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Benedetti's Puzzle

mizmar

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210
I'm fascinated by this stuff but I really don't have good ears, as they say, so I often don't quit get the "joke".
 

Jimmymack

Member
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358
I told you I was in tune, it's the piano that's out. Fascinating stuff, I knew nothing about it and I'm not sure that I'm much closer to grasping what's happening. It's a bit like the articles I read about the physics related to astronomical research. I start reading with such enthusiasm and by the end of the second paragraph I'm lost but I love reading it so that I can shake my head in amazement that somebody has discovered this and really does understand it (I always assume that they're not just kidding me). Thanks, that's so much fun I'll probably watch it again and maybe begin to get a grasp.
 

jbtsax

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Tenor Viol

Full of frets in North Shropshire
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I think the takeaway from the video is that we know that equal temperament is imperfect and that it is less 'in tune' than just intonation. However, it 'evens out' the out-of-tuness by distributing the syntonic comma equally amongst the 12 semi-tones of the octave. Accordingly, ET is more-or-less in tune for everything (major thirds are the biggest problem).

Trying to tune 'properly' using some sort of just intonation breaks down once you start moving away from the home key which you have tuned for. For example, if you have tuned for C, then F, Bb, G, and D will be 'OK' but move away any further and tuning will become more and more outrageous.

There is no way to tune an instrument (other than the computerised one used in the video) to dynamically 'correct' the tuning. Not only that, but it will result in a pitch shift.

So, if you want to play in all keys and be mostly in tune, there is currently no viable alternative to equal temperament.
 

Jimmymack

Member
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358
I suspect the saxophone is the perfect instrument for playing the non-tempered scale, being completely non-tempered itself, not to say bad tempered at times, but very adjustable on the fly. Maybe the trumpet beats it. I often wonder how people with perfect pitch cope, this suggests more pain for them than I had imagined.
 

jbtsax

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What I remember from playing in and conducting wind ensembles when there is not a fixed pitch instrument present is to lower the 3rd a bit in minor chords, raise the 3rd a bit in major chords, and lower the b7 in dominant 7th chords. The rest of the intonation pretty much takes care of itself as long as you match octaves, unisons, fourths, and fifths to avoid those "pesky" beats.
 

Colin the Bear

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13,558
Unfretted strings can also be flexible. It was a relief to me to finally understand just and equal temperament. The tuner was driving me mad. I could tune a guitar etc in an open tuning but struggled in standard tuning. All those numbers and fractions leave me cold. It's a great frustration that music is not mathematical. Science and technology helped get my head round it. Here's the video that did it for me. To summarise...play nice.
View: https://youtu.be/6NlI4No3s0M
 

mizmar

Member
Messages
210
So, if you want to play in all keys and be mostly in tune, there is currently no viable alternative to equal temperament.
Surely it's the other way round, historically? ET is a technology (along with more chromatic instruments) that enabled the development of larger ensembles and a broader pallet of harmonic structures.
 

Jimmymack

Member
Messages
358
It would be interesting to get three people together with perfect pitch with each one pitching A at 438, 440 or 442. I just take it on trust that some people have it, they may be imagining it for all I know.
 

Pete Effamy

Senior Member
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2,751
It would be interesting to get three people together with perfect pitch with each one pitching A at 438, 440 or 442. I just take it on trust that some people have it, they may be imagining it for all I know.
I know a few who have it, but it must vary in accuracy. Or does it? True perfect pitch. Yep, I’m back at the original question - perfect to what?
 

mizmar

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210
I know a few who have it, but it must vary in accuracy. Or does it? True perfect pitch. Yep, I’m back at the original question - perfect to what?
I thought / supposed it was; to the musical tradition they developed in?
The name of a pitch - eg 440hz is called "A" - is only meaningful within some culture or other. Even "220 and 880 are also A" is conventional.
 

Tenor Viol

Full of frets in North Shropshire
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Surely it's the other way round, historically? ET is a technology (along with more chromatic instruments) that enabled the development of larger ensembles and a broader pallet of harmonic structures.
Music until roughly the late baroque era, especially if played with fretted or fixed pitch instruments, avoided wandering too far from a core group of related keys.

You could do exotic modulations, but you would get the 'howling wolf' as the notes of the remote key would not be in tune. For example F# should be sharper than G flat. So if you were tuned to be 'in tune' for C major +/- a sharp or flat or two and played in G major, the F# would be tuned for that purpose. If you then modulated to say D flat major - the G flat would be hideously sharp because it had been tuned to be an F# in G major, not a G flat in Db major.

There were loads of experiments with keyboards that had 'split' notes to enable playing F#/Gb as separate notes. none really caught on.

On the viola da gamba (viol) it is common to 'split' the double loop of certain frets so that you can sharpen/flatten depending which note you're meant to be playing (e.g. F# / Gb).

This is why in older music you rarley stray beyond about 2 sharps/flats.

The baroque got a lot more adventurous....

Equal temperament was not an invention of the baroque era or of Bach, it was known before then. Eventually, it was 'settled' upon as the only realistic way for fixed pitch, fretted, and fretless instruments to be able to play together in any key.

Good musicians are aware of these issues and as @jbtsax says they will adjust the thirds in particular to improve the tuning (the major third is the least in tune interval as it is way too wide). When I tune my cello I use a 'violin tuning' option which provides pure fifths and not the slightly narrower ET ones.
 
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