equal tempered and tuning

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

Post by ipso facto » Sun Oct 23, 2016 7:34 pm

It's not the same, no - Young came up with his version of well temperament after Bach's death. I'm not sure whether we know exactly what tuning system Bach was using, and I expect it would have changed during his lifetime. It's a fair bet that Young's tuning is closer to ET than the Werkmeister tuning discussed above, or whatever tuning Bach was actually using, so I understand your point. I had tried to cover this by saying "implied". Of course you can logically say that, as you get further and further from ET, you reach a point where your alternative tuning produces a different affect. I think though that the mind recognises the true note, in the same way that you recognise a slighly mispronounced word, say, or a stretched version of a familiar photo. That is why I don't think a tuning system (based on our standard 12 tones) will ever have enough of an impact that you could say it produces a different affect. The mind has got in behind the tuning system before the affect is produced, I would say, at least if what we mean by "affect" is something fundamental, perhaps on a par with a change from major to minor. Of course many people think that music is purely a learned or conventional system, in which case that view wouldn't make sense - that's a different thread though.

amezcua
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Re: equal tempered and tuning

Post by amezcua » Mon Oct 24, 2016 12:03 pm

ipso facto The alteration of the nut line comes about because I will reposition an awkwardly placed note in the middle of the fretboard . Then setting that (central fretboard ) frequency with the tuning peg and Korg I have the foundations to relocate all the notes on that string. The open string position will be decided by the Korg . Only the permanent bridge line is left intact through all this . That process may be the one that upsets many people but it`s done to reduce the misalignment of the frets. And I have done it myself without needing help from anywhere else .
All the quotes (requoted) above were impressions taken from "Historical ETs and Modern ETs" on Pianoworld .
The two quotes I liked best were "The piano world is gradually loosening it`s grip on the security of the familiar in favour of the challenge and beauty of the original .
Also "I equate ET with the pollution that covered the Sistene Chapel ceiling. The true colours of the artist ."
Its a long and fascinating read .
What players and listeners prefer to hear is governed by what they find beautiful . That is as mysterious to predict as defining what a beautiful face is .

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Re: equal tempered and tuning

Post by Alan Carruth » Mon Oct 24, 2016 5:23 pm

ET allows for free modulation from one key to another because all of the keys are out of tune in the same way. G major sounds like E major that has been capoed up three frets(or, even better, with the recording speeded up). In the recordings of Bach's 'Well Temperament' that heard that was not the case; each major or minor key, while clearly being 'major' or 'minor', had it's own affect. Organists will often say that one or another key is more 'heroic' or 'introspective' or whatever, and that's to be expected with an unequal temperament such as they are used to hearing. All of the semitones are different sizes, and since each semitone is in a fixed position on the keyboard the mis-tunings stack up in different ways depending on which note is your tonic. I have to say that I don't remember where the person doing the demonstration got his data on Bach's tuning. This was something I heard on the radio in 1985, on the anniversary of Bach's birth: I remember being struck by the differences, without recalling any of the specifics.

amezcua
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Re: equal tempered and tuning

Post by amezcua » Mon Oct 24, 2016 8:35 pm

Is there a record of all the favourite keys used for guitar music? I get the impression they are mostly on the easier keys. It would be due to physical limitations. How many keys would a jazz guitarist modulate into ? Composers would already avoid using impossibly difficult keys if they want to make a living .The desire for such ultimate flexibility makes more sense with pianos when ten fingers are being used. In a much more physically limited instrument such as a harmonica you can buy a chromatic harmonica or stick to a more "natural" temperament and get one harmonica for each key .They would match a singer`s voice better .There is a reason Harmonica makers keep both types in stock . "Natural" temperament as opposed to artificial or synthetic. But how many discuss this sort of subject without trying to listen to performances from both sides . Maybe the hard liners in opposition to any change cannot hear or feel the difference . Then they would not see the point. That`s understandable .
When I compared Casals and Rostropovitch it was no more than admitting that fretless instruments cannot pitch every note to agree with a tuner. They have great skills in bringing the intonation back in line without making it sound like a mistake .That`s a huge skill . Sometimes that comes about in high notes where there is hardly any room for the fingers .

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guitarrista
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Re: equal tempered and tuning

Post by guitarrista » Mon Oct 24, 2016 8:56 pm

Alan Carruth wrote:I have to say that I don't remember where the person doing the demonstration got his data on Bach's tuning. This was something I heard on the radio in 1985, on the anniversary of Bach's birth: I remember being struck by the differences, without recalling any of the specifics.
In 2005 Lehman suggested that the recipe for Bach's well temperament is contained within the squiggles at the top of the Well-Tempered Clavier manuscript's title page(*), if you turn them upside-down, and the idea is that when you turn the page upside-down you get a set of instructions for tuning by the number of squiggles inside the big loops. One gets the sequence 2-2-2-2-2-0-0-0-1-1-1 which tells a tuner by how many twelfths of a Pythagorean comma to make the eleven fifths narrow from a perfect fifth as you go on a circle of fifths starting from C.
larips_640_ed.gif
You can read through his research and hypothesis, and even listen to sound samples, at http://www.larips.com/

(*) The original title page looks like this:
wtc-1722-from-1911-grove.gif
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amezcua
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Re: equal tempered and tuning

Post by amezcua » Tue Oct 25, 2016 4:26 pm

What is the translation of the triangle of words below the squiggles ?

ipso facto
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Re: equal tempered and tuning

Post by ipso facto » Tue Oct 25, 2016 4:46 pm

guitarrista wrote:In 2005 Lehman suggested that the recipe for Bach's well temperament is contained within the squiggles at the top of the Well-Tempered Clavier manuscript's title page(*), if you turn them upside-down, and the idea is that when you turn the page upside-down you get a set of instructions for tuning by the number of squiggles inside the big loops. One gets the sequence 2-2-2-2-2-0-0-0-1-1-1 which tells a tuner by how many twelfths of a Pythagorean comma to make the eleven fifths narrow from a perfect fifth as you go on a circle of fifths starting from C.
What a spot! I'm sceptical by nature and tend to think these things are more likely to be coincidences, but would love to think this was intentional - and it does fit with Bach having encoded his initials in his music, his fondness for puzzle canons, etc.
Alan Carruth wrote:In the recordings of Bach's 'Well Temperament' that heard that was not the case; each major or minor key, while clearly being 'major' or 'minor', had it's own affect. Organists will often say that one or another key is more 'heroic' or 'introspective' or whatever, and that's to be expected with an unequal temperament such as they are used to hearing. All of the semitones are different sizes, and since each semitone is in a fixed position on the keyboard the mis-tunings stack up in different ways depending on which note is your tonic.
It would be great to research this properly / empirically. We know that in an unequal temperament some intervals and progressions will be further out of tune than others, pretty much regardless of how we define "in tune". A good composer - and certainly a great composer like Bach - will avoid the dodgy ones and favour the ones that are closer to the ideal. Thus music composed for a particular well-tempered key may well have a particular character, and it is not surprising if musicians come to associate that character with that key... but is this really due to the temperament? I would say not, or only in the very limited sense that the temperament influenced the actual content of the music by restricting the composer's options. I think the heroic piece would still be heroic if played in ET. As I say, I think it would be very interesting to investigate this empirically, but you would probably have to use music that your subjects weren't already familiar with, otherwise you would just be testing how strong their convictions were. You would also have to find a way to control for the effects of absolute pitch - so not really very practical... which brings us back to our own personal convictions - in my case that we grasp music by relating it to an intuitive pitch structure, and that this mental process has a "snap to grid" function that means that mistunings are ironed out very early on,

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guitarrista
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Re: equal tempered and tuning

Post by guitarrista » Tue Oct 25, 2016 4:49 pm

amezcua wrote:What is the translation of the triangle of words below the squiggles ?
It says this:
Capture6.JPG
If I ever publish an own composition, I would like to introduce it with "For the Use and Profit of the Musical Youth Desirous of Learning
as well as for the Pastime of those Already Skilled" :guitare:
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Re: equal tempered and tuning

Post by omlove » Tue Oct 25, 2016 5:16 pm

Has anyone mentioned this article?

http://www.juilliard.edu/journal/ongoin ... emperament

The Ongoing Quest for Bach's Temperament
By TAMAR HALPERIN
March 2009
In 1722, when Bach compiled his ingenious collection of preludes and fugues in all 24 keys, he gave it the title Das wohltemperirte Clavier (The Well-Tempered Clavier), a term that became as familiar as the music itself. But what does the name actually mean? And what led Bach, whose 324th birthday is on March 31, to choose this peculiar title?


Figure 1: The circle of fifths (the rising fourths in this example are equivalent to descending fifths). If each interval is tuned to be pure, the last note will be lower than the first.

Figure 2: The title page of Bach’s Well-Tempered Clavier from 1722, with labels added by the author of this article. D denotes double loops, S indicates a simple loop, and convoluted loops are marked C.

Figure 3: Lehman’s interpretation of the scroll; the same graphic as in Figure 2, turned upside down, with note names above their respective loops.

Figure 4: O’Donnell’s interpretation of the scroll.

Figure 5: Francis's interpretation of the scroll.
Related Stories

Scaling Bach's Olympus, Key by Key
The term “well-tempered clavier” obviously refers to keyboard temperament. At the core of any discussion of temperament lies a mysterious acoustic phenomenon, which was discovered by Pythagoras in the sixth century B.C.: When a string is divided in half, the interval between the pitch of the full string and the pitch of its half-length is a pure octave; when a string is divided in the ratio 3:2, the interval between the pitch of the full string and the pitch of its two-thirds is a pure fifth.

As we know, a circle of 12 fifths begins and ends with the same note. Strangely, however, 12 pure fifths do not really “fit” within the octave, but actually exceed the octave, so that the final pitch is slightly different from the original one (it is lower than the original if the circle goes down, and higher if the circle goes up). The amount by which 12 fifths exceed the octave is called the “Pythagorean comma.” Therefore, to put it simply, temperament is the method of “compromising” the purity of the fifths in order to keep the octave pure. (Figure 1)

In practice, singers and most instrumentalists solve the problem spontaneously by intuitively adjusting their intonation. But this cannot be done on instruments of fixed pitch, like harpsichord, piano, organ, harp, lute, or guitar. These instruments require a choice of temperament, which dictates the degree of “falseness” of the intervals, and therefore the “usability” of the various keys. (If one of the fifths is very false, the keys that include this fifth will be unplayable.)

The standard temperament nowadays is “equal temperament,” in which all 12 fifths are narrowed by the same amount (1/12th of the Pythagorean comma). In this temperament (which is used in all modern pianos, for example), all the keys are equally false, but the degree of their “falseness” is relatively small. However, “equal temperament” did not become the standard until late in the 18th century or later (and well after Bach’s death). Until then, a large variety of non-equal temperament systems were used, and it is probably to one of these systems that Bach referred in the title of his 1722 collection.

The terms “well-tempered” and “well-temperament” were first coined in 1691 by German theorist Andreas Werckmeister, referring to a sort of temperament in which the fifths are of different sizes, but none of the fifths is too false to use. In “well-temperament,” all the keys are playable, yet they vary in their purity and timbre. The concept of variety was an important part of Baroque aesthetics, and the different colors of the various keys (created by non-equal temperament) was considered an advantage. Perhaps this is what Bach meant to demonstrate in his collection of preludes and fugues. While Werckmeister gives clear instructions for his proposed temperament, there is no direct evidence of the exact kind of temperament that Bach himself used.

Over the years, attempts to reconstruct Bach’s temperament were made by a number of musicologists, from Kirnberger and Marpurg in the 18th century to 20th-century musicologists Herbert Kellner and John Barnes. But perhaps the most curious and unusual approach to the question of Bach’s temperament was presented in the February and May 2005 issues of Early Music by harpsichordist/mathematician Bradley Lehman. (The illustrations in Figures 2-5 in this article appeared in that publication.)

Examining the title page of the W.T.C. from 1722, Lehman noticed that the decorative scroll above the text features 11 loops of three different kinds (simple, double, and convoluted). (Figure 2) It occurred to him that 11 is the number that would describe the temperament of 12 fifths (if the first note is given, it would also be the last note of the circle, rendering the “12th loop” unnecessary). He noticed the letter C attached to the first loop from the right, and then decided to turn the loops upside down. (Figure 3)

Lehman states that in the Baroque period the normal amount of tempering a fifth is 1/6th of a Pythagorean comma, which, he believes, is represented in the squiggle by the convoluted spiral. He interprets the double-spiraled loops to represent the tempering of 1/12th of a Pythagorean comma, and the simple loops to indicate pure fifths. He then devises a temperament he believes to be the one used by Bach, and which, according to Lehman, brings out qualities of Bach’s composition that are hidden in equal temperament.

Lehman’s theory received tremendous attention, being highly praised by some and severely criticized by others. The November 2006 issue of Early Music featured several responses to Lehman’s article on Bach’s temperament. One of them, written by John O’Donnell, challenges Lehman’s interpretation and offers an interesting alternative for the reading of the title-page scroll.

O’Donnell claims that, although tuning is done with fifths, its customary notation in the Baroque period was actually chromatic, rising from C. This is also the order chosen by Bach for the 24 preludes and fugues. Examining the scroll of the Well-Tempered title page once again, O’Donnell notices that the “D” of Das intersects with the third loop and that, to its left, there is a squiggle that looks like “Es,” the German notation for E flat, while to its right appears the German tablature symbol of “Dis” (D sharp). O’Donnell mentions the E-flat/D-sharp Prelude and Fugue, which, curiously, is the only piece in the collection that refers directly to the enharmonic phenomenon.

O’Donnell assigns pitch names to the scroll, starting from C on the left and ascending chromatically until reaching B on the right side. (Figure 4)

Based on his reading of the scroll, O’Donnell constructs a temperament system which, according to him, represents Bach’s own.

In 2007, in response to these articles, musicologist John Francis posted an article on the Internet that offered a systematic analysis of the scroll, considering the diagram’s orientation (left to right; right to left), tuning direction, starting position, and degree of temperament, so that, in total, 144 possibilities were mathematically analyzed. Francis compared his results with 52 existing tuning systems, and finally came up with a reading of the scroll, as seen in Figure 5.

However, with all of this said and done, to this day no one knows what Bach’s temperament really was. Whether the decorative scroll of the Well-Tempered Clavier is indeed a temperament prescription or not, it now provides a subject for endless interpretation, almost like the music that follows the title page. The quest for Bach’s temperament will most likely continue to preoccupy musicians and theorists for generations to come.

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Re: equal tempered and tuning

Post by Alan Carruth » Tue Oct 25, 2016 6:45 pm

"As we know, a circle of 12 fifths begins and ends with the same note. Strangely, however, 12 pure fifths do not really “fit” within the octave, ..."

It's not strange at all.

For every octave you double the frequency of vibration. The fifth is gotten by multiplying the original frequency by 3/2. The fifth implicit in vibrating systems like strings and air columns, which have upper partials that are in harmonic relationships with the fundamental. For example, when you pluck your A string, tuned to 110 Hz, it also sounds at whole number multiples of that frequency; 110, 220, 330, 440, and so on. We can hear these partials by touching the string at various points to suppress some and allow others to sound. Thus, touching the string just over the 12th fret suppresses all of the odd numbered partials (110hz, 330, 550. and so on) and only allows the even numbered ones (220, 440,660) to sound out. These are the partials that would be produced by a string tuned an octave higher, and that's what we hear. Touching the string at 1/3 of it's length suppresses the fundamental as well, and only allows the odd numbered partials divisible by three to sound: 330Hz, 660, 990, etc. This is a 12th above the fundamental, and shifting that down an octave gives the fifth.

In going through the 'circle of fifths' you're iterating a process of always multiplying a number by three, and dividing it by two. There is no way mathematically that you can do this any number of times and get back to the original number. That's why there's a difference in the pitch after you've run through twelve fifths, always shifting down an octave when you need to in order to stay within the octave. You can do this as many times as you want; 53 iterations gets you closer the starting pitch than 12, but there's still a difference.

Since the interval if the fifth is implicit in musical sound, it makes sense to try to preserve fifths as 'pure' tones, but it can't be done for all fifths. Temperament is the process of shifting the tuning of various intervals to some compromise that is 'close enough' to be satisfactory for the music to be performed, that preserves the octave which must be preserved. As with any compromise, what works for one may not be satisfactory for another.

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Re: equal tempered and tuning

Post by guitarrista » Tue Oct 25, 2016 7:08 pm

Alan Carruth wrote:"As we know, a circle of 12 fifths begins and ends with the same note. Strangely, however, 12 pure fifths do not really “fit” within the octave, ..."

It's not strange at all.
That caught my eye as well - what a weird thing to say - unless this was some sort of "popular" article written by a journalist without a musical background.

Why on earth would it be strange that (3/2)^12 is not an integer, never mind being exactly equal to a power of 2? As it happens, (3/2)^12 = 129.7463.. which is close, but a bit more than, 2^7 = 128. So 12 perfect fifths get you to a place a bit sharper (a Pythagorean comma worth, as a ratio, i.e. 3^12/2^19) than seven octaves.
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Re: equal tempered and tuning

Post by ipso facto » Tue Oct 25, 2016 8:31 pm

Alan Carruth wrote:Since the interval if the fifth is implicit in musical sound, it makes sense to try to preserve fifths as 'pure' tones, but it can't be done for all fifths.
Sure it can - you just have to give up the idea that Ab and G# are supposed to be exactly the same. It's funny because we insist they are different when it comes to spelling and analysis, but when it comes to tuning we say they are the same because there is only one key on the piano for both.

The statement you quoted is even less strange when you think that 12 fifths don't take you back to the same pitch class as you started with anyway.

From this point of view the problem is not with the fifths - the problem is that if you make all the fifths right you will find that the major thirds do not harmonise well... but is this because the scale tones you get tuning by fifths are out, or is it because chord thirds are not scale tones? The latter, in my opinion.
Temperament is the process of shifting the tuning of various intervals to some compromise that is 'close enough' to be satisfactory for the music to be performed, that preserves the octave which must be preserved. As with any compromise, what works for one may not be satisfactory for another.
Well I know you are in good company in holding that view, but for reasons that will be clear from the above, I prefer to see temperament as a compromise that allows one key to be used for what are really several (slightly) different notes.

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Re: equal tempered and tuning

Post by amezcua » Wed Oct 26, 2016 1:50 pm

There was a Radio 3 demonstration a few years ago where a pianist played the music in ET and then had the same piano retuned to demonstrate the unequal versions of the same music . It may have been a harpsichord . That totally convinced me that the older temperaments were better . Better musically. More expressive. Not getting away with any ugly sounds .120% better in every way .You can`t improve on 120% .

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Re: equal tempered and tuning

Post by pogmoor » Wed Oct 26, 2016 4:21 pm

amezcua wrote:That totally convinced me that the older temperaments were better . Better musically. More expressive. Not getting away with any ugly sounds .120% better in every way .You can`t improve on 120% .
OK if you don't modulate to a different key. Have a look at (and more importantly listen to) this video illustrating one of the common meantone tunings, 1/6 comma meantone:


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Re: equal tempered and tuning

Post by Alan Carruth » Wed Oct 26, 2016 5:56 pm

ipso facto wrote:
"Sure it can - you just have to give up the idea that Ab and G# are supposed to be exactly the same. It's funny because we insist they are different when it comes to spelling and analysis, but when it comes to tuning we say they are the same because there is only one key on the piano for both."

They used to make keyboards with the extra keys. I would not want to try to put the extra frets into a guitar. There is evidence that the move to ET was pushed in part by lute and viol players, who, cursed with straight frets, could not play in tune with the keyboards.

I'ts clear that we have come to an impasse based on different interpretations of the meaning of 'better'. Since what's 'better' is a matter of opinion there's no general answer to that question.

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