Author Topic: A rationale for finding a good temperament  (Read 2441 times)

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David Pinnegar

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A rationale for finding a good temperament
« on: August 07, 2011, 12:57:48 PM »
Hi!

I wonder how many people simply accept our keyboards as they appear to be and look at pitch as the obvious octave divided into 12 semitonal notes of which each will be an equal step along the way?

The origin and construction of our musicscape is different.

Why 12 notes?

It's really based upon the simplest complexity accessible to us which provides an adequate infinity of permutations for the purposes of making music.

If we take a note, that note is the result of a certain number of vibrations per second. The octave is perhaps the simplest interval, as it's twice the number, and then the fifth is three times the number.

We can derive notes from harmonics - as intervals the fourth, the third and minor third and second derive from the next notes on this spectrum of the series of harmonics obtained by multiplying the fundamental frequency by successive amounts.

But these harmonically derived notes, in combinations sound beautifully pure but don't always fit in exactly with harmonic series derived from any of those resultant notes when taken as a fundamental note.

I referred above to the "simplest complexity". The simplest relationship is the octave and the next simplest is the fifth, being a ratio of 1 1/2 times the fundamental frequency, 3/2. When we take 12 fifths, they coincide with 7 octaves and we get 12 notes as a result.

But the problem is that 7 octaves, being 2x2x2x2x2x2x2 = 128 and 12 fifths, being (3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2) = 129.7 . . . just recognisably being the same note but being that much slightly different.

Perhaps the natural thing to do is to distribute the error equally between all of the fifths, but in terms of the development of music, it doesn't really work. When we sing or play in brass bands, we like to produce pure intervals and certainly the origin of polyphony in the 12th century, I understand, was in the experience of the sweetness of singing in pure thirds. It was for this reason that Meantone developed giving 8 pure thirds giving unparalled sweetness to the music but making 4 keys, G# B C# and F# unplayable.

If one wants to start to play in all keys, one has to start juggling.

I start from a premise of allowing good thirds to be pure or beating slowly enough to enjoy the beats rather than losing count of them per second.

If we want to be able to play in keys denied to us by Meantone, we need to be a little pragmatic in the extent to which we allow modest shifts to our thirds and fifths . . .

For the purpose of forming a rationale, let's start with the three thirds centred on C - F-A C-E G-B as being substantially pure.

With 3rds and 5ths, this means that we are starting with defining C-F-A-E-B-G and then if we want Bb-D-F#.

Can anyone start to look at how the implications might be of shifting these intervals as against their equal temperament counterparts so as to achieve better sounding 3rds?

Best wishes

David P
David Pinnegar, BSc ARCS

Barry Williams

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Re: A rationale for finding a good temperament
« Reply #1 on: August 07, 2011, 01:32:32 PM »
David, whilst all of this is interteresting, indeed fascinating, please do not forget that tuning the organ to most temperaments wil restrict the repertoire required for the organ to do its job.  That is why what is called 'equlat temperament' was developed.  Music is a living art, not a dead historical science.  The samee arguments apply to retaining highly unusual pedal boards 'for historic reasons'.

Music in worship needs to be acceptable.  An organ tuned to the vast majority of temperaments will not be able to render many hymns satisfactorily.  Organ sare provided in churches to lead worship, not to satisfy academic criteria on temperaments.  There is an pipe organ not far from where BF, BM & L and I live which is tuned to a temperament for 'historical reasons'.  The church uses an electronic for services.  The 'historic' organ (it is actually quite modern) is hardly used at all.  That is no way to encourage the pipe organ.  In fact, it works against the pipe organ.

Barry Williams

David Pinnegar

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Re: A rationale for finding a good temperament
« Reply #2 on: August 07, 2011, 01:45:18 PM »
David, whilst all of this is interteresting, indeed fascinating, please do not forget that tuning the organ to most temperaments wil restrict the repertoire required for the organ to do its job.  That is why what is called 'equlat temperament' was developed.  Music is a living art, not a dead historical science. 

Dear Barry

Yes - I quite agree - but that's no reason for the subject not to be explored . . .and at some stage you were going to come down to Sussex to try out different temperaments and see what would be precluded by their application and if any could be of more universal benefit . . . and look forward to seeing what we can do in that direction.

Anyone else wishing to do so is similarly welcome!

Best wishes

David P
David Pinnegar, BSc ARCS

David Pinnegar

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Re: A rationale for finding a good temperament
« Reply #3 on: August 07, 2011, 11:48:45 PM »
For the purpose of forming a rationale, let's start with the three thirds centred on C - F-A C-E G-B as being substantially pure.

With 3rds and 5ths, this means that we are starting with defining C-F-A-E-B-G and then if we want Bb-D-F#.

My mind went towards these problems during a concert recently using a variety of unequal temperaments.

To get purer than equal temperament thirds, the thirds have to be narrower than in equal temperament.

Whilst historically one should start from C, it really doesn't matter to be A-centric.

Starting from A therefore, F has to be a little sharp to achieve the narrower purer third. Historically A major has always been quite bright so C# can probably remain at around Equal Temperament. Similarly E to G#

Similarly C to E has to be narrower resulting either in a flatter E or a sharper C. If C is sharp, then G# to C is wide. The authority on temperaments, Jorgensen, noted that G# was the last note to be introduced historically into the scale and therefore less importance was placed historically in achieving the best intonation into the G# Ab key . . .

Ideally one does not want sharp fifths as this simply causes errors to be accumulated elsewhere. For this reason a purer F-A means a sharper F which then has to match up with a sharper C unless the F-C fifth is going to end up too narrow.

So sharper than equal temperament C and F notes make some sense.

Vallotti, D'Alembert and other historic temperments have in common C and F sharp by 6-8 cents so the methodology appears to be starting to treading a known path. . . .

Best wishes

David P
David Pinnegar, BSc ARCS

 


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