Quick algorithm to find the simplest intervals that differ by a particular comma

This is a pretty quick and dirty algorithm to find the simplest two
intervals that differ by a particular comma. This is useful because,
for instance, if you're working within a tuning where that comma
vanishes, it's good to know which intervals are equated.

I'll leave the derivation of this as an exercise for the reader. This
actually yields two pairs of commas that are related.

1) Start with a comma - let's say 81/80.
2) Come up with two intervals - a/b and c/d - which hypothetically
differ by that comma. The goal here is to solve for a/b and c/d. Let's
say a/b is the larger interval.
3) If a/b and c/d differ by 81/80, then that means that (a/b) / (c/d)
= 81/80, and hence a/b * d/c = 81/80.
4) Thus ad = 81, and bc = 80. This is a slight algebraic
simplification, but we're looking for the simplest intervals anyway.
5) To find the simplest intervals that correspond to this property,
get the two factors that are closest in size for ad = 81 and bc = 80.
6) To get the two factors that are closest in size for 81, and also
for 80, find some integer factorizer that lists all fractions of these
numbers.
7) Take the square root of the number. For 81, the square root is 9.
If it works out like this nicely, you're done, because the two
fractions that are closest in size for 81 will be 9 and 9. Sweet.
8) For 80, the square root is 8.944. This doesn't work out. So in this
case, pick the closest factors on either side of this number - 8 and
10.
9) So back to our example, if ad = 81, the simplest answers for a and
d are a=9, d=9. If bc=80, the simplest answers for b and c are b=10,
c=10.
10) Plug these numbers back into the original fractions. So a/d and
b/c correspond to 9/8 and 10/9, which are the two simplest intervals
differing by 81/80 and hence equated in meantone.

Sometimes this works out to give you two pairs, rather than one. For
instance, if you're trying to find the simplest intervals equated in
porcupine -

- The porcupine comma is 250/243. So ad=250, bc=243.
- The simplest factors here are ad = 10, 25, and bc = 9, 27.
- Depending on how you assign these numbers, the two fractions this
yields are 10/9 and 27/25, or 25/9 and 27/10.
- Both are equated in porcupine temperament.

That's it. Very simple and straightforward. I have to run now, but
I'll be back later with a list of intervals equated for some common
temperaments.

-Mike

As promised, here it is:

16/15 - 4/3 and 5/4
25/24 - 5/4 and 6/5
27/25 - 9/5 and 5/3
81/80 - 9/8 and 10/9
128/125 - 8/5 and 25/16, or 16/5 and 25/8
250/243 - 10/9 and 27/25, or 25/9 and 27/10
256/243 - 16/9 and 27/16
648/625 - 27/25 and 25/24
2048/2025 - 64/45 and 45/32
3125/3072 - 125/64 and 48/25, or 25/64 and 48/125
15625/15552 - 125/108 and 144/125
20000/19683 - 125/81 and 243/160, or 160/81 and 243/125
32805/32768 - 135/128 and 256/243, or 243/128 and 256/135

If anyone has a decent list of 7-limit commas, feel free to post them
here and I'll work something out.

-Mike

On Thu, Feb 10, 2011 at 2:23 PM, Mike Battaglia <battaglia01@...> wrote:
> This is a pretty quick and dirty algorithm to find the simplest two
> intervals that differ by a particular comma. This is useful because,
> for instance, if you're working within a tuning where that comma
> vanishes, it's good to know which intervals are equated.
>
> I'll leave the derivation of this as an exercise for the reader. This
> actually yields two pairs of commas that are related.
>
> 1) Start with a comma - let's say 81/80.
> 2) Come up with two intervals - a/b and c/d - which hypothetically
> differ by that comma. The goal here is to solve for a/b and c/d. Let's
> say a/b is the larger interval.
> 3) If a/b and c/d differ by 81/80, then that means that (a/b) / (c/d)
> = 81/80, and hence a/b * d/c = 81/80.
> 4) Thus ad = 81, and bc = 80. This is a slight algebraic
> simplification, but we're looking for the simplest intervals anyway.
> 5) To find the simplest intervals that correspond to this property,
> get the two factors that are closest in size for ad = 81 and bc = 80.
> 6) To get the two factors that are closest in size for 81, and also
> for 80, find some integer factorizer that lists all fractions of these
> numbers.
> 7) Take the square root of the number. For 81, the square root is 9.
> If it works out like this nicely, you're done, because the two
> fractions that are closest in size for 81 will be 9 and 9. Sweet.
> 8) For 80, the square root is 8.944. This doesn't work out. So in this
> case, pick the closest factors on either side of this number - 8 and
> 10.
> 9) So back to our example, if ad = 81, the simplest answers for a and
> d are a=9, d=9. If bc=80, the simplest answers for b and c are b=10,
> c=10.
> 10) Plug these numbers back into the original fractions. So a/d and
> b/c correspond to 9/8 and 10/9, which are the two simplest intervals
> differing by 81/80 and hence equated in meantone.
>
> Sometimes this works out to give you two pairs, rather than one. For
> instance, if you're trying to find the simplest intervals equated in
> porcupine -
>
> - The porcupine comma is 250/243. So ad=250, bc=243.
> - The simplest factors here are ad = 10, 25, and bc = 9, 27.
> - Depending on how you assign these numbers, the two fractions this
> yields are 10/9 and 27/25, or 25/9 and 27/10.
> - Both are equated in porcupine temperament.
>
> That's it. Very simple and straightforward. I have to run now, but
> I'll be back later with a list of intervals equated for some common
> temperaments.
>
> -Mike
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> If anyone has a decent list of 7-limit commas, feel free to post them
> here and I'll work something out.

1029/1000, 250/243, 36/35, 525/512, 128/125, 49/48, 50/49,
3125/3072, 686/675, 64/63, 875/864, 81/80, 3125/3087, 2430/2401, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 15625/15552, 225/224, 19683/19600, 16875/16807, 10976/10935, 3136/3125, 5120/5103, 6144/6125, 65625/65536, 32805/32768, 703125/702464, 420175/419904, 2401/2400, 4375/4374, 250047/250000, 78125000/78121827

On Fri, Feb 11, 2011 at 3:07 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > If anyone has a decent list of 7-limit commas, feel free to post them
> > here and I'll work something out.
>
> 1029/1000, 250/243, 36/35, 525/512, 128/125, 49/48, 50/49,
> 3125/3072, 686/675, 64/63, 875/864, 81/80, 3125/3087, 2430/2401, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 15625/15552, 225/224, 19683/19600, 16875/16807, 10976/10935, 3136/3125, 5120/5103, 6144/6125, 65625/65536, 32805/32768, 703125/702464, 420175/419904, 2401/2400, 4375/4374, 250047/250000, 78125000/78121827

1029/1000 - 49/40 and 25/21, or 21/40 and 25/49
250/243 - 10/9 and 27/25, or 25/9 and 27/10
36/35 - 6/5 and 7/6
525/512 - 21/16 and 32/25, or 25/16 and 32/21
128/125 - 8/5 and 25/16, or 16/5 and 25/8
49/48 - 7/6 and 8/7
50/49 - 10/7 and 7/5
3125/3072 - 125/64 and 48/25, or 25/64 and 48/125
686/675 - 49/27 and 25/14, or 14/27 and 25/49
64/63 - 8/7 and 9/8
875/864 - 35/32 and 27/25, or 25/32 and 27/35
81/80 - 9/8 and 10/9
3125/3087 - 125/63 and 49/25, or 25/63 and 49/125
2430/2401 - 54/49 and 49/45
2048/2025 - 64/45 and 45/32
245/243 - 35/27 and 9/7, or 7/27 and 9/35
126/125 - 9/5 and 25/14, or 14/5 and 25/9
4000/3969 - 80/63 and 63/50
1728/1715 - 36/35 and 49/48, or 48/35 and 49/36
1029/1024 - 49/32 and 32/21
15625/15552 - 125/108 and 144/125
225/224 - 15/14 and 16/15
19683/19600 - 243/140 and 140/81
16875/16807 - 125/49 and 343/135, or 135/49 and 343/125
10976/10935 - 98/81 and 135/112, or 112/81 and 135/98
3136/3125 - 56/25 and 125/56
5120/5103 - 64/63 and 81/80, or 80/63 and 81/64
6144/6125 - 64/49 and 125/96, or 96/49 and 125/64
65625/65536 - 375/256 and 256/175
32805/32768 - 135/128 and 256/243, or 243/128 and 256/135
703125/702464 - 1125/896 and 784/625, or 625/896 and 784/1125
420175/419904 - 1225/648 and 648/343
2401/2400 - 49/48 and 50/49
4375/4374 - 125/81 and 54/35, or 35/81 and 54/125
250047/250000 - 567/500 and 500/441
78125000/78121827 - 12500/11907 and 6561/6250, or 6250/11907 and 6561/12500

Keep em comin!

-Mike