Yahoo Tuning Groups Ultimate Backup tuning ranking ratios by harmonicity and all that

First, some URLs of interest:

/tuning/topicId_1506.html#2897
http://tonalsoft.com/enc/g/gradus-suavitatis.aspx
http://www.mathematik.com/Piano/index.html
http://www.musikwissenschaft.uni-mainz.de/~ag/q/scale.pdf
http://home.datacomm.ch/straub/mamuth/mamufaq.html
and so on.

As part of a "JI for Dummies" project I'm mainly writing for myself, I'm making a big list of ratios sorted by the gradus suavitatis (GS) of each, from lowest to highest. (The GS of a number is the sum of its prime factors minus the number of prime factors plus one; the GS of a ratio, chord or scale is the GS of the least common multiple of the set of numbers involved. The idea and formula was devised by none other than Leonhard Euler.) I know I could use other measurements of harmonicity - I'm partial to Barlow's - and I'll probably take that into consideration.

Question: how would you "rank" ratios in terms of consonance, importance, and such?

I also realize that I really need to get a MIDI controller to this synth of mine and rig it so I can play 24-tone scales (ET, JI, meantone etc.), set up like Vyschnegradsky's piano.

~D.

🔗Carl Lumma <clumma@yahoo.com>

In fact there are dozens of very deep threads on this issue
on this list, the tuning-math list, and the harmonic_entropy
list, dating back at least to 1997.

> As part of a "JI for Dummies" project I'm mainly writing for
> myself, I'm making a big list of ratios sorted by the gradus
> suavitatis (GS) of each, from lowest to highest. (The GS of
> a number is the sum of its prime factors minus the number
> of prime factors plus one; the GS of a ratio, chord or scale
> is the GS of the least common multiple of the set of numbers
> involved.
> Question: how would you "rank" ratios in terms of consonance,
> importance, and such?

I'd listen to them. Have you tried listening to your list?
One approach is to listen keeping all the fundamentals at the
same pitch, rank them, and then do it again keep the upper
note of the ratio the same pitch.
Another method is just to listen to each ratio on several
roots before moving on to the next. This is usually easiest
to do by making an audio file, though if you prepare keyboard
mappings in advance it can be done interactively.

If you do some sort of listening test, please report back
on what you experience!

-Carl

🔗Carl Lumma <clumma@yahoo.com>

🔗Herman Miller <hmiller@IO.COM>

Danny Wier wrote:
> First, some URLs of interest:
> > /tuning/topicId_1506.html#2897
> http://tonalsoft.com/enc/g/gradus-suavitatis.aspx

> http://www.mathematik.com/Piano/index.html
The formula on this page gives some strange results: e.g., 9/8 has the same result as 5/4 (5), while on the other hand, 10/9 is up there with 17/16 with a score of 17.

> http://www.musikwissenschaft.uni-mainz.de/~ag/q/scale.pdf
This looks more reasonable, but 9/8 still has a better than expected score, the same as 6/5 (or better in the case of the Barlow disharmonicity measure). This isn't the sort of result I'd expect for a good consonance measure.

> http://home.datacomm.ch/straub/mamuth/mamufaq.html
> and so on.
> > As part of a "JI for Dummies" project I'm mainly writing for myself, I'm > making a big list of ratios sorted by the gradus suavitatis (GS) of each, > from lowest to highest. (The GS of a number is the sum of its prime factors > minus the number of prime factors plus one; the GS of a ratio, chord or > scale is the GS of the least common multiple of the set of numbers involved. > The idea and formula was devised by none other than Leonhard Euler.) I know > I could use other measurements of harmonicity - I'm partial to Barlow's - > and I'll probably take that into consideration.
> > Question: how would you "rank" ratios in terms of consonance, importance, > and such?

I'd look for a measure that puts 9/8 and 10/9 in the same general category with each other, being more dissonant than thirds but not the most dissonant intervals. As a rough guide, the sum of the numerator + denominator is pretty good for ratios of small integers. But any formula is going to have problems with intervals like 32/27, which is more consonant than might be expected (e.g., when compared with 9/8). Your ears are the best guide (especially with more than two notes being played at once).

🔗Robin Perry <jinto83@yahoo.com>

Hi Danny,

I've been pondering something around this issue for a while and
thought I'd toss it your way to see what you might think.

It's a relatively simple concept. Odd number harmonics are a
certain distance from 1/1 on a number line. In this system of
ranking, however, you can "jump" to distant numbers by not only 1's,
but by 2's. (and others) by following simple rules. (1.) You have
to jump upwards by doubling the current number, by adding one to it,
or by adding a factor of the number. So, from 2, you can jump to 4
or just add one to get to 3. From 3, you can go to 6 or 4. From 4,
you can go to 8, 5, or 6. (the jump from 4 to 6 is by a factor of
2). From 6, you can go to 12, 7, 8, or 9. The jump from 6 to 8 is
by a factor of 2. The jump to 9 is by a factor of 3.

The minimum number of steps it takes to get to a given harmonic is
it's harmonicity ranking. The higher the ranking, the lower the
harmonicity. That number of steps up plus the number of steps it
takes to get the harmonic back down to between 1/1 and 2/1 is the
ranking of the ratio.

For example: The 7 harmonic can be reached by jumping from 1 to 2
to 4 to (either 6 or 8) and then to 7, for a total of 4 steps. It
takes another 2 steps to arrive at 7/4 by halving 7 twice (3.5,
1.75). So the ranking of 7/4 is 6.

For subharmonics, it's basically the same except that they will
always have an added step to get up into the range between 1/1 and
2/1. Example: (1/5 subharmonic) Go from 1/1 down to ½, then to ¼,
then to 1/5, for a total of 3 steps. Multiply 1/5 times 2 cubed to
arrive at 8/5. The total number of steps is 6. This puts both 7/4
and 8/5 into the same ranking. So, a perfect fourth is not as
harmonic as a perfect fifth and so on.

Some common ratios and rankings based on this system:

2/1 1
3/2 3 (2,3,3/2)
4/3 4 (1/2,1/3,2/3,4/3)
5/4 5 (2,4,5,5/2,5/4)
8/5 6 (1/2,1/4,1/5,2/5,4/5,8/5)
6/5 6 (1/2,1/4,1/5,2/5,4/5,6/5)
5/3 5 (1/2,1/3,2/3,4/3,5/3)
7/4 6 (2,4,6,7,7/2,7/4)
8/7 7 (1/2,1/4,1/6,1/7,2/7,4/7,8/7)
9/8 7 (2,4,8,9,9/2,9/4,9/8)
16/9 8 (1/2,1/4,1/8,1/9,2/9,4/9,8/9,16/9)
7/6 7 (1/2,1/4,1/6,2/6,4/6,6/6,7/6)
12/7 8 (1/2,1/4,1/6,1/7,2/7,4/7,8/7,12/7)
10/9 8 (1/2,1/3,1/6,1/9,2/9,4/9,5/9,10/9)
9/5 7 (1/2,1/4,1/5,2/5,4/5,8/5,9/5)
7/5 7 (1/2,1/4,1/5,2/5,4/5,6/5,7/5)
10/7 8 (1/2,1/4,1/6,1/7,2/7,4/7,5/7,10/7)

This goes on and on, of course. A simple scale based on all the
harmonic and subharmonic pairs up through level 6 would look like:

1/1, 6/5, 5/4,4/3,3/2,8/5,5/3,7/4,2/1.

I'm interested in hearing what you might have to say about this
idea especially if you listen to the pairs and can tell me if you
agree, or not, that those on the same `level' are equally consonant
or dissonant (as the case may be).

Regards,

Robin Perry

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>
> First, some URLs of interest:
>
> /tuning/topicId_1506.html#2897
> http://tonalsoft.com/enc/g/gradus-suavitatis.aspx
> http://www.mathematik.com/Piano/index.html
> http://www.musikwissenschaft.uni-mainz.de/~ag/q/scale.pdf
> http://home.datacomm.ch/straub/mamuth/mamufaq.html
> and so on.
>
> As part of a "JI for Dummies" project I'm mainly writing for
myself, I'm
> making a big list of ratios sorted by the gradus suavitatis (GS)
of each,
> from lowest to highest. (The GS of a number is the sum of its
prime factors
> minus the number of prime factors plus one; the GS of a ratio,
chord or
> scale is the GS of the least common multiple of the set of numbers
involved.
> The idea and formula was devised by none other than Leonhard
Euler.) I know
> I could use other measurements of harmonicity - I'm partial to
Barlow's -
> and I'll probably take that into consideration.
>
> Question: how would you "rank" ratios in terms of consonance,
importance,
> and such?
>
> I also realize that I really need to get a MIDI controller to this
synth of
> mine and rig it so I can play 24-tone scales (ET, JI, meantone
etc.), set up
> like Vyschnegradsky's piano.
>
> ~D.
>

🔗Danny Wier <dawiertx@sbcglobal.net>

----- Original Message ----- From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, August 23, 2007 3:08 PM
Subject: [tuning] Re: ranking ratios by harmonicity and all that

>> http://www.musikwissenschaft.uni-mainz.de/~ag/q/scale.pdf
>
> I've only skimmed this, but it's interesting. Good find.

I'll try get to the other replies to my OP soon.

In this PDF document, it gives a slightly different definition for gradus suavitatis: it doesn't add one to the sum of prime factors minus the number of prime factors. So the GS of 1/1 and 2/1 is given as 0 and 1 respectively, not 1 and 2. (Barlow disharmonicity grades the same intervals as 0 and 1.)

I still need to finish reading it myself.

~D.

🔗Danny Wier <dawiertx@sbcglobal.net>

From: "Carl Lumma" <clumma@...>

> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:
>>
>> First, some URLs of interest:
>>
>> /tuning/topicId_1506.html#2897
>> http://tonalsoft.com/enc/g/gradus-suavitatis.aspx
>> http://www.mathematik.com/Piano/index.html
>> http://www.musikwissenschaft.uni-mainz.de/~ag/q/scale.pdf
>> http://home.datacomm.ch/straub/mamuth/mamufaq.html
>> and so on.
>
> In fact there are dozens of very deep threads on this issue
> on this list, the tuning-math list, and the harmonic_entropy
> list, dating back at least to 1997.

I forgot that list existed... I need to rejoin.

>> As part of a "JI for Dummies" project I'm mainly writing for
>> myself, I'm making a big list of ratios sorted by the gradus
>> suavitatis (GS) of each, from lowest to highest. (The GS of
>> a number is the sum of its prime factors minus the number
>> of prime factors plus one; the GS of a ratio, chord or scale
>> is the GS of the least common multiple of the set of numbers
>> involved.
>> Question: how would you "rank" ratios in terms of consonance,
>> importance, and such?
>
> I'd listen to them. Have you tried listening to your list?
> One approach is to listen keeping all the fundamentals at the
> same pitch, rank them, and then do it again keep the upper
> note of the ratio the same pitch.
> Another method is just to listen to each ratio on several
> roots before moving on to the next. This is usually easiest
> to do by making an audio file, though if you prepare keyboard
> mappings in advance it can be done interactively.

I still want to write some sort of progressive ear training program, even if it's in old-fashioned BASIC. (I never got around to learning Java like I wanted to.) I do have Scala and FTS, and I made a bunch of mp3 files of intervals to play randomly on my portable mp3 player (a poor man's iPod, basically).

And my Roland synth does have a scale tuning function, so I can toy with that.

~D.

🔗Danny Wier <dawiertx@sbcglobal.net>

From: "Herman Miller" <hmiller@...>

> Danny Wier wrote:
>> First, some URLs of interest:
>>
>> /tuning/topicId_1506.html#2897
>> http://tonalsoft.com/enc/g/gradus-suavitatis.aspx
>
>> http://www.mathematik.com/Piano/index.html
> The formula on this page gives some strange results: e.g., 9/8 has the
> same result as 5/4 (5), while on the other hand, 10/9 is up there with
> 17/16 with a score of 17.

Oh, there's an error in the script then. I tried 128/117 and got a GS of 45, but the prime factorization of 128 is 2^7, and for 117, 3^2 * 13; 2*7 + 3*2 + 13 - 10 + 1 = 24. Scala also gave me 24.

>> http://www.musikwissenschaft.uni-mainz.de/~ag/q/scale.pdf
> This looks more reasonable, but 9/8 still has a better than expected
> score, the same as 6/5 (or better in the case of the Barlow
> disharmonicity measure). This isn't the sort of result I'd expect for a
> good consonance measure.

As an afterthought, I think I prefer Euler's simpler formula to Barlow's: the curve of weighting of increasing primes is a bit steep.

> I'd look for a measure that puts 9/8 and 10/9 in the same general
> category with each other, being more dissonant than thirds but not the
> most dissonant intervals. As a rough guide, the sum of the numerator +
> denominator is pretty good for ratios of small integers. But any formula
> is going to have problems with intervals like 32/27, which is more
> consonant than might be expected (e.g., when compared with 9/8). Your
> ears are the best guide (especially with more than two notes being
> played at once).

What I'm working on is putting a set of ratios to be learned in sequence from simplest to most complex, but I wanted to rank 32/27 before 13/11, for example, so one could be familiar with more complex ratios with a lower highest prime before introducing higher primes.

Though 10/9, with a GS of 10, comes after 9/8, a grade 8, 5/4 and 5/3 are both grade 7.

Here is the list of all ratios of the cross products of harmonics and subharmonics 1-16 of at least 1/1, ordered by GS, in no particular order after that:

1: 1/1
2: 2/1
3: 3/1 4/1
4: 3/2 6/1 8/1
5: 4/3 16/1 5/1 9/1 12/1
6: 10/1 5/2 8/3 9/2
7: 16/3 7/1 5/4 5/3 9/4 15/1
8: 9/8 6/5 7/2 14/1 8/5 10/3 15/2
9: 7/4 16/9 9/5 7/3 12/5 16/5 15/4
10: 10/9 7/6 8/7 15/8 14/3
11: 9/7 7/5 12/7 16/7 16/15 11/1
12: 10/7 14/9 14/5 11/2
13: 15/7 11/4 11/3 13/1
14: 15/14 11/8 11/6 13/2
15: 16/11 11/5 12/11 13/4 13/3 11/9
16: 11/10 13/8 13/6
17: 13/12 16/13 15/11 13/9 11/7 13/5
18: 13/10 14/11
19: 15/13 13/7
20: 14/13
23: 13/11

It's a rough system right now, obviously. I still haven't figured out how I want to order ratios with the same GS, other than putting those with the lowest highest primes before the others in each group.

~D.