equal tempered and tuning

Construction and repair of Classical Guitar and related instruments
Rasputin
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Re: equal tempered and tuning

Post by Rasputin » Tue May 30, 2017 7:31 pm

Not a joke then. Sorry, my bad.

I think before you work out how to present data you have to decide what aspects you are looking to bring out.
Smith wrote:
Tue May 30, 2017 6:06 pm
What I mean is I imagine a computer generated animated graphic that represents sound waves in motion and in color.
The waves would be a certain color when on equal temperament and gradually change color.

What, for example, would moving sound waves look like if the octave were blue, the fifth red, and all the way down the line.
If you really mean the sound waves, it would look like an animated Jackson Pollock with all the partials. I suspect you would only have a couple of shades available to show notes considered to be out of tune, because you need at least 8 basic colours for your in-tune notes, so they not are going to be very far apart on the colour spectrum. You also have to solve the problem of defining what is in tune.

I think the best graphical representations of pure and impure harmony are the Lissajous figures you can find on YT (no colours, I admit). You can't have a fixed pitch tuning system where every triad in the key is pure though, so you still have to decide what compromises to make.

Part of me also thinks that trying to watch music is like trying to listen to sculpture. If you can't hear a difference, it doesn't really matter. If you can hear a difference, you don't need to see it. For example, you could decide what tuning compromises to make using the Lissajous figures and an understanding of what chords are most important, but since we are concerned with how music sounds, it makes more sense to me to do it by ear.

StevSmar
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Re: equal tempered and tuning

Post by StevSmar » Fri Jun 02, 2017 12:58 am

Rasputin wrote:
Tue May 30, 2017 6:51 am
.... I don't think the graphics would be hard to do. The problem is defining what "out of tune" means. What is the reference?...
Good point Rasputin. What I meant was if you picked the octaves being in tune, the thirds fourths and fifths go out of tune and vice verse.

Id love to see a graphic which demonstrates how you need to choose your in-tune interval, and by making this choice you're also picking which intervals will be out of tune for a particular temperament.
Regards,

Steven from Winnipeg

StevSmar
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Re: equal tempered and tuning

Post by StevSmar » Fri Jun 02, 2017 1:05 am

Trevor Gore wrote:
Tue May 30, 2017 10:00 am
....Pretty hard to do on just one static graphic, but here's a slide I use...
Thanks Trevor.

Yes I agree, almost incomprehensible with a static graphic, it would need to be a animation that shows how every temperament and intonation is a compromise. Something that convinces those that are curious that "for everything that is gained, there is something you will lose"

(one of my favourite sayings, interesting how it also applies to the "what is the best temperament" debate)
Regards,

Steven from Winnipeg

Rasputin
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Re: equal tempered and tuning

Post by Rasputin » Fri Jun 02, 2017 8:22 am

StevSmar wrote:
Fri Jun 02, 2017 12:58 am
Rasputin wrote:
Tue May 30, 2017 6:51 am
.... I don't think the graphics would be hard to do. The problem is defining what "out of tune" means. What is the reference?...
Good point Rasputin. What I meant was if you picked the octaves being in tune, the thirds fourths and fifths go out of tune and vice verse.

Id love to see a graphic which demonstrates how you need to choose your in-tune interval, and by making this choice you're also picking which intervals will be out of tune for a particular temperament.
I think most people say that the octave needs to be spot on (which means you multiply by 2). There are stretched tuning systems in existence but they are not really to do with temperament.

If you define the fifth as 3/2 and the third as 5/4 (which are based on the harmonics,treating octaves as equivalent) you then find that:

If the fifths of a given key in all tune then your octaves are sharp, so at least one of the fifths has to be tempered downwards, i.e. made flat (which is the same thing as making a fourth sharp).

You can put all of this adjustment in one note, in which case you will have one fifth that is quite a bit flat, or you can spread it over a few, giving you a few fifths that are just slightly flat

But...

Flat intervals sound bad - much worse than sharp intervals - and flat fifths sound especially bad.
The fifths are in an order (the order of the circle of fifths). There will be a knock-on effect on all notes further round the circle of fifths than the first one you make flat.

If your fifths were in tune, your major thirds would be flat and your minor thirds sharp.

Any change to the fifths will interfere with this. It sounds as though it makes sense to flatten a fifth which is a minor third in the key, since they start off sharp - but then the knock-on effects can be worse. If we were in C we might flatten D, but then we compromise the important dominant chord. Same if you said we'll go for the one at the end of the circle of fifths and said B, which would already be flat when used as the third of the dominant.

All too complicated to be expressed on a graphic. Some kind of interactive thing maybe.

amezcua
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Re: equal tempered and tuning

Post by amezcua » Sun Jun 04, 2017 6:18 pm

So when did the idea of key colours being a fantasy come into fashion ? All that sincere hard work over hundreds of years being flushed away .

Rasputin
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Re: equal tempered and tuning

Post by Rasputin » Sun Jun 04, 2017 6:55 pm

The idea that key colours are a fantasy is a hobby horse of yours, as far as I remember nobody else has said that.

I'm quite sure they exist but I think you have fundamentally misunderstood what they are. In your previous posts you contradict yourself to the point where - as I said above - I just don't see how we can have a constructive discussion on the subject.

Alan Carruth
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Re: equal tempered and tuning

Post by Alan Carruth » Sun Jun 04, 2017 8:28 pm

Key colors may well be a fantasy in ET tuning, but it might be hard to demonstrate that in live setting. Most instruments have different tonal colors at different pitches, and that can't be separated from the key. The lowest two resonances on guitars, for example, tend to be near the pitch of G, so music played in G or related keys may well sound different from the same music played in an unrelated key that doesn't feature the G pitch prominently.

Violins tend to have their two most prominent output peaks near the pitches of the open middle strings: D~294 Hz and A=440. Violas are smaller relative to the pitches the strings are tuned to, so those same resonances tend to be about a third higher (iirc) than the two open middle string pitches of G~196 and D~294. My violin making teacher did most of the work to develop an octet of instruments that were properly scaled relative to their tuning pitches, so that each one would have the sort of timbre and balance of the violin within it's tuning range. The 'Alto violin', which is tuned like a viola, is about 4" longer in the body than the average viola, and is played upright on a peg like a 'cello, since you need to be about 6'9" tall and capable of scratching your knees without bending down to play it comfortably under the chin. When Yo Yo Ma used one to record the Bartok viola concerto some of the critics complained that it didn't sound like a viola at all, but, rather, like a big violin. That, of course, was the point. Similar problems were encountered when using an Alto to play some of Mozart's music written for two violas. He knew the sound of the instruments he was writing for, and tended to place important notes in the two viola parts close to the two main resonances, with the first viola prominently featuring the higher, and the second the lower pitch.

In those cases we're actually talking about 'instrument color' and not 'key color', but since music is played on instruments, most of which do have some sort of color, it's hard to separate the two things. Electronic instruments should posses no color, but you often can't say the same for the speakers they're played through. Those of us whose pitch sense is not absolute might have problems distinguishing, say Emaj from Fmaj when an electronic instrument is used to transpose a given piece of music, assuming sufficient time elapsed between the samples so that we didn't notice the actual pitch change. That sort of test could be done relatively easily, and could possibly answer the question in the abstract.

Rasputin
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Re: equal tempered and tuning

Post by Rasputin » Sun Jun 04, 2017 9:42 pm

I thought where Amezcua was coming from was that one of the problems with ET is that it destroys the key colours that are available in historical temperaments by making every key the same. That's why he's keen on saying that the other temperaments have particular colours.

I think there are probably a few aspects of key colour and it's likely to be difficult to tease them apart. The timbre of the human voice obviously changes with pitch, and this could be an aspect of the key colour perceived by the choirmasters you mentioned (Alan) in an earlier post. They may also have had a sense of absolute pitch, which could come into the equation as you say.

Still, from the point of view of temperament I think it really starts with the idea that slightly different intervals have a slightly different character, and therefore that in any form of JI or unequal temperament, where the interval sizes differ slightly from key to key, the keys themselves will have a different character. As far as I can see, that is just common sense and totally unobjectionable.

The harder question - coming at it from the point of view that being in tune is not just a matter of convention, but has a deeper basis in our musical nature - is whether a tempered interval can ever have more aesthetic value than a pure interval, i.e. one that is perfectly in tune. The idea that it can seems to be the driving force behind the view that the historical temperaments have special value. I am very sceptical about this - my instinct is that it doesn't get better than being in tune. If someone doesn't like the fact that, in ET, everything except the octave is slightly out of tune, the answer is to explore just intonation and how it might now be possible to build an instrument with active tuning so that every note can be spot on. I suspect this would be possible, but not enough people care enough about the issue to make it really worthwhile (I know I don't - my interest in tuning stems from an interest in what makes music tick, or what its basic building blocks are).

Since pretty much everyone agrees that temperaments are a compromise - which obviously means that there is some deeper tuning standard that they depart from - we never really get as far as discussing what the notes should ideally be. This is a shame because the subject has a lot of implications for theory, for example the near-universal belief that chords originate as stacks of thirds.

Dirck Nagy
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Re: equal tempered and tuning

Post by Dirck Nagy » Sun Jun 04, 2017 10:56 pm

Regarding tuning "controversies", I'm not sure what the big deal is.

In aulden Days of Yore, people used temperaments which were adequate to produce the music of the time.

As times changed, so did music. Tuning systems other than E.T. could be used until after the baroque, because music of the time simply didn't modulate to many different unrelated keys. By the mid 1800s, things were different, hence E.T.

Often overlooked are questions of instrument construction. Consider the manufacturing tolerances of the day, especially:
  • uniform diameters of organ pipe
  • Wire-drawing technology
  • Metallurgy technology
  • the uniformity of gut strings, and other natural fibers
  • standards and techniques of measurement
Now, how noticiable were the differences between, say, Kirnberger X and Werckmeister Y in light of the considerations above? I suspect much of this existed mainly as a laboratory curiousity.

Tuning systems approximating E.T. have been in use on fretted instruments since the early Renaissance. Zarlino and Galilei even alluded to it when they noted that lutes cannot play in tune with keyboard instruments. (i dont have this source at my fingertips, sorry...i'll try to find it later)

The "Rule of 18" is a geometric fret layout method that has been used since at least the late 1400s (again, i don't have sources handy, sorry) which does not involve calculations of ratios beyond the first fret, but requires only a set of dividers and a straight edge. This system, in theory, produced an octave that was about 12 cents off, but this could be adjusted by compensation at the bridge.

Keeping in mind the tolerances noted above, a symmetric fret scale was actually quite easy for a luthier of the day to lay out, and was, for all practical, measurable purposes, equal temperament.


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Alan Carruth
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Re: equal tempered and tuning

Post by Alan Carruth » Mon Jun 05, 2017 9:29 pm

Rasputin writes:
"Since pretty much everyone agrees that temperaments are a compromise - which obviously means that there is some deeper tuning standard that they depart from - we never really get as far as discussing what the notes should ideally be."

The ideal that is being departed from is 'pure' intervals, with no beats. In ET the only such interval is the octave. I'm sure I, or somebody, went into the math behind it in an earlier post. Basically, in theory, you'd like to be able to add up some number of 'pure' fifths to get back to an even multiple of the original stating frequency. To do this you multiply the starting frequency by 3 to get the interval of a 12th, and then divide that by 2 to drop it down an octave, obtaining the fifth. Then you use that new frequency as the starting pitch to find the next fifth up, and so on. When you've done that 12 times you're back around to the starting pitch, almost. Note that you always multiply by 3, and always divide by two.

The problem is that 3 and 2 are incommensurable: there is no way you can multiply a number by three and divide it by two any number of times and get back to the original number. The difference between the frequency you started with and the one you ended up with is the comma, and the whole issue of temperament is figuring out what to do with the comma. The compromise is not something that people chose to do out of shear cussedness; it's necessary. That's why you either get to listen to pure intervals, or modulate keys. You can't have both.

I don't think that this was a mere 'laboratory curiosity' to folks like Bacon or Bach, who put a lot of thought and effort into this question. The differences between temperaments are clearly audible, so it must have been possible for people to make the sort of adjustments they talked about to get them to work. Don't sell the old boys short. One of my students deals in gut strings, and he claims that in many ways they used to make them a lot better than we do now.

amezcua
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Re: equal tempered and tuning

Post by amezcua » Tue Jun 06, 2017 12:03 am

Update for Rasputin . Ipso Facto used the word fantasy when answering my question about key colours . But can you honestly say that any major scale on a guitar sounds the way a scale should sound? That`s a good basic start apart from key colours .
My grouse was about Howard Goodall making claims for ET. He shows an orchestra in the 38th minute of the video as if that would add credibility to his commentary . Well the bowed strings do not use ET . The drums in the background don`t. The brass instruments are the closest in that section to ET . It`s a very casual , lazy way to assist his grand assertions . People have been annoyed about disagreements over tuning musical instruments for centuries . Why expect this current topic to be a haven of peace and tranquility ?

Rasputin
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Re: equal tempered and tuning

Post by Rasputin » Tue Jun 06, 2017 9:37 am

Alan Carruth wrote:
Mon Jun 05, 2017 9:29 pm
Rasputin writes:
"Since pretty much everyone agrees that temperaments are a compromise - which obviously means that there is some deeper tuning standard that they depart from - we never really get as far as discussing what the notes should ideally be."

The ideal that is being departed from is 'pure' intervals, with no beats.
Yes I think that's often the background assumption, and I'm sure some truth it it, but my point is that we never get down to details. It can't be entirely true because some intervals (a seventh, say) will beat however you tune them.

The reason I think it is interesting to get into the details is that the intervals that do need to be pure are, as I see it, the basic building blocks of music, with the others being derived from them. Just like the distinction between root position chords and inversions, I think this can help us understand the whole phenomenon of music.

For example, many people say that to make a minor chord you start with a minor third and put a major third on top, giving you the fifth. However if you take your minor third from the harmonic series (19/16), you find that when you go up a major third (5/4) you do not end up with a perfect fifth (3/2). You can fudge it and say the minor third must therefore be 6/5, but 6/5 is not from the harmonic series (denominator not a power of 2) and does beat, so I don’t think that is a legitimate move. What this means as I see it is that not all of these intervals can be constitutive of the chord, and that this is a fact of nature rather just a practical problem (unlike the comma issue you raise). I think that if we were to get into tuning and experiment properly we would find that the minor chord that sounds just right (even if we are not used to hearing it, which is telling) is made by going up a perfect fifth and then down a major third. That suggests that the fifth, not the third, is the basis of the triad and that the attention given to stacks of thirds in the theory of harmony is misplaced. This then gives us a principled basis for saying that added notes beyond the seventh are not chord tones (rather than the accident of history theory) and also suggests that Rameau was right to say that six chords are functional – the orthodox theory ignores these completely and reanalyses them as having a different root, largely because they don’t fit with the idea that chords are built by stacking thirds.
The problem is that 3 and 2 are incommensurable: there is no way you can multiply a number by three and divide it by two any number of times and get back to the original number. The difference between the frequency you started with and the one you ended up with is the comma, and the whole issue of temperament is figuring out what to do with the comma.
I would rather say that because people think that temperament is about the comma, they never get deep enough into it to cast any light on how music really works. The comma is just a practical problem - it arises when we try to put the sharps and flats on the same key. If you follow the series of fifths from Gb to A#, you never get back to the same note - you're not supposed to. The problem only arises if you get as far as F# and assume you are supposed to be back where you started. What this is telling us is that Gb and F# are not supposed to be exactly the same note, and you have to continue up the series to get the sharps. If you had a fixed pitch instrument capable of playing all the notes from Gb to A# there would be no comma issue, so the problem is just the practical one that such an instrument would have lots of keys / frets, and would be difficult to play. At the same time, because this forces us into fudging the intervals, it masks the question of what the notes are really supposed to be.

With that in mind I only partly agree when you say:
The compromise is not something that people chose to do out of shear cussedness; it's necessary. That's why you either get to listen to pure intervals, or modulate keys. You can't have both.
amezcua wrote:
Tue Jun 06, 2017 12:03 am
Update for Rasputin . Ipso Facto used the word fantasy when answering my question about key colours .
OK, I found that post:
ipso facto wrote:
Mon Oct 17, 2016 2:31 pm
amezcua wrote:In the process musicians discovered the various emotional reactions and effects produced by using "Well Temperaments". They started by trying to root out negative effects and ended with many positive effects.
A-ha - so that's where you are coming from. I think this is a fantasy, but if you believe in it then that explains your interest in alternative tuning systems.
That is not saying that key colours are a fantasy - the fantasy is that they have positive effects. As I said above, I don’t think it gets better than being in tune – the effect of tempering an interval is always negative.
But can you honestly say that any major scale on a guitar sounds the way a scale should sound? That`s a good basic start apart from key colours .
Yes, I do think a major scale in ET sounds very close to the way it should. A perfect scale IMO is just a series of perfect fifths put into a single octave (using perfect octaves for the conversion, obviously). The fifths in ET are only very slightly out.

That said, I suspect that for you this is equivalent to asking whether the tonic chord of that scale sounds the way it should. For me that is a different question, but I appreciate that most people believe that chords are built out from scale tones, which would make it the same. The tonic chord is acceptable to my ear but is much improved if the third is flattened (I can do this experiment on my keyboard by switching temperaments). This is why I said above that some violinists will play scales in one tuning (pythagorean or just fifths) but arpeggios in another (using just thirds for the chord thirds), and that I thought they were on the right track. The implication is that the thirds of chords do not really come from the scale, but are so close to the nearest scale tone that we can generally get away with using the same pitch for both – that again is a practical compromise and not a fundamental problem with the mathematics of music.

amezcua
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Re: equal tempered and tuning

Post by amezcua » Wed Jun 07, 2017 10:00 am

There are a number of impressive looking explanations of music on the net . But reading them attentively discovers some odd statements.One good example is "History of Tuning and Temperaments" presented by Howard Stoess .It`s not clear to me if he wrote it l himself .It`s an "annotated outline" whatever that means . So , near the bottom of the page it states "ET is the only practical tuning for a modern piano ". Is that an attempt to stay inside the lines to avoid upsetting the majority ? It just seems a silly thing to say .
There are some youtube videos demonstrating unequal temperaments on pianos . Sadly one grand piano is placed near a window onto a garden and the sunlight had warmed up the strings and upset the exact tuning. That is the explanation given . It added a certain charm to the subject . We can all smile at that . Maybe musical artists should use North facing rooms like painters .

Alan Carruth
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Re: equal tempered and tuning

Post by Alan Carruth » Wed Jun 07, 2017 6:26 pm

On thinking about this whole thread it seems to me that we are taking different approaches and thus, to some extent, talking past each other. I'm a guitar maker, and my task is to produce an instrument with more or less fixed pitches that will be useful to a wide range of players. Given all the limitations of the case ET is a good place to start for me. It's the common system, which means that it will be useful for the widest range of players. It also uses straight frets, which greatly simplifies making the fretboard. It is, of course, a compromise, and unsatisfactory in any ways, but it is the usual compromise, and doesn't land me out on a limb. If I take the time to get it all to work as well as possible (with the understanding that 'perfection' is beyond reach on the guitar, or almost any other non-digital instrument) it's not too bad.

Although I am certainly interested in music I'm not primarily a performer or composer. Those of you with far better hearing than mine, who are primarily performers of music, are rightly more interested in questions of how harmonies are built and dissonances avoided. Although I have given some thought to this in the past, it's far beyond my poor powers to provide the necessary hardware or software to make it happen, particularly on the guitar. Since this thread has been on the luthier's page of the forum I've been trying to give the luthier's perspective. Perhaps it should be moved? ;)

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bacsidoan
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Re: equal tempered and tuning

Post by bacsidoan » Wed Jun 07, 2017 7:46 pm

To put things in perspective, I wonder if anyone on this forum beside amezcua will pursue a different temperament other than ET on a classical guitar after 9 pages and 134 replies.

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