equal tempered and tuning

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guitarrista
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Re: equal tempered and tuning

Post by guitarrista » Tue Oct 18, 2016 7:01 pm

Alan Carruth wrote:You also get beating between related notes that are 'close'. For example, your A string would be tuned to 440 Hz at standard pitch. The high E in 'just' tuning would be at 660 Hz, or a 3:2 ratio. In ET the high E is at 329.6 Hz. .4 Hz. off, and you can hear a slight beating a little slower than once every two seconds if you listen very carefully.

Some harps, and most pianos, use 'stretch' tuning. The upper partials of the lower strings are not very close to harmonic: the stiffness of the strings shifts them sharp. They will beat with the higher notes that they're supposed to agree with. It's useful in those cases to stretch the octaves, making the notes above the center of the instrument progressively sharper, and the low note progressively flatter, so that things are not too much out of line. One of my students found that the harp she'd made sounded much nicer when the tuning was stretched by about 3 cents per octave.
Yup. Even with ideal strings (i.e. no stiffness to make overtones sharp due to the extra restoring force), overtones of fundamental frequencies which are not particularly close can get quite near another and have a beating effect. In fact, this is part of the basis for perceiving dissonance in intervals - more dissonant intervals have more overtones which are near one another creating interference resulting in a perception of 'roughness'.
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Re: equal tempered and tuning

Post by ipso facto » Tue Oct 18, 2016 7:13 pm

guitarrista wrote:Yup. Even with ideal strings (i.e. no stiffness to make overtones sharp due to the extra restoring force), overtones of fundamental frequencies which are not particularly close can get quite near another and have a beating effect. In fact, this is the basis for perceiving dissonance in intervals - more dissonant intervals have more overtones which are near one another creating interference resulting in a perception of 'roughness'.
If that were true then a given interval - a minor second say, would be more dissonant in higher registers than in lower ones. Maybe to you it is - if so, fair enough. It's not to me though - or to Helmholtz, who came close to holding this theory, but ended up having to do some special pleading because of the effect of register (see the webpage I linked to above).

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Re: equal tempered and tuning

Post by guitarrista » Tue Oct 18, 2016 7:20 pm

ipso facto wrote:
guitarrista wrote:Yup. Even with ideal strings (i.e. no stiffness to make overtones sharp due to the extra restoring force), overtones of fundamental frequencies which are not particularly close can get quite near another and have a beating effect. In fact, this is the basis for perceiving dissonance in intervals - more dissonant intervals have more overtones which are near one another creating interference resulting in a perception of 'roughness'.
If that were true then a given interval - a minor second say, would be more dissonant in higher registers than in lower ones.
Why? Could you elaborate on this?
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Re: equal tempered and tuning

Post by ipso facto » Tue Oct 18, 2016 8:03 pm

It's because pitch is a geometric progression i.e. to go a constant distance you multiply by a constant, rather than adding.

If you take the minor second between B2 (123.5 Hz in ET) and C3 (130.8 Hz), the difference in about 17 Hz, so you will have 17 beats per second. If you take it up an octave everything doubles, including the difference, so you now have about 34 beats per second.

Another proof that dissonance can't be equated with beating is that this rate of 34 beats per second is approximately the same as you find between C2 (65.4 Hz) and G2 (98.0 Hz), and yet that is a consonance, even in ET.

It is not strictly true to say everything will be heard to beat more as you go up the register - it would have been more accurate if I'd said that beating depends on register. At some point the individual beats get too fast to hear as separate beats, in the way a flipbook stops flickering if you move it fast enough. At that point you get an illusory third pitch.

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Re: equal tempered and tuning

Post by guitarrista » Tue Oct 18, 2016 9:01 pm

ipso facto wrote:It's because pitch is a geometric progression i.e. to go a constant distance you multiply by a constant, rather than adding.

If you take the minor second between B2 (123.5 Hz in ET) and C3 (130.8 Hz), the difference in about 17 Hz, so you will have 17 beats per second. If you take it up an octave everything doubles, including the difference, so you now have about 34 beats per second.
Perhaps you forget that overtones are all integer multiples of the fundamental, i.e. fundamental times 2,3,4,5,6,7.... not just fundamental times 2, 4, 8.. so you can get other combinations. (BTW you mean 7, not 17, above). Also, doesn't your argument say the opposite of what you said before (which is why I wasn't sure what you meant) - I think here you are saying that minor seconds in higher registers will be perceived as less, not more, dissonant), whereas earlier you said "minor second say, would be more dissonant in higher registers than in lower ones."

A more illustrative example about the role of close overtones in dissonance is to take fundamentals which are not close, as in an interval which is more than an octave (a minor second is not a good illustrative example re: overtones' role because relative loudness matters too and the biggest contribution to dissonance in the case of a minor second is the clash of the fundamentals themselves).

Let's take A-Bb with A=110Hz and Bb = 233.082Hz (13 semitones wide). You get the following overtones, and I've highlighted pairs whose frequencies are closer than the original fundamentals:
Capture5.JPG
What I am saying about the relationship between perceived dissonance and closeness of overtones is not controversial - there are articles in peer-reviewed science journals explaining it.. I'll try to find some later for you if you are interested.
ipso facto wrote:It is not strictly true to say everything will be heard to beat more as you go up the register
Of course - I did not say that - the preferred term is roughness in part for that reason; which is what I used.
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Re: equal tempered and tuning

Post by ipso facto » Tue Oct 18, 2016 10:07 pm

guitarrista wrote:You forget that overtones are multiples of the fundamental, i.e. fundamental times 2,3,4,5,6,7.... not just fundamental times 2, 4, 8.. so you can get other combinations.
I didn't forget - I used the fundamentals because they produce the loudest beats. The same principle will apply to the overtones.
guitarrista wrote:(BTW you mean 7, not 17, above).
Quite right - so you'd to go up 2 octaves rather than one before you got beating at approximately the same rate as the perfect fifth I mentioned.
guitarrista wrote:Also, doesn't your argument say the opposite of what you said before (which is why I wasn't sure what you meant) - I think here you are saying that minor seconds in higher registers will be perceived as less, not more, dissonant, whereas ealier you said "minor second say, would be more dissonant in higher registers than in lower ones."
I don't think so - until you reach the point where the beating becomes an illusory tone, it gets worse as you go higher. But the direction doesn't really matter - if dissonance was roughness and roughness was beating, than the same interval (overtones and all) would be more or less dissonant depending on the octave, and that is not what we find.
guitarrista wrote:What I am saying about the relationship between perceived dissonance and closeness of overtones is not the least bit controversial - there are articles in peer-reviewed science journals explaining it.. I'll try to find some later for you if you are interested.
I disagree there - if you look for example at http://www.music-cog.ohio-state.edu/Mus ... ories.html you will find a survey of various theories of dissonance. On my reading of the site, this particular theory (or rather Helmholtz's refinement of it) is mentioned more for its historical interest than anything else. On any reading though, it is not presented as the orthodoxy. To be honest, I wouldn't be much deterred if it was the orthodoxy - I don't think it works. Obviously, any theory of dissonance has to account for the fact that notes which contain frequencies that are nearly but not quite the same do beat and are dissonant. That is what made the theory so appealing in the first place. What I am saying is only that when you work it through you can't explain dissonance this way, as Helmholtz accepted back in the 19th century when he refined the idea.
guitarrista wrote:
ipso facto wrote:It is not strictly true to say everything will be heard to beat more as you go up the register
Of course - I did not say that - the preferred term is roughness in part for that reason; which is what I used.
This is a misunderstanding (my fault for not being clearer in my earlier post). I wasn't saying you said it - I said it, and was correcting myself. I'm not sure it gets any better if you substitute the term roughness, especially if you have defined roughness in terms of beating.

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Re: equal tempered and tuning

Post by guitarrista » Tue Oct 18, 2016 11:02 pm

Thanks for your reply. Looks like you worked off of an earlier version of my post - perhaps you can take a look at the current version, though not sure if it would change anything substantive you write as a response. I did provide what I consider a better example of overtones being closer than original fundamental frequencies. I will look up the research I had marked before (don't remember Helmholtz being mentioned).

EDIT: OK I did some digging and in the interest of not adding another new post in what is a tangent to the original topic, will add to this one.

Looks like my main source for claiming that "What I am saying about the relationship between perceived dissonance and closeness of overtones is not controversial - there are articles in peer-reviewed science journals explaining it" and earlier "..the basis for perceiving dissonance in intervals - more dissonant intervals have more overtones which are near one another creating interference resulting in a perception of 'roughness'" was Dave Benson's book "Music: A Mathematical Offering" which has proper references to peer-reviewed papers included. It does discuss a modified Helmholtz (the main treatment is off of Plomp and Levelt, 1965) and more recent add-ons and alternatives to that. So basically the sensory dissonance theory is the one I was referring to.

I looked around for more recent papers. Some relevant samples:

1. Peretz, I., & Zatorre, R. J. (2005). Brain organization for music processing. Annu. Rev. Psychol., 56, 89-114..
Despite the saliency of dissonance and its initial account in terms of the poor spatial resolution of the basilar membrane (Plomp & Levelt 1965), its functional origins and neural consequences are still open questions (Tramo et al. 2003). Current evidence suggests that dissonance might be further computed bilaterally in the superior temporal gyri by specialized mechanisms. Assemblies of auditory neuron populations in Heschl’s gyrus exhibit phase-locked activity for dissonant chords but not for consonant chords, as measured with implanted electrodes in both humans and monkeys (Fishman et al. 2001). Such cortical responses to dissonance can be disrupted by bilateral lesion of the auditory cortex, resulting in a loss of sensitivity to dissonance (Peretz et al. 2001).
2. Johnson-Laird, P. N., Kang, O. E., & Leong, Y. C. (2012). On musical dissonance. Music Perception: An Interdisciplinary Journal, 30(1), 19-35.
Many theories, however, allow for both sensory and tonal factors to affect dissonance (McDermott et al., 2010). Such theories need to make predictions about the relative dissonance of any chords (cf. Parncutt, 2006), and so they need to integrate sensory dissonance with principles of tonality that listeners can tacitly acquire.
3. Cousineau, M., McDermott, J. H., & Peretz, I. (2012). The basis of musical consonance as revealed by congenital amusia. Proceedings of the National Academy of Sciences, 109(48), 19858-19863.
Helmholtz (20) is usually credited with the idea that dissonant chords are unpleasant because they contain the sensation of roughness, a notion that was fleshed out by ensuing generations of psychoacousticians (5, 6, 21, 22, 26). Theories of dissonance based on beating have been dominant in the last century and are now a regular presence in textbooks (1–3). However, a second acoustic property also differentiates consonance and dissonance: the component frequencies of the notes of consonant chords combine to produce an aggregate spectrum that is typically harmonic, resembling the spectrum of a single sound with a lower pitch (Fig. 1B). In contrast, dissonant chords produce an inharmonic spectrum. Such observations led to a series of analyses and models of consonance based on harmonicity (23, 27–29). Although beating-based theories are widely accepted as the standard account of consonance, harmonicity has remained a plausible alternative.
4. Large, E. W., Kim, J. C., Flaig, N. K., Bharucha, J. J., & Krumhansl, C. L. (2016). A neurodynamic account of musical tonality. Music Perception: An Interdisciplinary Journal, 33(3), 319-331.
[Helmholtz] proposed that as the auditory system performs a linear analysis of complex sounds, proximal harmonics interfere with one another and produce a sensation of roughness, which he equated with dissonance. Small integer ratios yield more consonant musical intervals, he surmised, because they have more harmonics in common and thus fewer harmonics that interfere. When extrapolated to complex tones, the interaction of pure tone components predicts the perception of consonance well (Kameoka & Kuriyagawa, 1969). Recently, there has been a renewed interest in the neurophysiological basis of consonance. Theories based on the neural processing of pitch relationships have led to the development of concepts such as harmonicity and dynamical stability. Harmonicity is the degree to which the frequency spectrum of two complex tones resembles the harmonic spectrum of the difference tone of the fundamental frequencies (Gill & Purves, 2009; Tramo, Cariani, Delgutte, & Braida, 2001). Dynamical stability is based on the synchronization properties of ensembles of coupled neural oscillators (Shapira Lots & Stone, 2008), which we describe in further detail below. Interestingly, both theories relate consonance and dissonance to simple integer frequency ratios. Integer ratio-based predictions have been shown to account for generalizations about musical scales cross-culturally (Gill & Purves, 2009) and to account for the standard ordering of consonances as described in Western music theory (Shapira Lots & Stone, 2008).
All in all, I'd say that sensory dissonance theory is not controversial or disproved, but certainly does not seem (and is not meant to be) the whole story. In that sense, you were right to point out that it is not the only basis for consonance/dissonance and I modified my earlier post accordingly. Thanks for spurring me to revisit this subject!
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Re: equal tempered and tuning

Post by amezcua » Tue Oct 18, 2016 11:28 pm

Why I decided to change to an unequal temperament comes from being most of my life a violin player . I had no problem listening to other guitarists but playing the notes myself I overlapped into the way a violinist is continually monitoring the pitch of every note . A similar urge to move away from ET happened with our piano . My daughter played and that was fine. When I started to learn the piano the same overlap happened . But listening to famous players on their Steinways was spoiled when I became more conscious of the pitches and intervals. Sadly there was no going back . So this part of my personal journey .

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Re: equal tempered and tuning

Post by ipso facto » Wed Oct 19, 2016 6:43 pm

guitarrista wrote:All in all, I'd say that sensory dissonance theory is not controversial or disproved, but certainly does not seem (and is not meant to be) the whole story. In that sense, you were right to point out that it is not the only basis for consonance/dissonance. Thanks for spurring me to revisit this subject!
OK, well I understand your point of view. I think that there are fundamental problems with this as a theory and that, to the extent that it is still current, it is only because no-one has come up with anything better.

Looking at your example, I am not convinced you would hear (I would hear) beating of the 8th harmonic of the A against the 3rd harmonic of the Bb, because on most instruments at least these will be very quiet. Also I think this would produce a difference tone rather than beating - the Australian site I linked to higher up the thread puts the threshold at about 40 Hz.

More fundamentally though, I would say the same principle applies. Dissonance can't just be the fact of beating, or all dissonant intervals would be equally so - the idea has to be that the dissonance is in the intensity of the beating, its loudness, the pitch relationship between the beating pairs and the fundamental, or some combination of those factors. The trouble is that these all change when you move a given interval up or down. Looking at your example, the initial pattern is:

A(1st) vs Bb (Fnd) - 13 beats per second
A(3rd) vs Bb (1st) - 26 beats per second
A(5th) vs Bb (2nd) - 39 beats per second
A(8th) vs Bb (3rd) - 58 Hz (difference tone)

Helmholtz thought that 35 beats per second was hardest on the ear, so in terms of intensity, this is fairly bad. In terms of loudness, probably not so much.

If you now move the whole shebang up an octave, you will still have one at 26 beats per second, but this relates to a different pair and will therefore be quieter, as well as having a different pitch relationship to the fundamental. The other pairs are now producing difference tones.

For me the essential reason why the beating pattern changes when you transpose is that the beating relationship is an arithmetical one (one frequency minus the other) whereas the relationship between a given interval and that same interval transposed is a geometric one, based on multiplication / division rather than addition / subtraction. It follows that the beating pattern associated with two pitches separated by a given interval will change when you tranpose the interval. That means that, unless you say that the dissonance or consonance of the interval also changes, you can't equate dissonance with the pattern of beating, any more than with the simple fact of beating.

If you go up another octave (which takes you to concert A on the bottom) you have:

A(1st) vs Bb (Fnd) - 52 Hz
A(3rd) vs Bb (1st) - 104 Hz
A(5th) vs Bb (2nd) - 156 Hz
A(8th) vs Bb (3rd) - 232 Hz

So at this point all of the pairs are producing difference tones, with the beating having been replaced by a more complex cluster of (perceived) pitches. The relationship between the pitch of the difference tones and the pitch of the sounded notes will now remain constant if the interval is transposed upwards. However, this this fact can't be used to explain the dissonance of the interval, because the explanation would itself contain the problem we started out with, i.e. how to account for the fact that some pitch relationships are dissonant. Also, as you move back down, perhaps in smaller steps, the difference tones will disappear one by one and be replaced with beating - but not only does the interval remain dissonant, it remains dissonant in the same way.

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Re: equal tempered and tuning

Post by ipso facto » Wed Oct 19, 2016 6:47 pm

amezcua wrote:Why I decided to change to an unequal temperament comes from being most of my life a violin player . I had no problem listening to other guitarists but playing the notes myself I overlapped into the way a violinist is continually monitoring the pitch of every note . A similar urge to move away from ET happened with our piano . My daughter played and that was fine. When I started to learn the piano the same overlap happened . But listening to famous players on their Steinways was spoiled when I became more conscious of the pitches and intervals. Sadly there was no going back . So this part of my personal journey .
That must be annoying! Maybe it's something you can learn to switch off. If I recall Mozart had every opportunity to become a violinist but chose the piano - or its precursor - and seemingly wasn't bothered by this despite his incredible ear. At any rate, I don't remember it being mentioned in his published letters.

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Re: equal tempered and tuning

Post by amezcua » Sun Oct 23, 2016 8:04 am

What I have noticed in just one simple Bach Sarabande BWV 1012 is that the Kirnberger III guitar tuning is more in tune than Casals or Rostropovitch . Apart from that they seem to use a massive overkill amount of volume for such a delicate composition .
Imagine that .A Guitar more in tune than 2 world famous cellists .

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Re: equal tempered and tuning

Post by amezcua » Sun Oct 23, 2016 9:56 am

In "An introduction to Historical Tunings " by Kyle Gann he talks about his piano being tuned to the Young temperament ."This is a subtle tuning , quite usable in all keys , and the differences from Equal Temperament are more evident to the pianist playing than to the listener ". I tried to find the site with the Gaelic harp being tuned just now as there were some very precise descriptions of intervals . Some were very unpleasant and avoided and others were particularly beautiful . I could not remember the exact wording for the beautiful ones . There was a milder description for one interval that had "character ". These words are seldom heard when referring to anything in Equal Temperament . Kyle Gann says that ET has painted it all drab gray . This the reason much ancient and historical music is not heard any more .
"bones" mentioned the "predecessor of ET "and it sounded as if that was the golden trophy we had all been striving for.
I read through the True Temperament page again and the video had me struggling to turn the volume off. That was only a cleaned up ,more accurate ,version of ET on an electric guitar . Not a great leap forward musically but full marks for making some waves with ripply frets . But the main reaction was against the appearance . The fret ripples not his hairstyle . To me it was like JCB music. If composers worked with earthmoving machines it would be perfect . Well my groundbreaking idea is to ditch the straight line Nut. That`s pretty fundamental .

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Re: equal tempered and tuning

Post by PeteJ » Sun Oct 23, 2016 11:55 am

ipso facto wrote:
PeteJ wrote:Azalais - I'm confused by the idea of out-of tune notes beating. Surely they can only beat when they're in tune.
I don't get this - beating is an interference phenomenon more associated with dissonance than consonance - see http://www.animations.physics.unsw.edu.au/jw/beats.htm
Yes, you're right.

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Re: equal tempered and tuning

Post by ipso facto » Sun Oct 23, 2016 1:00 pm

amezcua wrote:What I have noticed in just one simple Bach Sarabande BWV 1012 is that the Kirnberger III guitar tuning is more in tune than Casals or Rostropovitch . Apart from that they seem to use a massive overkill amount of volume for such a delicate composition .
Imagine that .A Guitar more in tune than 2 world famous cellists .
OK but doesn't that just take us back to the point that there is scope for argument about what the pitch of the notes should ideally be (which is not the same issue as what tuning system we should use for fixed pitch instruments). Is it that Casals and Rostropovitch were going for the pitch approximated by Kirnberger III and missed, or is it that they had a different idea of what the pitch ought to be?

Did you have a specific passage in mind by the way, or is this more of a general comment?

In your violin playing, did you come across the idea that scales ought to be played in pythagorean tuning but arpeggios should have the flatter major thirds given by just intonation? I do not know how orthodox / unorthodox this is. I could even be what Casals and Rostropovitch were doing, which would explain why they were going for a pitch that you consider to be out of tune.
amezcua wrote:In "An introduction to Historical Tunings " by Kyle Gann he talks about his piano being tuned to the Young temperament ."This is a subtle tuning , quite usable in all keys , and the differences from Equal Temperament are more evident to the pianist playing than to the listener ".
There is recording of the Well-tempered Clavier, I think by Derek French, in Young temperament. It does sound different - I quite like it, but I wouldn't say it has a different "affect", as was implied higher up the thread. It seems to me that word suggests a much more dramatic difference.
Kyle Gann says that ET has painted it all drab gray . This the reason much ancient and historical music is not heard any more .
A sweeping claim. How much ancient and historical literature is heard these days? Times change.
"bones" mentioned the "predecessor of ET "and it sounded as if that was the golden trophy we had all been striving for.
Wishful thinking. There are more notes than there are keys on the keyboard or frets on the fretboard, so the keys /frets have to do double duty, making every system a fudge of some kind. People strike the balance in different ways, but it is a matter of personal preference. Fundamentally it is not a soluble problem (nor is it a very big problem, in my opinion).
I read through the True Temperament page again and the video had me struggling to turn the volume off. That was only a cleaned up ,more accurate ,version of ET on an electric guitar .
Yes, but you could use the same principle to get a cleaner version of KIII, or any other tuning system.
Not a great leap forward musically but full marks for making some waves with ripply frets . But the main reaction was against the appearance . The fret ripples not his hairstyle .
:lol:
Well my groundbreaking idea is to ditch the straight line Nut. That`s pretty fundamental .
I don't understand - surely this only differs from retuning the open string in that the interval between the open string and the first fret will be slightly different.

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Re: equal tempered and tuning

Post by Alan Carruth » Sun Oct 23, 2016 6:38 pm

ipso facto wrote:
"There is recording of the Well-tempered Clavier, I think by Derek French, in Young temperament. It does sound different - I quite like it, but I wouldn't say it has a different "affect", as was implied higher up the thread."

Is Young temperament the same as Bach's Well temperament? If it's not, why would you expect the one to sound like the other? We need to compare apples with apples.

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