# Pythagorean-esque question about tunings



## Argus

Okay, so a series of 12 just perfect fifths up from a pitch and 7 octaves up from the same pitch are seperated by a Pythagorean comma. These intervals, when brought into the range of an octave, give us 12 slightly unequal tones per octave.

My question is is what happens if you were to use just major thirds rather than perfect fifths. I know a series of 3 major thirds gives us the interval 125/64 which is seperated from the octave by a diesis (128/125, 41 cents). However if you were to keep stacking thirds what comma would you reach that is closer than the diesis, and how many major thirds would be used in the series to get there? Then when all the intervals from these thirds are brought into the span of an octave, how many would there be per octave?

I'm mainly interested in how many tones would be in the scale. I've worked out the first 12 or so and it looks like somewhere in the region of 30-36 because there are still a lot of gaps.


----------



## matsoljare

You would only get further from the octave anyway, but the answer to your second question would be 31 equal, as it has a major third very close to the just 5/4.


----------



## Argus

matsoljare said:


> You would only get further from the octave anyway, but the answer to your second question would be 31 equal, as it has a major third very close to the just 5/4.


Cheers.

I know you'd get further from the octave but I meant if you brought everything down into the span of one octave. So after the 125/64 you'd transpose the 625/256 down to 625/512.

I'd worked it out up until the interval 48828125/33554432 but then got tired of such annoyingly large ratios. I guessed it'd be around 30 steps because I thought you'd get a tone closer to the 25/16 before the 5/4.


----------



## matsoljare

The comma would be 4722366482869645213696/4656612873077392578125, to be exact. That's 2^72 / 3^31, to put it another way!


----------



## Argus

matsoljare said:


> The comma would be 4722366482869645213696/4656612873077392578125, to be exact. That's 2^72 / 3^31, to put it another way!


Is that the comma closest to the 1/1, the 5/4 or the 21/16?

Not that it really matters, those kind of ratios are too large/precise to be trying to use. 31 TET is a lot simpler.


----------



## Philip

actually the closest after 3 thirds is 28. then 59, 146, 643, further it overflows... the ratios are:

0, 1.0 = (5/4)^0 / 2^0
3, 0.976562500000000 = (5/4)^3 / 2^1
28, 1.00974195868290 = (5/4)^28 / 2^9
59, 0.995682444457783 = (5/4)^59 / 2^19
146, 0.998959536101118 = (5/4)^146 / 2^47
643, 0.999837132392563 = (5/4)^643 / 2^207
...


----------



## Argus

Philip said:


> actually the closest after 3 thirds is 28. then 59, 146, 643, further it overflows... the ratios are:
> 
> 0, 1.0 = (5/4)^0 / 2^0
> 3, 0.976562500000000 = (5/4)^3 / 2^1
> 28, 1.00974195868290 = (5/4)^28 / 2^9
> 59, 0.995682444457783 = (5/4)^59 / 2^19
> 146, 0.998959536101118 = (5/4)^146 / 2^47
> 643, 0.999837132392563 = (5/4)^643 / 2^207
> ...


If I read your statistics right they are all showing commas close to the unison. So 28 thirds would give you an interval with a smaller comma in relation to the 1/1? Is there any commas between the other intervals before the comma close to the 1/1 is reached? i.e. commas close to the 5/4 or the 25/16.

What I had started to do is map all these intervals into the single octave to use them as scale steps. After the first 12 or so it appears they should all be (fairly) equally spaced up until an interval that is very close (a small comma) to the 25/16, then after that you get the coomas to the 5/4 and 1/1 respectively until you reach the second set of commas (after the 59 thirds you say). So descending from the octave 2/1, you have 125/64, 15625/8192, 1953125/1048576 etc until you reach the interval close to the 25/16 (which I've yet to work out because the ratios weretoo large to work with).

I'm not good at maths at all but I've managed to create scales like this using two different intervals (3/2 & 5/4, 3/2 & 7/4 etc) but these arranged themselves into neater series of intervals. I'd also started this same scale idea with 7/4's which seems to have slightly less tones per octave before the comma is reached but it could be the same.


----------



## Philip

Argus said:


> If I read your statistics right they are all showing commas close to the unison. So 28 thirds would give you an interval with a smaller comma in relation to the 1/1? Is there any commas between the other intervals before the comma close to the 1/1 is reached? i.e. commas close to the 5/4 or the 21/16.
> 
> What I had started to do is map all these intervals into the single octave to use them as scale steps. After the first 12 or so it appears they should all be (fairly) equally spaced up until an interval that is very close (a small comma) to the 21/16, then after that you get the coomas to the 5/4 and 1/1 respectively until you reach the second set of commas (after the 59 thirds you say). So descending from the octave 2/1, you have 125/64, 15625/8192, 1953125/1048576 etc until you reach the interval close to the 21/16 (which I've yet to work out because the ratios weretoo large to work with).
> 
> I'm not good at maths at all but I've managed to create scales like this using two different intervals (3/2 & 5/4, 3/2 & 7/4 etc) but these arranged themselves into neater series of intervals. I'd also started this same scale idea with 7/4's which seems to have slightly less tones per octave before the comma is reached but it could be the same.


not exactly. for example, the ratios should be read:

(1.) 3, 0.976562500000000 = (5/4)^3 / 2^1
(2.) 28, 1.00974195868290 = (5/4)^28 / 2^9
(3.) 59, 0.995682444457783 = (5/4)^59 / 2^19

(1.) "three major thirds yield an error of exactly 2.4% below the first octave."
(2.) "28 major thirds yield an error of less than 1% above the ninth octave."
(3.) "59 major thirds yield an error of less than 0.5% below the 19th octave." etc.

when you scale down the nine octave span of 28 thirds into a single octave, the intervals are:

frequency ratio, interval from previous
1.0
1.03397576569128 1.03397576569128
1.05879118406788 1.024
1.08420217248550 1.024	
1.11022302462516 1.024	
1.13686837721616 1.024	
1.16415321826935 1.024	
1.19209289550781 1.024	
1.22070312500000 1.024 
1.25000000000000 1.024	
1.29246970711411 1.03397576569128
1.32348898008484 1.024	
1.35525271560688 1.024 
1.38777878078145 1.024	
1.42108547152020 1.024
1.45519152283669 1.024
1.49011611938477 1.024
1.52587890625000 1.024
1.56250000000000 1.024	
1.61558713389263 1.03397576569128 (*)
1.65436122510606 1.024	
1.69406589450860 1.024	
1.73472347597681 1.024	
1.77635683940025 1.024	
1.81898940354586 1.024 
1.86264514923096 1.024
1.90734863281250 1.024
1.95312500000000 1.024
2.0 1.024 -> should be 1.03397576569128, but truncated to "wolf" major third from (*)

intervals:

128 / 125 = 1.024
(5/4)^25 / 256 = 298023223876953125/288230376151711744 = 1.03397576569128...


----------



## Philip

if you can compose music in this 28 note system, i would be interested in hearing it. but as with the common 12 note equal temperament, perhaps you could use a 28th root of 2 (1.02506421196587) interval all around.


----------



## Argus

Philip said:


> if you can compose music in this 28 note system, i would be interested in hearing it. but as with the common 12 note equal temperament, perhaps you could use a 28th root of 2 (1.02506421196587) interval all around.


Thanks, that's cleared things up. Also, I used 21/16 before when I meant 25/16.

So apart from the intervals directly after the 1/1, 5/4 and 25/16, the scale is basically made up of intervals seperated by the diesis. Tempering it out to 28 TET makes sense because I doubt the ear would be able to differentiate between the two kinds of just interval in actual music.

I'll have to have a mess around in it but I'm not sure I'll be able to work with it yet. I got the idea for building scales like this from La Monte Young. He uses a scale built from intervals built from a 3/2 and a 7/4 and further extensions of these two ratios. Anyway, you end up with clumps of tones around the 1/1, 9/8, 21/16, 3/2 and 7/4, which gives it a nice simple pentatonic sound but with microtonal inflections and varying beating partials possible. These scales with clusters of close tones seem to be most usable for me because you can be as simple or complex as you like. This being a very even scale may mean I 'miss out' certain steps and in effect use an uneven scale of less than 28 tones built from it.

By the way, what formula did you use to work out all the mathematical gubbins?


----------



## Philip

Argus said:


> By the way, what formula did you use to work out all the mathematical gubbins?


(5/4)^a = 2^b ; a = 0,1,2...

if you solve for b and round it, 2^round(b) is the nearest octave, eg.

5/4^28 = 2^b => b = 28*ln(5/4)/ln(2)

b = 9.01398665684614 rounded to 9

thus ratio of interval to nearest octave is:

(5/4)^28 / 2^9 = 1.00974195868289 (or less than 1% above)


----------



## Philip

so... if (5/4)^a = 2^b then

(5/4)^a / 2^floor(b) ; a=0,1,2,...,27 sorted are the interval ratios between (1...2)


----------



## matsoljare

I'd say just use 31-equal instead, there's really no benefits to 28 that i can think of....


----------



## Philip

matsoljare said:


> I'd say just use 31-equal instead, there's really no benefits to 28 that i can think of....


You're right because in 31-TET the major third is +0.8 cents vs 28-TET at -0.6 cents... which are both very close. Although 28 non-TET does yield absolutely pure major thirds and different color tonalities.


----------



## pianozach

Yes, but it sounds like this.

A playlist of four videos comparing *Mean, Pythagorean*, and *Equal* tunings

https://www.youtube.com/playlist?list=PL-0jRwu33pSid5VUnw_3SLy_jwawmZGLJ 

Here's just *Pythagorean* tuning.


----------



## adrien

I understand the 5th has a basis in physics and natural resonances of vibrating objects like strings or air-columns in pipes, that being the vibration mode of 1/3 of the string and then an octave down (so 3:2). I don't know of such a thing for a 3rd, and so to stack 3rds where does the 3rd come from? Is it then arbitrary, or is it a 3rd derived from stacking 5ths? I guess I fail to see the purpose.


----------

