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Unequal temperament

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Torctgyd 14 Dec 04 - 12:24 PM
s&r 14 Dec 04 - 12:35 PM
IanC 14 Dec 04 - 12:42 PM
GUEST,punkfolkrocker 14 Dec 04 - 01:11 PM
s&r 14 Dec 04 - 02:01 PM
Peace 14 Dec 04 - 02:06 PM
Peace 14 Dec 04 - 02:09 PM
belfast 14 Dec 04 - 03:14 PM
MaineDog 14 Dec 04 - 03:27 PM
s&r 14 Dec 04 - 04:16 PM
Cluin 14 Dec 04 - 04:33 PM
MaineDog 14 Dec 04 - 08:41 PM
M.Ted 15 Dec 04 - 12:49 AM
pavane 15 Dec 04 - 05:10 AM
s&r 15 Dec 04 - 05:20 AM
pavane 15 Dec 04 - 06:25 AM
Peter K (Fionn) 15 Dec 04 - 11:09 AM
pavane 15 Dec 04 - 11:17 AM
pavane 15 Dec 04 - 11:18 AM
s&r 15 Dec 04 - 11:54 AM
s&r 15 Dec 04 - 12:07 PM
GUEST,punkfolkrocker 15 Dec 04 - 12:09 PM
M.Ted 15 Dec 04 - 01:10 PM
s&r 15 Dec 04 - 02:04 PM
M.Ted 15 Dec 04 - 06:20 PM
The Fooles Troupe 15 Dec 04 - 10:56 PM
pavane 16 Dec 04 - 03:27 AM
s&r 16 Dec 04 - 04:17 AM
pavane 16 Dec 04 - 05:21 AM
RichM 16 Dec 04 - 09:47 AM
JohnInKansas 16 Dec 04 - 02:33 PM
The Fooles Troupe 17 Dec 04 - 05:25 AM
M.Ted 17 Dec 04 - 09:31 AM
s&r 17 Dec 04 - 09:55 AM
M.Ted 17 Dec 04 - 01:09 PM
GUEST,Claudio 15 May 08 - 04:19 PM
Jack Campin 16 May 08 - 05:37 AM
MikeofNorthumbria 16 May 08 - 06:58 AM
Mr Happy 16 May 08 - 07:31 AM
Jack Campin 16 May 08 - 07:41 AM
Paul Burke 16 May 08 - 08:47 AM
Highlandman 16 May 08 - 11:48 AM
M.Ted 16 May 08 - 03:03 PM
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Subject: Unequal temperament
From: Torctgyd
Date: 14 Dec 04 - 12:24 PM

Is this a good example of the problem of unequal temperament (just to get it clear in my head what it was)

If you get a piano tuned in the old, unequal, temperament say C. Then the scale of C is correct, as are the sharps and flats (is this correct?). If you then tried to play in G or F then the F# in G and the Bb in F are not quite correct but you could get away with it. If however you tried to play in keys further from C the less the piano would be in tune and therefore unplayable in these keys. Is this a correct example of the problem?

The Well Tempered Clavier came out in the 1700's so I presumed this was the start of the move to equal temperament. However, fretted instruments predate this so did they evolve an equal temperament of their own or were they also unable to play in only a few keys?

Cheers


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Subject: RE: Unequal temperament
From: s&r
Date: 14 Dec 04 - 12:35 PM

That's about it but the piano is a strange beas in its own right; the high end is tuned sharp and the low end is tuned flat to compensate for the 'non-string' behaviour of the thick bass strings

Stu


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Subject: RE: Unequal temperament
From: IanC
Date: 14 Dec 04 - 12:42 PM

Probably worth saying, also, that before the pinaoforte appeared there was a very limited demand for people to play music which wasn't in the natural key of the instrument.

This is still so today in most "folk" environments, which is why most folk tunes are in G or D Major, or their relative minor keys.

:-)


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Subject: RE: Unequal temperament
From: GUEST,punkfolkrocker
Date: 14 Dec 04 - 01:11 PM

i sort of understand this..

so how does it apply to autoharp..

i tune every string to concert pitch with electronic chromatic tuner..
[well as close within 5 cents either way
as i can get each string to stay in tune..]

but should the bass & treble strings be 'adjusted'
as in the case of piano explained above..

or is the slightly delinquent tuning of autoharp
accepted as part of its 'traditional' musical character..???


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Subject: RE: Unequal temperament
From: s&r
Date: 14 Dec 04 - 02:01 PM

Autoharp should be OK as the bass strings arent anywhere near as thich as on a piano. The harmocs on a piano are not 'in tune. since the low strings dont vibrate as a simple string - something in between a strin and a bar

Stu


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Subject: RE: Unequal temperament
From: Peace
Date: 14 Dec 04 - 02:06 PM

Over a space on the instrument--that is, over lots of octaves--the relative tuning of the instrument to itself begins to differ from the exact tuning of the instrument.


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Subject: RE: Unequal temperament
From: Peace
Date: 14 Dec 04 - 02:09 PM

Google this site which explains it very well.

Music Scales - Frequency, Notes, Octaves, Tuning, Scales, Modes ...


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Subject: RE: Unequal temperament
From: belfast
Date: 14 Dec 04 - 03:14 PM

It is generally said that the "well tempered clavier" refers to the use of what was then a novel form of tuning keyboard instruments. But although this is repeated from book to book and from document to document it is not universally accepted. Some say that "well tempered" means just what it appears to mean, "well tuned". Look here CLICK for example.

I assume that if that is true one would need to retune the instrument when starting in a new key. Otherwise you would end up with the sound that is known as the "wolf tone". Perhaps that phrase will strike a resonance with some Irish Republicans. This site CLICK will give you some barely comprehensible info on the subject.


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Subject: RE: Unequal temperament
From: MaineDog
Date: 14 Dec 04 - 03:27 PM

I have heard of autoharp players keeping several instruments tuned diatonically to different keys. Each one can be used only in 2 or 3 keys, but they sound really sweet that way.
MD


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Subject: RE: Unequal temperament
From: s&r
Date: 14 Dec 04 - 04:16 PM

Wolf tone is normally the sum of unwanted resonances in string instruments notably the Cello.

Different systems of arriving at the twelve notes in an octave have been put forward: what seems true to me is that the ear is rather more tolerant than the dictionary. If it weren't the piano would be rather less popular than it is I think.

Stu


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Subject: RE: Unequal temperament
From: Cluin
Date: 14 Dec 04 - 04:33 PM

I've always found that the human ear was a little more tolerant of a slight sharpness than it was of a slight flatness.


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Subject: RE: Unequal temperament
From: MaineDog
Date: 14 Dec 04 - 08:41 PM

This is certainly true.
Another effect to consider is that a loud sound often seems sharp to the listener. When the fiddler is trying to match pitch with others, his own sound is loud in his left ear. If he is not careful, he will tend to play flat because of his own loudness. At a distance, that will be noticable.
My teacher always encouraged me to play much sharper than I thought was right, partly for this reason, and partly because many people seem to expect a fiddle to be a little sharp.
MD


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Subject: RE: Unequal temperament
From: M.Ted
Date: 15 Dec 04 - 12:49 AM

Pianos were never tuned in "the old, unequal temperament"--the modern systems of temperament were developed to make the piano possible(ungodly and untunable monster that it is)--

A lot of people don't realize it, but but string players play in what is pretty much "natural" intonation--when you listen to a piano concerto, the piano always sounds out of tune with the orchestra, and it is--most other instruments end up being in tune because the players adjust the pitch of each note a bit to make it fit--even though the guitar is tempered, we tend to squeeze those notes back to the natural pitch without even thinking about it--


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Subject: RE: Unequal temperament
From: pavane
Date: 15 Dec 04 - 05:10 AM

The fiddles being played sharp was given as the reason that 'Classical' tuning pitch for orchestras tended to increase over time, as everyone else tuned sharper to catch up.

I am informed that harmonies sound much better in 'natural' tuning, and I am intending to try some experiments with MIDI files. MIDI (at least, my Roland player) provides a fine tune facility, to adjust the tuning of each pitch a cent at a time.

I intend to calculate the 'perfect' (or Just) pitch for each note in the current key, and then use MIDI controls to have the player change the tuning whenever the MIDI file calls for a change of key.

This should be quite simple to program, once I have calculated the tunings for each possible key - which will all be different, of course.


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Subject: RE: Unequal temperament
From: s&r
Date: 15 Dec 04 - 05:20 AM

This is a mess of worms

1640 Vienna Franciscan Organ A457.6

1699 Paris Opera A404

1711 John Shore's tuning fork, a pitch of A423.5 He invented the tuning fork, one of which still exists today.

1780 Stines, for Mozart, A421

1780 Organ builder Schulz A421.3

1714 Strasbourg Cathedral organ A391

1722 Dresden's chief Roman Catholic church organ A415

1759 Trinity College Cambridge organ A309

1762 Stringed instruments at Hamburg A405

1772 Gottfried Silbermann built the organ in the main Roman Catholic church in Dresden, and it had a pitch of A 415 at the time.

1780 Organ builder Schulz A421.3

1780 Stein's tuning fork A422.6

1751 Handel's own fork A422.5

1800 Broadwood's C fork, 505.7, which is about half a semitone lower than that of today

1811 Paris Grand Opera A 427

1812 Paris Conservatoire A440, as modern pitch

1813 George Smart adopted for the Philharmonic Society the pitch of A423.3.

1820 Westminster Abbey organ and possibly Paris Comic Opera used a pitch of A422.5.

1828 Philharmonic Society A 440

1834 Vienna Opera A 436.5

1835 Wolfels piano maker A443

1836 Pleyel's Pianos A446

1846 Philharmonic pitch was A452.5 (very high) which lasted till 1854

1846 Mr Hipkins piano tuner (Meantone) A433.5 (Equal) A436.0

1849 Broadwood's medium pitch was A445.9 which lasted till 1854

1858 New Philharmonic pitch C522

1860 Cramer's piano makers of London A448.4

1862 Dresden Opera A 440

1871 Covent Garden Opera House A 440

1877 Collard's piano maker standard pitch was A 449.9

1877 St. Paul Cathedral organ A446.6

1877 Chappell Pianos A455.9

1877 Mr Hipkins piano tuner A448.8

1878 Her Majesty's Organ A436.1

1878 Vienna Opera A447

1879 Covent Garden Opera A450

1879 Erard's factory fork 455.3

1879 Steinway of England A 454.

1879 British Army regulation pitch for woodwinds A451.9

1880 Brinsmead, Broadwood, and Erard apparently used a pitch of A455.3

1880 Steinway may have been using a pitch of A436. According to Steinway of New York, 1880 is right around the time they switched from three piece rims to the continuous rim that is used today. So it is unlikely the pitch was any higher before 1880, yet Steinway of London had a fork A454.7.

1885 In Vienna a pitch of A435.4 was adopted at a temperature of 59 degrees Fahrenheit for A.

1885 At an international exhibition of inventions and music in London a pitch of A452 was adopted.

1896 Philharmonic pitch A439, giving C522

1925 On the 11th of June the American music industry adopted A440.

1936 American Standards Association adopted A440.

1939 At an international conference A440 was adopted.



Stu


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Subject: RE: Unequal temperament
From: pavane
Date: 15 Dec 04 - 06:25 AM

That 1759 Trinity College A309 looks WAY out - is it a typo for 409?

Scientific pitch, C=512 would give A about 407, I believe


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Subject: RE: Unequal temperament
From: Peter K (Fionn)
Date: 15 Dec 04 - 11:09 AM

s&r's selective history implies that this is a relatively recent issue, but in fact the problem was well understood by Pythagoras. It comes down to fairly basic maths (OK math) which I will try briefly to explain, since I don't think other posts have really dealt with it.

In western music the basic building block is the semitone. A semitone is one twelfth of an octave. Thus, according to the currently prevailing standard for concert pitch, the A above "middle" C vibrates at 440Hz; and the A one octave higher than that vibrates at 880; the gap between these two As is filled with 11 other notes, the intervals between which are known as semitones.

A string which sounds A (440) when open will sound an octave higher - A'- if stopped halfway along its length. (Just as a 4ft organ pipe will be pitched one octave higher than an 8ft pipe, other things being equal.) Stop the string at the 2/3 point and the note produced will be E.

The interval from A to E is known as a fifth. After the octave itself, the fifth is the second strongest interval in the harmonic sequence - the pattern of overtones heard when any note is sounded, or the notes that can be derived from a fixed-length tube. The fifth has a "pure" quality about it that sounds almost hollow. So much so that classical and romantic composers for orchestra, organ, piano etc, usually sought to avoid successive fifths in harmony (except for special effect, as in Vaughan- Williams' orchestration of Greensleeves for instance). This interval is instantly recognisable to the tuned ear, and to many untrained ears too. Indeed violinists - once they have tuned the A string to an external source (tuning fork, piano, oboe or whatever) - use the unique sound of the fifth to tune the three other strings, G, D and E. (A feat they usually accomplish within a second or two for each note.)

But from A, move to its fifth, then the fifth above that and so on, until you come to the next A. The note you arrive at will have a frequency that is not divisible by the 440 you started from, because it is an unequal number of octaves removed from the original A. You have arrived at a note that would sound noticeably out of tune.

For an instrument confined to a range of an octave or so, this problem is not likely to be a factor, in the same way as it would suffice to divide a year into 365 days if there was only one year. But as the range extends, the problem becomes increasingly evident and is particular evident on piano and organ, with their ranges of several octaves.

If a sequence of notes is tuned in conformance with harmonic intervals, there will come a point where a significant correction is needed - an interval so unnatural to the ear as to be unusable. Such an interval is known as a "wolf tone". But if the mathematical discrepency is shared equally across all intervals, harmonies will be compromised.

Tempering, be it well tempering, meantime tempering or whatever, are all attempts to square the circle. In effect they are musical equivalents of the leap year. Some systems give priority to the most crucial intervals - fifths and major thirds. Some favour particular keys at the expense of others. Hence a tendency for some instruments to sound better in one key than another, and in the classical era for subjective values such as "warm," "brilliant," "plangent" etc to be attached to specific keys.

Instruments of infinite pitch - the strings and trombones in an orchestra, and of course the human voice - can arrive at compromises almost on a note-by-note basis, but other instruments are contrained to their pre-determined semitones, and it is between these, as someone observed, that discrepencies will sometimes be noticed (but not by me, because my ear is not good enough).


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Subject: RE: Unequal temperament
From: pavane
Date: 15 Dec 04 - 11:17 AM

The progressive error is actually due to the mathematical fact that no power of 2 can equal a power of 3.

Therefore no series of fifths (3/2 multiples) can ever return to a true octave.


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Subject: RE: Unequal temperament
From: pavane
Date: 15 Dec 04 - 11:18 AM

And before some mathematician tells me I am wrong, let me exclude the power zero.


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Subject: RE: Unequal temperament
From: s&r
Date: 15 Dec 04 - 11:54 AM

I'm more in line with Pavane. If it were possible to crystallize tuning and its associated problems into simple maths in a few paragraphs some of the world's best musicians and mathematicians wasted a lot of time.

I made no implication in my post above - you may have drawn inferences

Stu


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Subject: RE: Unequal temperament
From: s&r
Date: 15 Dec 04 - 12:07 PM

The whole question of tuning is theoreticall much more complex than playing/singing/writing demands.

There's lots of detailed info on the web (not all reliable) but this one took my eye Try to follow this lot

Stu


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Subject: RE: Unequal temperament
From: GUEST,punkfolkrocker
Date: 15 Dec 04 - 12:09 PM

Bloody hell..!!!!

i became a musician because at school i was crap at maths and physics.

..and now 30 years later it seems one of the reasons
i've always been a crap musician
is because i'm no good at maths and physics..?????

..oh well as long as any song my band plays
still starts with a shouted simple count-in
of " One two free fo...oerr..!!!!

i should be ok..


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Subject: RE: Unequal temperament
From: M.Ted
Date: 15 Dec 04 - 01:10 PM

The reason that we have tried to develop "even" or "equal" temperament scales in the first place is to allow composers to use the same major/minor scales starting on any step of the scale--this allows us to build the same kind of chords on any note when we harmonize, and it allows us to move through the circle of fifths melodically--This is the basic stuff of "Western" classical music, but there are many fine musical traditions that have done just fine without it--

While it may be impossible to construct a chromatic scale without compromises based only on the fifth--Paul Hindemuth showed that it can be done using other harmonics--

Also, the "just" scale is a tempered scale, as well, Pavane--if you want a "natural" scale, grab a set of bagpipes--


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Subject: RE: Unequal temperament
From: s&r
Date: 15 Dec 04 - 02:04 PM

Do you mean Hindemith?

Stu


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Subject: RE: Unequal temperament
From: M.Ted
Date: 15 Dec 04 - 06:20 PM

Yep--sorry for the typo--about half the time, I write "Merry Christmans", too--


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Subject: RE: Unequal temperament
From: The Fooles Troupe
Date: 15 Dec 04 - 10:56 PM

'If a sequence of notes is tuned in conformance with harmonic intervals, there will come a point where a significant correction is needed - an interval so unnatural to the ear as to be unusable. Such an interval is known as a "wolf tone". '

I thought this was called a 'Caesura', or similar, may have bodgied the spelling.

I thought 'wolf tones' were those horrible notes that badly made violins badly played, made - due to uneanted resonaces getting out of control.

Didn't they also release a few records?... :-)


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Subject: RE: Unequal temperament
From: pavane
Date: 16 Dec 04 - 03:27 AM

Punkfolkrocker

Yes, music is VERY much based on maths and physics - did you see the reference to Pythagoras above? One of the foremost mathematicians of
classical times.

It was he who showed that the interval of a fifth was based on a 3/2 relationship.

Oh yes, my reference to 'natural' above was meant to refer to perfect 5ths - sorry if it is out of line with common usage.

I agree that even-tempered tuning is a compromise necessary for western music. It involves making each interval correspond to a multiple which is the 12th root of 2, so that multiplying by it 12 times brings you to a note of double the one you started with, i.e. a perfect octave.

According to Excel, this should be 1.059463094

Multiplying A=440 by this 7 times gives a fifth, the actual value of which will be 659.2551138

A perfect 5th would be 3/2 times 440, which is 660, therefore the even-tempered note is 0.74 too flat

If you could play the two together, you may hear a 'beat note' of the difference - like the ones Guitarists use when tuning.


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Subject: RE: Unequal temperament
From: s&r
Date: 16 Dec 04 - 04:17 AM

Pavane - do you mean a Just fifth?
Perfect in my understanding is a term that denotes intervals which do not change their nature when inverted, so

Major when inverted becomes minor
Minor when inverted becomes major
Diminished when inverted becomes augmented
Augmented when inverted becomes diminished
Perfect when inverted stays perfect.

Stu


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Subject: RE: Unequal temperament
From: pavane
Date: 16 Dec 04 - 05:21 AM

In which case we are talking of different things.

By 'perfect' I mean the 3:2 ration is exact. If that is what you understand by a 'Just' 5th then OK

What exactly do you mean by the term 'Inverted' in this context?
Can you give an example?


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Subject: RE: Unequal temperament
From: RichM
Date: 16 Dec 04 - 09:47 AM

Though it may not be politically correct, I have always tuned my autoharp to separate-but-equal temperament...


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Subject: RE: Unequal temperament
From: JohnInKansas
Date: 16 Dec 04 - 02:33 PM

Foolestroupe:

If you apply integer ratios to multiply a starting frequency repeatedly, as when using the 3/2 ratio and moving by "fifths," when you "come back" to the original "note" (at some higher octave) the "error" from perfect octave pitch is commonly called the "comma." The usual value is given as 81/80, or about 1/10 of a semitone.

The difficulty with "just" tuning (everything in ratios of integers) is that the F you get when you start from D and proceed "by fifths" unitl you land on F isn't the same as the F you get if you start from G and proceed "by fifths." In theoretical analyses, the "errors" are generally stated as 1-comma, 2-comma, etc, and theoretically, starting from any two notes, I think I recall that you should never be more than 3-commas off for any given note determined from the two separate "starting points."

One of the "best" and most thorough discussions of this subject I've seen is in Herman Helmholtz's On the Sensations of Tone. Dover has (or had) a cheap reprint of the 2d Ellis English translation (1885) of the Helmoltz Fourt Edition (1877) that should be available. Be advised that it's about 600 pages of "theory and measurements," and may be more than most will want to go through, but you don't have to read it all.

John


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Subject: RE: Unequal temperament
From: The Fooles Troupe
Date: 17 Dec 04 - 05:25 AM

A great TV documentary I have mentioned in the past here (I think!) is the "Howard Goodall's Big Bangs in Music" (BBC I think). He goes into considerable detail with visual examples of the 'perfect temperament to equi-temperament' historical voyage - his opinion is that it was the Piano Accordion that spread the 12 tone equi-temperament everywhere and wiped out other older 'perfect' style tunings.

(See - I mentioned Piano Accordions again!)


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Subject: RE: Unequal temperament
From: M.Ted
Date: 17 Dec 04 - 09:31 AM

With all due respect, they didn't use "perfect" style tunings before the piano accordian was introduced--


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Subject: RE: Unequal temperament
From: s&r
Date: 17 Dec 04 - 09:55 AM

Pavane
Inversions of intervals, so (say)
c - e = Major Third; e - c = minor sixth
c - F# = Augmented fourth; F# - c = diminished fifth
c - g = perfect fifth; g - c = perfect fourth

And here in more detail

I would call 3:2 a just fifth because I know 'perfect' as above


Stu


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Subject: RE: Unequal temperament
From: M.Ted
Date: 17 Dec 04 - 01:09 PM

In order to really understand the harmonic overtone series and how it is used in created scale systems, you have to do the math--none of it really makes sense any other way--It is very important to understand that within the harmonic overtone series, there are a lot of options for each of the intervals--

Here is a link that explains all the different tuning systems, including the math--

The Development of Musical Tuning Systems


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Subject: RE: Unequal temperament
From: GUEST,Claudio
Date: 15 May 08 - 04:19 PM

Hi friends.
I am the author of a book (1978)that was the first attempt to systematically cover the acoustics and history of temperament.

Very well received in the 80's is now out of press and obsolete.
So many new discoveries have been made in the last 3 decades by the scholars, that I am now fully re-writing it. It should be printed (or available online) within a few months.

It includes critical reviews of many of the latest discussions (e.g. recent "Bach" temperaments and the Victorian temperaments).

Details will be published in my site where you can also read the new introduction. Stay tuned!

Claudio

Bray, Ireland

http://temper.braybaroque.ie/


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Subject: RE: Unequal temperament
From: Jack Campin
Date: 16 May 08 - 05:37 AM

So, what did your book do first?

Helmholtz's "On the Sensations of Tone" (1863) includes a history of temperament. So, I think, does Athanasius Kircher's "Musurgia Universalis" (1655). So did pretty near every comprehensive music theory text of the last 100 years.


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Subject: RE: Unequal temperament
From: MikeofNorthumbria
Date: 16 May 08 - 06:58 AM

A helpful book on this topic is "Temperament: the Idea that Solved Music's Greatest Riddle" by Stuart Isacoff (published by Alfred A Knopf, New York 2001). It deals with the technical issues thoroughly, but in a relatively jargon-free manner. It also sets the search for better tuning systems in historical context, though readers who just want the musical information can skip these passages and still get the fundamental message.

Howard Goodall's TV programme is also very useful – and entertaining – though some of the technical points flew past so quickly that I needed to replay them a few times to follow him. (The DVD is advertised as available from Amazon.)

However, I've found nothing in print or online which applies this technical information to the tuning problems which afflict the average folk guitarist. Even when your guitar is properly set up and accurately tuned, when you play the standard first-position C chord, the top E string sounds a bit sharp. So, you twist the tuning peg till it sounds better and play. So far, so good.

But later, when you play a first-position E chord, the top E string sounds flat.   You twist the peg again to pull it up – and then you notice that the G# on the 3rd string sounds a bit sharp, so you adjust that, and play away happily. All is well while you keep playing in the key of E, but when the next song in C comes along, your 3rd string sounds flat, while your 1st string sound sharp. And so it goes on – and on – and on.

This problem haunted me for decades, until I eventually learned that the major third interval (C – E in the key of C, E – G# in the key of E) is 17% sharp of the "natural" major 3rd. Even those of us with fairly insensitive or untrained ears can tell the difference – but when we try to put it right, we often make things worse. So, what's the solution? Basically, there isn't one - but there are techniques that can alleviate the problem somewhat.

1) Make sure your guitar is properly set up, put good strings on it, change them regularly, and tune them carefully with a good electronic tuner.   After that, don't keep tweaking to try and improve things further, as this is more likely to hinder than help.

2) When you strum chords, try to avoid sounding the 3rd note in the chord (E in C, G# in E, etc) on the first or second strings, where it seems to be more noticeable. For example, try playing the 1st position C chord holding down a G note (1st string, 3rd fret) with your little finger, instead of having an open-string E.   (There are similar work-arounds for other common chords.)

3) Try playing in open tunings like DADGAD and using chords without the third interval in them – when you have to fret a 3rd as a passing note, it's less noticeable against the jangling background of root/5th drones. Or play in open tunings with a slide and avoid the fretting problem altogether.

4) When playing jazz, blues and rock music (and also some kinds of East European and Asian music) hit the note one fret below the one you want, and bend it up until it sounds right. (If it was good enough for Broonzy, Django and Hendrix …)

I could go on, but this post is too long already.

Wassail!


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Subject: RE: Unequal temperament
From: Mr Happy
Date: 16 May 08 - 07:31 AM

http://temper.braybaroque.ie/

Loads've fascinating stuff there Cloudy-oh Deave-Erroly, thanks!


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Subject: RE: Unequal temperament
From: Jack Campin
Date: 16 May 08 - 07:41 AM

5) give up and play something fretless like an ud or fretless bass.


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Subject: RE: Unequal temperament
From: Paul Burke
Date: 16 May 08 - 08:47 AM

Play in the nude? It's not a pretty site, especially the piccolo. But this book, Harmonious Triads, has a very good section on the practical problems of actually tuning to equal temperament, that were not solved completely until the second decade of the 19th century.


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Subject: RE: Unequal temperament
From: Highlandman
Date: 16 May 08 - 11:48 AM

Great thread.
I've done a lot of reading and experimenting on this subject myself. There may be more 'technically correct' answers but most of the discussion here is much more sound than I have read in other places. MofN, your explanation of the woes of guitar players is the best-presented I have ever seen.
One thing I wanted to throw in, for the mathematically-minded, is this: most of the development in temperament and tunings was done, historically, BEFORE the discovery of the properties of logarithms. We now have math tools to make calculating this stuff almost trivial, but in the days before logs, it took a pretty keen mind to get a grip on what was going on there.
Cheers
-Glenn


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Subject: RE: Unequal temperament
From: M.Ted
Date: 16 May 08 - 03:03 PM

Problems with intonation on a guitar are pretty much resolved if you tune the A string to A-440, then sound the octave harmonic on the 12th fret, and tune so that the A's on each string are in tune with it.

And Mike's"explanation" as to why certain guitarists bend notes is a bit of drollery.


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Subject: RE: Unequal temperament
From: MikeofNorthumbria
Date: 16 May 08 - 04:30 PM

M Ted, you misunderstood my intention. I didn't mean to imply that correcting problems caused by equal temperament was the reason why those quitarists bent notes - only that if bending notes was OK with such as the then it was OK with me.

Nevertheless, it is a fact that bending upwards from the note below (as they did, for whatever reason)does help to moderate the harshness of that awkward third note in the equally tempered major scale.

Incidentally I've also heard that some jazz guitarists of the swing era would often play the note below the third of the scale and then slur up a semitone, achieving a similar mellowing effect.

And a third possibility - if the tune requires you to hold that harsh third note for a bit longer you can blur its edges with a touch of vibrato.   

Wassail!


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Subject: RE: Unequal temperament
From: JohnInKansas
Date: 16 May 08 - 05:46 PM

Napier "wrote the book on logarithms" in his Minifici Logarithmorum Canonis Descriptio in 1614. Prior to that, and in fact very much prior to that there were many mathematicians who had the knowledge needed to do so. They just didn't feel the need to "give a name to" the function.

Other "functions" that were used for very long times include the now well-known trigonometric functions. Understanding and use of some "non-linear" trigonometric functions is evident in the construction of astronomical and navigation instruments dating back to "near pre-history." The common names for the relationships known and used, and intelligible mathematical "proofs" (ways of laying them out on a scale), for trig functions is fairly recent, with one source asserting that the well known "tangent" was first given a name ca. 1583.1

"Harmonic" or "Just" tuning is a natural way of tuning, and has some appeal, but the recognition that transposing to a different key on the same instrument made the tune sound different had been recognized for centuries. The explanation of the "coma" error is quite old, and a number of different "accommodations" were proposed. Most of the "differently tempered" scales permitted key changes over a range of keys, with lesser change in "flavor" of the tune for some adjacent keys, but in most cases even more drastic changes at keys further around the circle.

Beethoven was among the first to recognize that if the ratio of frequencies for each semitone pair was identical, it wouldn't make any difference what key you played in. This "adjustment" lost a bit of the "strength" of a few tone-pairs, but arguments notwithstanding, those "resonances" are only subtly heard in complex chords of the kinds commonly used by "pianists" like Mr. B.

While only one specific constant ratio works if you want to have an octave with 12 semitones, it wasn't really necessary for Beethoven to have a name for the ratio, or to know an exact numerical value for it. Graphical methods, almost trivially simple, existed since Euclid for constructing the sequence of "locations" where each location is "x times the distance" (from a fixed point) of the adjacent point. Regardless of the "fixed ratio" chosen to construct the sequence of points (string length increments or fret positions) the "correct" ratio can be found by moving the fixed point (moving the bridge) until some thirteenth (twelve intervals up) position is an octave from an arbitrarily chosen first one.

Knowing that the exact ratio needed was the "twelfth root of two," and having the "newly invented logartithm" function to make the explanation a bit more convenient probably was helpful in spreading Beethoven's advocacy for his "new theory;" but that probably wasn't really necessary to his implementing and demonstrating it, or to it's (eventual) fairly general adoption.

That Beethoven learned of logarithms, and the existence of "the twelfth root of two" from a mathematician friend (whose name is generally known but that I've forgotten for now) should absolutely amaze us as being the proof positive that in Beethoven's day friends did gather at the local tavern (and probably got well and properly swacked) and did trade letters, in which they discussed something other than football, politics, and religion - and the brawls were not so violent that nothing was remembered by the time they got home to think about it.

1 TANGENT (as a trigonometric ratio) was introduced by Thomas Fincke (1561-1656) in his Thomae Finkii Flenspurgensis Geometriae rotundi libri XIIII, Basileae: Per Sebastianum Henricpetri, 1583. He wrote tangens in Latin. In English, TANGENT is found in 1594 in Exercises by Thomas Blundevil.

John


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Subject: RE: Unequal temperament
From: Darowyn
Date: 17 May 08 - 09:05 AM

If you are puzzled by the twelfth root of two concept, look at it like this:-
It's the number you have to multiply by itself twelve times to make two.

Logically, if you apply this number to dividing the scale length of a guitar into frets or multiplying the frequency of a tonic note, do it twelve times and you are half way along the string, or double the frequency of the note-i.e the octave frequency.
Cheers
Dave


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