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Evolution of Equal Tempered scale
      #990840 - 01/06/12 03:56 PM
Hi Guys,

I've been wondering how the major 7th note became the standard in the Equal tempered major scale and who standardised it?

I ask this because the minor 7th (which would make it Mixolydian) exists in the harmonic series. I know it's to do with the dominant chord being major so it creates the leading note tension needed but I am trying to find out the person whom actually standardised this. I know Bach's well tempered clavier established this but was Bach also involved in the mathematical process?

Thanks


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Scramble
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Re: Evolution of Equal Tempered scale new [Re: ]
      #990845 - 01/06/12 04:18 PM
The history of equal temperant is a very involved story. This is a good book on it:
Temperament

(I see it has one, bad, review saying it's insuffiently technical, but in fact it's a good introductory read that is understandable by anyone who knows a bit about harmony).

Edited by Scramble (01/06/12 04:22 PM)


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: Scramble]
      #990863 - 01/06/12 05:53 PM
A good primer on the subject of temperament (despite the provocative title) is:

'How Equal Temperament Ruined Harmony -and Why You Should Care'

However, I'm not sure your question would be answered by acoustics alone, so...

Nobody "decided" as such. What really happened is all the ecclesiastical modes became crystallised into major, melodic minor and harmonic minor scales (although a rare use of the mixolydian mode can be heard in the credo of Bach's Mass In B minor masterwork!)

To understand this evolution, you could do worse than getting a book called 'Sketching at the Keyboard'. It's just a brilliant book for anyone who's got the humility to work through the simple little tunes. I don't agree with the author on the interpretation of the harmonic series as 'a major scale and dominant 7th chord' though. To me, that's just pure pareidolia. The true series sounds only vaguely like a tempered major scale or dominant 7th chord, except to those who are too fixated on tempered pitches and notation.

For a start, the 1:7 in the harmonic series is sharper than in equal temperament and actually sounds quite 'at rest' depending on the context. This natural b7th was common in American 'barber shop' quartet singing at the end of songs, without any tension to resolve. (They used this cliche in an episode of The Simsons with 'the be sharps').

The harmonic series represents acoustic clarity. In reality, the only true acoustic consonances occur between the fundamental and each successive harmonic, that is: 1:2, 1:3, 1:4, 1:5, 1:6, 1:7, and so on. Even if these intervals occurred between two sounded tones (e.g. on a piano) there would be some acoustic dissonance because of the clashing overtones from each tone.

However, because these clashes are high up in the series, we tolerate them. We even tolerate overtone clashes between 2:3 (true 5th) and 3:4:5 (1st inv. true major triad) -unless they are played low down on an instrument like the piano, in which case we can hear the clashing overtones too well. (Composers like Beethoven used this to powerfull effect though!)

The thing is, not everything has to grow literally out of the series for it to 'work' acoustically. A lot of theoreticians (like George Russell, Hindemith, Schenker, etc) have wasted thought trying to justify musical materials this way. There's more at play here than the overtone series.

First, we prefer scales that have an ideal balance of intervals, like the diatonic scales (in semitones = 2212221). (We usually reject the Locrian mode only because the tritone 'insults' the fifth coming from the key note. Similarly, the wholetone scale presents this issue, except now none of the overtone 5ths are supported by the scale. This lack of tonality in the wholetone scale gives it a detached, dreamy colour. Also, the 'String of Pearls' works better as 12121212 than 21212121 because the latter doesn't support the 5th from the tonic).

The diatonic scales can be derived in many different ways that do not contradict the series because they're still made up of acceptably consonant intervals between harmony notes. The most common ways are...

1)A cycle of fifths: F C G D A E B

2)The natural major chord as tones and their fifths: G C E Bb + harmonics D G B F (gives C,D,E,F,G,A,Bb,B in one rotation).

Cheers.


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Gone To Lunch
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Re: Evolution of Equal Tempered scale new [Re: ]
      #990899 - 02/06/12 12:04 AM
This is another good source


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: ]
      #990965 - 02/06/12 02:55 PM
...I forgot to add that it's a complete myth that J.S. Bach wrote the Well-Tempered Clavier to demonstrate equal temperament. Bach intended a 'Well' temperament, hence the title, which would've been an irregular temperament.

There have been suggestions that the squiggles at the top of Bach's original manuscript page reveal his intended tuning:



Most Bach scholars agree that it means something, but it's still all very Dan Brown at the moment. (That said, the scientist at the centre of that particular contoversy might be about to reveal Leonardo's lost 'Battle of Anghiari'!)

Equal temperament really took hold as a piano convention with the introduction of cast iron frames. This meant that the strings could be tightly strung, which produces a much brighter and louder sound. This also means pianos became more difficult to tune.

Gradually, musicians began to accept the evenly 'out of tune' system of equal tuning because of huge practical benefits of standardisation. It remains an unsatisfactory tuning in general, especially for 3rds** and 6ths, and because of the lack of key colour which was once an important part of tonality. (For example, when a classical composer repeated a passage of piano music in a different key, that new key would've offered a slightly different key colour in the same way different modes have different colour. That's largely lost in equal temperament, although I suppose tuning imprefections still produce some difference between keys.)

(**Just to give you some idea, 4 semitones makes up an equal tempered major 3rd. A natural major 3rd is 3.86 semitones.)


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: ]
      #990993 - 02/06/12 05:30 PM
Thanks J.A.S and all. A lot of food for thought there, and it amazes me the lengths mathematicians have gone to to create scales and "justify" them as set standards. Even the Arab scales are a different world, which tried before the Renaissance West to decipher scales by using maths. It shows a lot of philosophy has also gone into it and it is a very subjective matter too.


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Chaconne



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Re: Evolution of Equal Tempered scale new [Re: ]
      #990994 - 02/06/12 05:31 PM
Some DAW's, and VST's or samplers (Kontakt eg) allow you to change the temperament. Anyone interested in the evolution of temperament should have a go using simple organ or piano patches, or a good sampled Harpsichord. It really helps to understand whats going on, and certainly playing early music - Byrd or Couperin say - sounds more engaging since you get a lot more purity in the intervals.

In fact its amazing how much of the keyboard was useless at certain periods. Given you can change things at the click of a menu though, its a great way to 'hear' the theory and the compromises involved.

--------------------



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zenguitarAdministrator
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Re: Evolution of Equal Tempered scale new [Re: ]
      #990999 - 02/06/12 06:39 PM
And just to add, there were many Well Temperaments, not just one. The Well Tempered Klavier was written for Werkmeister III and it is constructed using 4 different size semitones.

Andy

--------------------
When the going gets weird, the Weird turn Pro.


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: zenguitar]
      #991113 - 03/06/12 02:01 PM
Quote zenguitar:

And just to add, there were many Well Temperaments, not just one.




I did say Bach intended a 'Well' temperament (though).

Quote zenguitar:

The Well Tempered Klavier was written for Werkmeister III and it is constructed using 4 different size semitones.




I think the jury is still out on what the intended tuning really was. Werckmeister temperament is often used for J.S. Bach's music, but only because it is suited to playing music with a lot of chromaticism. It is likely that whatever Bach used wouldn't have been too far from it anyway.

His son C.P.E. Bach wrote that J.S. Bach always insisted on tuning his instruments himself, that it never took him more than 15 minutes, that "most of" the fifths were tempered. So if some of the 5ths were not tempered, then we know it couldn't have been equal temperament, that's for sure. (Also, his other son, Johan Christian, was later importing split-key pianos into Britain to support various meantone temperaments.)

The reason I don't dismiss the title page interpretations (swiggles) is that they are very precisely drawn, changing direction in a way that does not seem random and it was definitely drawn upside down. The page was unfortunately cropped, so Bach may have made it clearer than it appears now. Also, Bach was fascinated by codes and numerology, like composing on the notes of his name, and including biblical number references in his sacred works. He had extensive knowledge of instruments, he could build organs, he could play most (if not all) the instruments in his orchestra to some extent (certainly proficient in all string and keyed instruments) and could read any kind of music notation or tabliture. It's difficult to know what a mind like his intended about anything. I think even his sons were confused.

Cheers.

(P.S. Bach could even have used a tuning that involved minimal retuning between certain prelude performances (i.e. a variety of slightly different tunings based on one). It was quite common then to retune slightly between performances (otherwise it became a bad tempered clavier). I find this highly unlikely though, because his chromatic fantasia would have required a single tuning throughout.)


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Daniel Davis



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Re: Evolution of Equal Tempered scale new [Re: ]
      #991285 - 04/06/12 10:26 PM
Just to point out that the development of tonality i.e. the movement from the old modes to major and minor is rather distinct from the developments in temperament. The OP seems to confuse the two issues.

The major scale is as old as all the other scales of western music. For scale formation you might as well look up wikipaedia as here, and look for such items as tetrachords.

The question might well be 'when did the major scale become popular'? In this case it is true that the major scale existed through the dark ages, middle ages, and Renaissance, but became far more popular in the widespread in the Baroque, and overtook minor pieces in popularity during the Classical period.

As for the harmonic series - it doesn't have a major 7th, but then it doesn't have a perfect fourth either. There is some eastern European music which uses the overtone scale if that's what you like, and there is no reason why you shouldn't use that scale if you choose. Of course it drags you very much to one key, and almost to one chord, but if you want to depict stasis its awesome. (Bagpipes are tuned to an overtone scale with the exception of the fourth which is perfect - they consequently have the strengths of pure intervals over the drones, and weakness of being very limited harmonically.)

Where tonality shines over modality is that it has strength, richness, and flexibility as indeed demonstrated by the Well-Tempered Clavier.

--------------------
Daniel Davis
Edinburgh Recording Studio Windmill Sound


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: Daniel Davis]
      #991382 - 05/06/12 03:12 PM
The fourth does exist but it relates to the fifth - Perfect fourth .It is relative to context and can be consonant in one context and dissonant in another.

I am questioning how the equal tempered scale evolved to become such a dominant standard.
Historically We believed, at one point, there were only 5 planets so there could only be 5 notes in a scale.

Edited by Music Manic (05/06/12 03:14 PM)


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: Daniel Davis]
      #991389 - 05/06/12 03:52 PM
Music Manic Quote:

As for the harmonic series - it doesn't have a major 7th, but then it doesn't have a perfect fourth either.




The only perfectly consonant intervals occur between the fundamental and overtones. All other intervals are derivative. There is a major 7th in the third octave of the series 1:15 (minus 1:8). A perfect fourth can be derived by subracting a 1:3 from the 1:4.

Music Manic Quote:

There is some eastern European music which uses the overtone scale if that's what you like, and there is no reason why you shouldn't use that scale if you choose. Of course it drags you very much to one key, and almost to one chord, but if you want to depict stasis its awesome. (Bagpipes are tuned to an overtone scale with the exception of the fourth which is perfect - they consequently have the strengths of pure intervals over the drones, and weakness of being very limited harmonically.)




There's Jaw harp music, overtone singing, musical (mouth) bow music, didjeridoo music, music based on overblown flute and horns, etc. There is African music that uses vocal scales (and intonation) based on overtones (the Aka of Central Africa use a pentatonic scale Bb, C, D, E, G with overtone intonation, usually descending yet completely without key. They also use a scale often interpreted as a wholetone scale Ab(ish), Bb, C, D, E, F#(ish) based on overtone intonation.

Indian music is heavily influenced by overtones, as all the tuning systems involve harmonic and Pythagorean intervals. Their 'sruti' are derived from a chain of ascending 5ths and another chain of ascending 4ths merged from the same fundamental.

Quote Music Manic:

The fourth does exist but it relates to the fifth




Daniel meant in relation to a scale formed from the fundamental. 1:11 (minus 1:8) is closer to a #4th.

Quote Music Manic:

I am questioning how the equal tempered scale evolved to become such a dominant standard. Historically We believed, at one point, there were only 5 planets so there could only be 5 notes in a scale.




It was really a piano convention as I said above. It was just too impractical to retune cast-iron framed pianos for individual pieces of music. All tempered scales were really intended for fixed pitch instruments. Equal temperament was also used for guitars, although many lutes had moveable and/or doubled frets to support meantone temperaments.

For other instruments, Mozart advocated a 21-note meantone chain of 5ths:

Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx

In the cases of Ebb, Fb, and Bbb, Mozart wrote: "...these tones the harpsichord has not, but all other instruments have...".


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Chaconne



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Re: Evolution of Equal Tempered scale new [Re: ]
      #991833 - 07/06/12 05:16 PM
The thing to keep in mind about equal temperament, at the risk of sounding simplistic, is that all intervals are out of tune by the same amount, i.e they are equally tempered. All keys therefore sound the same, there are no key colours beyond relative pitch, ie 'middle G' is higher than middle C.

Before this, tuning was an attempt to keep as much purity in a keyboard as possible. This is impossible though with the 12 note division, you cannot have all perfect thirds and fifths - if you go round the cycle tuning in perfect fifths - it does not close the circle - you cannot 'fit it in' 12 notes.

So some intervals were tempered, made out of tune, so that some intervals could remain more pure. Some of these intervals were put in intervals that where not needed often, in less used keys.

However over time, freedom to use all keys, and to modulate to and from keys, became the norm and music became more complicated. So the change in colour between keys - because the intervals were not the same depending on what key you were in -became less desirable.

The solution? Make all keys equally in or out of tune.
Make the interval of a fifth in C major, the same as in C# major. It is not in any unequal temperement, so transposition or modulation introduces 'key colour', or at worst unusable intervals.

--------------------



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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: Chaconne]
      #991834 - 07/06/12 05:19 PM
I do think that even if we use other temperaments, equal temperament is still an extremely useful yardstick with which to understand all other temperaments and tunings.

(BTW, I didn't mean to imply Mozart used equal temperament for keyboards. He would've used meantone temperaments suited to the particular composition, but there is surprisingly little discussion on tuning and temperament in the writing of musicians in that period.)


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zenguitarAdministrator
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Re: Evolution of Equal Tempered scale new [Re: ]
      #991887 - 08/06/12 12:11 AM
Quote J.A.S:

I do think that even if we use other temperaments, equal temperament is still an extremely useful yardstick with which to understand all other temperaments and tunings.




Sorry, but NO, that is hindsight and it gives us a perspective that the original composers just didn't have. Equal temperament gives us a framework against which we can reference earlier works, but to understand them we have to put equal temperament to one side and listen to them in their original context. While equal temperament does provide an academic reference point for anyone prepared to look that far, in the real world it's ubiquity becomes a barrier to understanding. I think you are looking through the wrong end of the telescope here.

Andy

--------------------
When the going gets weird, the Weird turn Pro.


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Re: Evolution of Equal Tempered scale new [Re: zenguitar]
      #991961 - 08/06/12 03:00 PM
Quote zenguitar:

Sorry, but NO, that is hindsight and it gives us a perspective that the original composers just didn't have.




But hindsight is such a valuable thing. There are a lot of things we have a clearer perspective on today, using metric and decimalised currency, for instance.

I use equal temperament combined with cents as a kind of 'metric system' for music. This enables me not simply to make comparisons with equal temperament but I also know the values of all the pure harmonic intervals for comparing tuning systems too -but only because I use equal temperament as a yardstick!

If we take true harmonic intervals for example...

True Perfect 5th =7.02 semitones
True Major 3rd =3.86 semitones
True Minor 3rd =3.16 semitones

Their inversions are so easily worked out by subtracting them from the octave (12)...

True Perfect 4th = 12 minus 7.02 (perfect 5th) =4.98 semitones
True Minor 6rd = 12 minus 3.86 (major 3rd) =8.14 semitones
True Major 6rd = 12 minus 3.16 (minor 3rd) =8.84 semitones

The system of cents (where the octave is divided into 1200 equal parts) combined with equal temperament derived intervals, is definitely the easiest and most accessible way I know to understand different tunings and intervals. I really can't imagine having a clearer understanding than I do now.

I have spent quite a few years now using this system to understand the Indian, Turkish and Arabian tonal systems as well as western temperaments and intonation systems. There has also been much of the confusion within these traditions over the centuries, especially between their own proposed equal temperament systems and those evolved by practical working musicians. Ironically, equal temperament is only ever proposed because it's such a practical system, which is why it is better used as a yardstick.

Quote zenguitar:

Equal temperament gives us a framework against which we can reference earlier works, but to understand them we have to put equal temperament to one side and listen to them in their original context.




Again, the system allows me to know and judge the values for harmonic, meantone, pythagorean intervals and not just equal. So they are not really 'filtered' through equal temperament any more than a draughtsman is restricted to the centmetre or metre marks on his ruler.

For instance, I understand a Pythagorean major 3rd as 4.08 semitones. Instantly, I know this is significantly broader than in equal temperment. But I can also work out the difference between this and a natural 3rd is 0.22 semitones, which is a comma (4.08 minus 3.86). Since all the Pythagorean fifths are pure (7.02 semitones) I can easily work out a Pythagorean minor third as 7.02 minus 4.08 = 2.94. Another comma therefore exists between a Pythagorean minor third and a true minor 3rd (3.16 minus 2.94). I also know the different 'acceptable' meantone 3rds, which are usually narrower than 4, because they were trying to preserve their consonance at the expense of 5ths. (BTW this is because 3rds are narrower and produce more noticeable beating than do 5ths.)

So I'm not restricted by equal temperament at all, because I know these values as individual reference points too. This is all about theoretical comparisons, because we can't really "listen" to different temperaments in anything other than their original context. I don't think our brains are capable of fixing equal temperament in the ear for comparison whilst listening to music! Equal temperament just becomes a theoretical measure. (Of course, piano tuning requires counting beats and its own methodology.)

Quote:

While equal temperament does provide an academic reference point for anyone prepared to look that far, in the real world it's ubiquity becomes a barrier to understanding. I think you are looking through the wrong end of the telescope here.




I find it is not necessarily the case that the exact way things were once understood (like imperial measurements) is always better or even necessary for a proper understanding -except perhaps if you're a musical historian. Later composers covered ground much more quickly than earlier composers with the use of roman numerals for harmony, which is very useful to understanding earier music when combined with figured bass, but they were not originally used.


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: ]
      #991962 - 08/06/12 03:11 PM
I've done quite a bit of reading and have noticed that we have needed something to relate to as to deciphering what music is. This all started with Geometry and ended up with Helmholtz's harmonic series, but even with that we still can't verify what is right without using our ears. Pythagoras's Harmony of the spheres and the use of simple ratios doesn't seem so outlandish. Ptolemy furthered his theories and gave us Just intonation from which we relate our equal tempered scale. Temperament dealt with the "Wolf" notes to make music more palatable. Further to this equal temperament gave us the possibility to "move" around more rather than be stuck in one key.

The Science isn't clarified by numbers but how our ear works, this is why things are so varied. Melodically we can do what we want but harmonically things become challenged.

52 note scale. Would have loved to see that piano!


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Anonymous
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Re: Evolution of Equal Tempered scale new [Re: ]
      #991966 - 08/06/12 03:40 PM
For me, numbers can serve us without it becoming a way of thinking. I'm certainly not that way inclined naturally, but for years I became quite obsessed with escaping equal temperament and understanding alternatives. This has now turned into a practice I hardly even need to think about anymore. I use it in my own music all the time: I use fretless guitars, a moveable fret guitar, violin, alternative keyboards (tuneable 6-6), thumb pianos, etc.

Quote:

52 note scale. Would have loved to see that piano!




Mozart's prefered method really corresponded to 55 equal divisions of the octave really. But for keyboards, of course there is only 12, except in the case of split-key pianos (where sharps and flats were split in two on the black keys) there were 17 notes.


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Bitsumishi



Joined: 26/10/11
Posts: 34
Re: Evolution of Equal Tempered scale new [Re: ]
      #992172 - 10/06/12 10:19 AM
The intoduction to Schoenberg's 'Theory of Harmony' is worth a look, total classic text -there's a (weirdly formatted) translation to the first three chapters here, as well as the later essay 'Problems with Harmony':

http://tonalsoft.com/monzo/schoenberg/harm/monzo-intro.aspx

'Structual Functions of Harmony' is also pretty badass:

http://books.google.is/books/about/Structural_Functions_of_Harmony.html?id =mt-RLtfhIfMC&redir_esc=y

Schoenberg -what a dude!


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