhaps they are correct with pure sine wave tones only (no overtones).

BTW, using my formula the 33/16 has a value of 0.804 and the 47/19 has a value of 0.8348 and the latter does indeed sound sweeter (both played with the sine lead voice).

47/19 is very "non periodic" yet sounds beautiful (using a sine lead voice) so perhaps you can't be "too non periodic".

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"If I'm right, all intervals (in harmony) that are wider than an octave are good (e.g. 23/47)."
> Even then (much as I champion the idea of avoiding critical band dissonance)...I think even with "near perfect" critical band dissonance formed by far-apart tones you can go wrong by being too non-periodic.
>
> Try the interval 33/16 (IE even with just playing two tones)and you'll hear anti-periodic beating fierce enough to ruin any gains you got from critical band consonance (even with sine waves as the notes and no overtones...the beating period takes so long to repeat it makes it sound dissonant).
> Note: the example 47/23, despite having a very high "prime" limit, DOES work (at least to my ears), but your brain pretty much rounds the clashing overtone to the 2/1 harmonic and it "close enough" to being periodic that your brain doesn't notice as the high prime 47/23 rounds well to the nearest low-prime of 2/1 (kind of works a bit like temperament does in that way: like comparing 160kb/s mp3 quality to CD quality it's "just close enough to trick you").
>
> Sadly, I don't think there's a way to nail consonance just by either concentrating on pure periodicity theory (IE JI and "harmonic entropy") or the concept of critical band dissonance...to get it right, IMVHO, you have to take both into account.
>

John>"Thanks Michael,
I clearly have a lot more work to do. "

It's all good...I haven't figured it all out either. However I've figure out enough to be sure that you can't take either the critical-band-consonance theory or JI theory alone and solve nearly all tuning/consonance-related problems. It's great we have people around with your attitude though IE more dedicated to solving the problems at hand than showing that they know a lot or taking too much ego in their work.

>"I played the 33/16 interval using
a piano voice and it clearly sounded "jarring". With a sine lead voice
however the beating was obvious but (to my ear and my ear has fooled me
before) not too unpleasant."
>"but just out of curiosity I played a 47/19 interval and again, with a
piano voice it didn't sound great but with the sine lead voice it
sounded beautiful."
it would be nice if I had a second person on here verify (or "de-verify"?) this, but I'm guessing in the 47/19 piano case a low overtone of the first tone on the piano is clashing with the first root or first overtone of the second (higher) base tone.
This type of overtone/root clashing I've found is generally the only thing that can make an interval played with a sine wave sound better than with a full instruments. This goes hand-in-hand with the criticism given before of (paraphrased) "a weakness is your theory's not take timbre into account".

Also to note...aligning overtones to match root tones (IE making the timbre and scale match and change for each different root tone) and using Sethares-style timbres are the two ways to make "instruments act like sine waves" and achieve sine-wave like consonance.
Both of these (of course) involve electronic processing/manipulation of sound, though, so it is a bit of a barrier for non-electronic music (unless there was a guitar-pedal or "real-time rack processor" designed to perform the processing in real-time...really wish there was). ;-)

One thing though...I actually think "just optimizing for sines" is not a bad idea because if you start optimizing for all possible overtones you are very likely to just run straight into JI diatonic (as so many theorists, including Sethares, have).

>"I need to look into this but I haven't ruled out my formulas just yet.
Perhaps they are correct with pure sine wave tones only (no overtones)."

I think, at least with fairly low prime limit ratios your formula should work pretty well...but I'm pretty sure temperaments with high-limit fractions fairly near low-limit ones will confuse it (due to things like the "so close to the octave" mis-match you ran into).
The thing I believe you formula would ultimately need (even with regards to "just" sine wave) is some way to start exponentially (and seperately) subtracting the critical-band and JI/"periodicity" types of consonance past a certain point. For example, for you numbers around 6/5 might be one "limit" on periodic consonance while around 1.09 might be one for "critical band consonance".
Another plus I figure would be to assume that two notes in a scale very close to each other need to be more periodic than those, say, two notes much further apart because the first two are fighting against more critical band dissonance and need less periodic dissonance in order to compensate for this (here things like super-particular ratios, of course, come in handy).

>"47/19 is very "non periodic" yet sounds beautiful (using a sine lead voice) so perhaps you can't be "too non periodic"."
Well, come to think of it, 47/19 = 2.47, which is VERY close to 2.5 (AKA 5/2) which is very periodic. Same issue as you ran into with the being too close to the 2/1 (octave) before, only instead you're too close the to the "octave plus the major third". Here it seems temperament again appears to comes into play.

The only conclusion I'm about 99% sure of is you only have a certain amount of slack to far as periodicity and critical band...and if you either grossly violate one of these limitations or start pushing one without giving a corresponding amount of lack on the other you run into trouble.

-Michael