Cambridge paper about Pythagoras

ToneControl

Firefox suggested I read this

https://www.cam.ac.uk/research/news/pythagoras-was-wrong-there-are-no-universal-musical-harmonies-study-finds?utm_source=pocket-newtab-en-gb

AFAIK they are saying "normal chords don't work on Bonang notes, therefore Pythagoras was wrong"

But as they say themselves, the Bonang overtones are not at the octave etc.
more detail here:
(PDF) SOME STUDIES ON THE UNDERSTANDING THE DIFFERENT TONES QUALITY IN A BONANG SET (researchgate.net)

Pythagoras did talk about overtones. 

I think it's interesting that some instruments have unusual overtones, but wouldn't go as far as saying Pythagoras was "wrong".

The fact that different chords sound good on the bonang does not surprise me

Cranky

Sounds like it’s exploring Pythagoras’s role in creating a musical ethnocentrism that’s lasted a long time.

So, Pythagoras didn’t like jazz.  And he didn’t know anything about instruments and music from around the world.  So he overextended his thesis.

Happens all the time.

Stratavarious

I’ve ‘sweetened’ tunings for years… he’d be turning in his grave.

jpfamps

Pythagoras is not "wrong".

Pythagorian intervals with low integer frequency ratios reduce beat notes, and so sound consonant; this is simply a function of the laws of physics, which when I last looked are not socially constructed.

However using Pythagorian intervals as the basis for tuning results in some very out of tune notes unless you only play in one key, so the tuning of all instruments deviates from Pythagorean intervals.

The history of tuning predominantly concerns the compromises made to try to make instruments sound as consonant as possible in all keys.

This is all very well know and should not really come as a surprise to anyone.

jpfamps

Stratavarious said:

I’ve ‘sweetened’ tunings for years… he’d be turning in his grave.

Actually you are "sweetening" them to make the intervals closer to the Pythagorean ratios.

The evenly tempered scale, which is nominally employed on the guitar, deviates significantly from the Pythogorian intervals.

The worst interval is the major 3rd which is 13 cents sharp for example.

The evenly tempered scale is a compromise that is equivalently out of tune in all keys.

Also check out piano tuning, which is "stretched" from even temperament so the bass notes are flatter and treble notes sharper.

As the deputy head of jazz at a very prestigious music school said to me, "everything is always out of tune, you just need to deal with it."

jpfamps

ToneControl said:

Firefox suggested I read this

https://www.cam.ac.uk/research/news/pythagoras-was-wrong-there-are-no-universal-musical-harmonies-study-finds?utm_source=pocket-newtab-en-gb

AFAIK they are saying "normal chords don't work on Bonang notes, therefore Pythagoras was wrong"

But as they say themselves, the Bonang overtones are not at the octave etc.
more detail here:
(PDF) SOME STUDIES ON THE UNDERSTANDING THE DIFFERENT TONES QUALITY IN A BONANG SET (researchgate.net)

Pythagoras did talk about overtones. 

I think it's interesting that some instruments have unusual overtones, but wouldn't go as far as saying Pythagoras was "wrong".

The fact that different chords sound good on the bonang does not surprise me

Presumably the "normal" chords don't work because the overtones cause significany dissonance, ie they deviate significantly from the Pythogorean ratios.

viz

^ exactly.

And I really like your "everything is always out of tune, you just need to deal with it." - that exact process has been used for centuries by composers using various key / tuning relationships and capitalising on the harmonic and melodic opportunities they afford, to make the most of the compromises that must be made in pitches and intervals. Awesome stuff.

guitarjack66

jpfamps said:

Stratavarious said:

I’ve ‘sweetened’ tunings for years… he’d be turning in his grave.

Actually you are "sweetening" them to make the intervals closer to the Pythagorean ratios.

The evenly tempered scale, which is nominally employed on the guitar, deviates significantly from the Pythogorian intervals.

The worst interval is the major 3rd which is 13 cents sharp for example.

The evenly tempered scale is a compromise that is equivalently out of tune in all keys.

Also check out piano tuning, which is "stretched" from even temperament so the bass notes are flatter and treble notes sharper.

As the deputy head of jazz at a very prestigious music school said to me, "everything is always out of tune, you just need to deal with it."

That last paragraph has validated my playing!

jpfamps

OK, I have to apologise here as I have got some of the details of this wrong.

I was posting from (fading!) memory and on reflection thought I'd not got it correct so I have consulted the relevent chapters in The Science of Sound.

Pythagoras did indeed make the observation that consonant intervals have low integer frequency ratios.

Octave 2:1

Perfect 5th: 3:2

Perfect 4th: 4:3

If you ascend by a Pythagorean fifth then a forth you acheive a ratio of 2:1, ie an octave, and all is well in the fermament.

(3/2 x 4/3 = 12/6 or 2/1 so an octave).

If you look at the cycle of 4th / 5ths you should realise that you can derive the frequencies of all 12 notes simply be moving up or down a 4th/5th

If you do this using the Pythagorean ratios you encounter a number of problems.

1) If you got completely around the cycle of 5th (ie increase frequency by 3:2 12 times) you get back to the starting note, and should end up with a frequency ratio that is a power of 2 greater than the original frequency (ie it's a power of the octave 2:1 ratio). However you don't, you end up about 24 cents sharp, suggesting that there will be tuning problems with the Pythagorean system.

2) The thirds derived from Pythagoeran fiths sound dissonant. The major third is about 22 cents sharp and the minor third is flat.

The Pythagorean major third has a frequency ratio of 81/64; hardly a low integer ratio.

3) The semitones in the Pythagorean major scale are a different ratio to the semitones in the chromatic notes.

Interestingly early music predominantly used 4th and 5th as the other intervals were regarded as dissonant.

Several attempts have been made to try to make thirds consonant.

Just Intonation starts from adjusting the major triad to sound consonant.

If you do this frequency ratios end up as: 4:5:6 ie low integer ratios again.

This major third is 5:4 (so the 81/64 Pythogorean ratio is sharp of this); the fifth comforms to the Pythagorean 3:2.

However Just Intontion has a whole host of problems.

In the major scale one of the fifths is out of tune (D-A in the C major scale).

Even worse are the chromatic notes, as you end up with a situation where G# and Ab are different frequencies.

Meantone tuning was one attempt to try to resolve this issue, and flattening the 5ths slightly to get the major 3rd more in tune.

These days most intruments use an evenly tempered scale were the ratio between semitones is the 12th root of 2; this is still a compromise with the major 3rd still being the most out of tune note by about 14 cents.

Conclusions:

1) The Pythagorean observation that consonant intervals have low integer ratios seems to be correct.

2) It is impossible to tune an instrument where by all the intevals are the correct ratios for most consonant tuning.

3) All tuning is a compromise. It's impossible to make something better without making somthing else worse.

AS an aside steel guitarist often use a "sweetened" tuning (there some quite extensive and heated threads on steel forum about tuning....).

As an excercise I've tried tuning my steel by ear from a single pitch reference.

For A6th (top to bottom F# A C# E F# A C# E)

Tuning the top E to pitch, I then tune the octave E string and A strings to that E by ear by reducing beat notes.

I then tune the C# in the top A major triad to sound best.

Finally I tune the othe C# and F#s to the top C#.

Invariably when I test the accuarcy of the tuning on a tuner the C#s and F#s are flat of even temperement.

Sorry for the long post.........

viz

Nice one JFA.

I think quarter comma meantone is the best of all the tunings, so long as you stay in key. It's gorgeous. Needs a bit of judgement to get the first four fifths precisely the same amount of out-of-tune-ness, but after that every one of the major 3rds is just. It's awesome. 

guitarjack66

jpfamps said:

OK, I have to apologise here as I have got some of the details of this wrong.

I was posting from (fading!) memory and on reflection thought I'd not got it correct so I have consulted the relevent chapters in The Science of Sound.

Pythagoras did indeed make the observation that consonant intervals have low integer frequency ratios.

Octave 2:1

Perfect 5th: 3:2

Perfect 4th: 4:3

If you ascend by a Pythagorean fifth then a forth you acheive a ratio of 2:1, ie an octave, and all is well in the fermament.

(3/2 x 4/3 = 12/6 or 2/1 so an octave).

If you look at the cycle of 4th / 5ths you should realise that you can derive the frequencies of all 12 notes simply be moving up or down a 4th/5th

If you do this using the Pythagorean ratios you encounter a number of problems.

1) If you got completely around the cycle of 5th (ie increase frequency by 3:2 12 times) you get back to the starting note, and should end up with a frequency ratio that is a power of 2 greater than the original frequency (ie it's a power of the octave 2:1 ratio). However you don't, you end up about 24 cents sharp, suggesting that there will be tuning problems with the Pythagorean system.

2) The thirds derived from Pythagoeran fiths sound dissonant. The major third is about 22 cents sharp and the minor third is flat.

The Pythagorean major third has a frequency ratio of 81/64; hardly a low integer ratio.

3) The semitones in the Pythagorean major scale are a different ratio to the semitones in the chromatic notes.

Interestingly early music predominantly used 4th and 5th as the other intervals were regarded as dissonant.

Several attempts have been made to try to make thirds consonant.

Just Intonation starts from adjusting the major triad to sound consonant.

If you do this frequency ratios end up as: 4:5:6 ie low integer ratios again.

This major third is 5:4 (so the 81/64 Pythogorean ratio is sharp of this); the fifth comforms to the Pythagorean 3:2.

However Just Intontion has a whole host of problems.

In the major scale one of the fifths is out of tune (D-A in the C major scale).

Even worse are the chromatic notes, as you end up with a situation where G# and Ab are different frequencies.

Meantone tuning was one attempt to try to resolve this issue, and flattening the 5ths slightly to get the major 3rd more in tune.

These days most intruments use an evenly tempered scale were the ratio between semitones is the 12th root of 2; this is still a compromise with the major 3rd still being the most out of tune note by about 14 cents.

Conclusions:

1) The Pythagorean observation that consonant intervals have low integer ratios seems to be correct.

2) It is impossible to tune an instrument where by all the intevals are the correct ratios for most consonant tuning.

3) All tuning is a compromise. It's impossible to make something better without making somthing else worse.

AS an aside steel guitarist often use a "sweetened" tuning (there some quite extensive and heated threads on steel forum about tuning....).

As an excercise I've tried tuning my steel by ear from a single pitch reference.

For A6th (top to bottom F# A C# E F# A C# E)

Tuning the top E to pitch, I then tune the octave E string and A strings to that E by ear by reducing beat notes.

I then tune the C# in the top A major triad to sound best.

Finally I tune the othe C# and F#s to the top C#.

Invariably when I test the accuarcy of the tuning on a tuner the C#s and F#s are flat of even temperement.

Sorry for the long post.........

While I didn't understand a lot of that post it was still a fascinating read. If I can ask a stupid question though? For those many musicians who are tuning by ear (I cannot)  what exactly are they tuning to?

ToneControl

Some digital tuners offer sweetened tunings for guitar

Most digital pianos have a setting for just intonation if you select it from the arcane menus, definitely worth a little play with it to see the difference in sound.
If you have a piece in one key, there's no harm in recording it in just intonation if you feel like it.

Sometimes I detune my G string to better suit certain pieces

I have a guitar with something a bit like just intonation.
It's basically an optimised compromise to sound good with as many keys, as many chords as possible.
Frankly I think it is amazing, and I hardly touched my other guitars for a couple of years. I'd happily have it on all my guitars.
https://truetemperament.com/fretboards/

ToneControl

guitarjack66 said:

jpfamps said:

While I didn't understand a lot of that post it was still a fascinating read. If I can ask a stupid question though? For those many musicians who are tuning by ear (I cannot)  what exactly are they tuning to?

you use a digital tuner (or in the old days a tuning fork) to get one string on the guitar tuned, then tune the other strings to that one, by using intervals: fretted notes, harmonics, etc. There are many favourite ways.

I usually tune the top E, and then use the tuner to get the others about right, which saves time, then tune all strings directly to top E (If you tune each string to the next, you get cumulative errors):
First  I tune B 5th fret to the E, check E 7th fret to open B
G open to  3rd fret, the 9th fret on G is often way off E on some guitars
D 2nd fret to E
A open string with open E, listen for the beats around the perfect interval
bottom E open and 12th fret harmonic to top E

Then check again because guitars are made of wood, and change shape when you tighten up bits of wire tied to each end.

guitarjack66

Fascinating read is this whole thread.

viz

ToneControl said:

guitarjack66 said:

jpfamps said:

While I didn't understand a lot of that post it was still a fascinating read. If I can ask a stupid question though? For those many musicians who are tuning by ear (I cannot)  what exactly are they tuning to?

you use a digital tuner (or in the old days a tuning fork) to get one string on the guitar tuned, then tune the other strings to that one, by using intervals: fretted notes, harmonics, etc. There are many favourite ways.

I usually tune the top E, and then use the tuner to get the others about right, which saves time, then tune all strings directly to top E (If you tune each string to the next, you get cumulative errors):
First  I tune B 5th fret to the E, check E 7th fret to open B
G open to  3rd fret, the 9th fret on G is often way off E on some guitars
D 2nd fret to E
A open string with open E, listen for the beats around the perfect interval
bottom E open and 12th fret harmonic to top E

Then check again because guitars are made of wood, and change shape when you tighten up bits of wire tied to each end.

ICBM does that too, but to the G. I’ve moved to the G too now, it’s ace. 

ToneControl

viz said:

ToneControl said:

guitarjack66 said:

jpfamps said:

While I didn't understand a lot of that post it was still a fascinating read. If I can ask a stupid question though? For those many musicians who are tuning by ear (I cannot)  what exactly are they tuning to?

you use a digital tuner (or in the old days a tuning fork) to get one string on the guitar tuned, then tune the other strings to that one, by using intervals: fretted notes, harmonics, etc. There are many favourite ways.

I usually tune the top E, and then use the tuner to get the others about right, which saves time, then tune all strings directly to top E (If you tune each string to the next, you get cumulative errors):
First  I tune B 5th fret to the E, check E 7th fret to open B
G open to  3rd fret, the 9th fret on G is often way off E on some guitars
D 2nd fret to E
A open string with open E, listen for the beats around the perfect interval
bottom E open and 12th fret harmonic to top E

Then check again because guitars are made of wood, and change shape when you tighten up bits of wire tied to each end.

ICBM does that too, but to the G. I’ve moved to the G too now, it’s ace. 

what's better about it?
I trust the G less than the others, especially on a wound string on an acoustic

jpfamps

guitarjack66 said:

..

While I didn't understand a lot of that post it was still a fascinating read. If I can ask a stupid question though? For those many musicians who are tuning by ear (I cannot)  what exactly are they tuning to?

That's an interesting question.

Studies looking at stringed instruments (ie not fixed pitch) and vocal groups show that there is indeed a tendancy to deviate away from the evenly tempered scale.

Interestingly there is often a tendancy towards Phythagorean intervals, suggesting that 4th and 5th are very important intervals.

My guess is those people tuning by ear would gravitate to the tuning that sounds best to them in the musical context that they are operating.

jpfamps

I think there are 2 tuning issues with guitar that are accomodated by tuning adjustments.

1)

Whilst the guitar is nominally designed to be tuned to an evenly tempered scale, it doesn't work out in practice.

One solution is the True Temperament fretboard.

I've played a guitar with True Temperament fretboard and it does indeed sound very good. Bending strings can be interesting as on some frets the pitch of the bent string initial goes down before raising as you bend further.

So adjustments could help to get the guitar into a better approximation of the evenly temperscale.

2)

The evenly tempered scale is a compromise, and adjustments could also be made towards more perfect intervals, although this approach is likely to be more useful when playing in a particular keys.