An Alternative Method for Building Scales

An Alternative Method for Building Scales

Part I: Melody in Scales

No commas here, no generators, no circles or chains of intervals.

After 14 years of number crunching I arrived at two likely formulas for quantifying the strength/sweetness of melodic and harmonic intervals using sine wave tones (no overtones) only and not tones with a regular harmonic series.

Starting with Harmony I got...

(2 + 1/x - 1/y - diss(x,y) )/2

The diss(x,y) is a function.
If y/x <= 0.9375 the diss(x,y) is simply y/x
If y/x > 0.9375 the diss(x,y) is (1 - y/x)*15
x should be greater than y and both should be less than 256. Beyond 256 you get some intervals that the formula says are bad but some of these are within 6.776 cents (256/255) of a *good* interval and by virtue of this are in themselves also good.

Testing the above formula it was consistent with sine wave tones but not consistent with complex tones. Intervals that should have been stronger sounded weaker and vice versa. So I wrote a program using the formula above that paired the first 32 partials (harmonics) of each note in an interval with the first 1024 partials of both notes avoiding doing any pairing twice. Why not pair *all* 1024 partials of both notes? I'm not sure but it doesn't work. Perhaps it's because as you progress along the harmonic series the 1/x + 1/y part of the formula approaches zero (e.g. 1/256 + 1/512 = 0.0059) but the dissonance value does not diminish significantly (e.g. 256/512 = 0.5).

Right or wrong, the program I wrote (called IntCalc7) seems consistent when testing harmony intervals whose notes have a regular harmonic series/timbre (i.e. the frequencies of the partials are approximately x, 2x, 3x, 4x etc and the amplitudes of the partials are approximately y, y/2, y/3, y/4 etc).

What about melody? The formula I worked out was 2/x + 2/y. Why not
1/x + 1/y? This is purely cosmetic: using 2/x + 2/y any melodic interval with a value of 1.0 or higher sounds like a Major progression and any melodic interval with a value less than 1.0 sounds like a Minor progression.

I had a suspicion that the 2/x + 2/y formula might be good for complex tones as well as sine wave tones and indeed it is. I wrote two programs that used the 2/x + 2/y formula. The first program paired the first 32 partials of each note with the other 1024 partials and the second paired *all* of the first 1024 partials of both notes. The results were different in magnitude but in all three cases the proportions were pretty much exactly the same. In other words the 2/x + 2/y formula for melody is equally good for both sine wave tones *and* complex tones with a `regular' timbre.

For me this is relevant to scale construction. Say you want a scale exactly one octave wide. The tonic is 1/1 and the octave is 2/1. Now you select a few notes (I stick with eleven but you can choose any number) in between 1/1 and 2/1 so as to build a scale. It seems to me that (if 1/1 is the tonic), every note in the scale should "go" with 1/1 according to a *good* melodic interval within 6.776 cents (256/255) accuracy. What are the good notes that *go* with 1/1? Here they are in order of strength (use fixed width font)...

Interval, Strength, Width
1/1 4.000 0.0000c
2/1 3.000 1200.0000c
3/2 1.6667 701.9550c
4/3 1.1667 498.0450c
5/3 1.0667 884.3587
5/4 0.9000 386.3137
7/4 0.7857 968.8259
6/5 0.7333 315.6413
7/5 0.6857 582.5122
8/5 0.6500 813.6863
9/5 0.6222 1017.5963
7/6 0.6190 266.8709
8/7 0.5357 231.1741
11/6 0.5152 1049.3629
9/7 0.5079 435.0841
10/7 0.4857 617.4878
9/8 0.4722 203.9100
11/7 0.4675 782.4920
12/7 0.4524 933.1291
13/7 0.4396 1071.7018
11/8 0.4318 551.3179
10/9 0.4222 182.4037
11/9 0.4040 347.4079
13/8 0.4038 840.5277
15/8 0.3833 1088.2687
11/10 0.3818 165.0042
13/9 0.3761 636.6177
14/9 0.3651 764.9159
13/10 0.3538 454.2140
12/11 0.3485 150.6371
16/9 0.3472 996.0900
17/9 0.3399 1101.0454
13/11 0.3357 289.2097
14/11 0.3247 417.5080
13/12 0.3205 138.5727
17/10 0.3176 918.6417
15/11 0.3152 536.9508
16/11 0.3068 648.6821
19/10 0.3053 1111.1993
17/11 0.2995 753.6375
14/13 0.2967 128.2982
18/11 0.2929 852.5921
15/13 0.2872 247.7411
19/11 0.2871 946.1951
17/12 0.2843 603.0004
20/11 0.2818 1034.9958
16/13 0.2788 359.4723
21/11 0.2771 1119.4630
15/14 0.2762 119.4428
19/12 0.2719 795.5580
17/13 0.2715 464.4278
18/13 0.2650 563.3823
17/14 0.2605 336.1295
19/13 0.2591 656.9854
16/15 0.2583 111.7313
20/13 0.2538 745.7861
23/12 0.2536 1126.3193
17/15 0.2510 216.6867
--------------------------------
17/16 0.2426 104.9554

Anyone want to test these melodic intervals against their values?

I propose 5 categories of Melodic intervals...

(i) Value 1.0 or higher is Major (sweet).
(ii) Value between 0.5 and 0.9999 is a Blue Minor (sweet).
(iii) Value between 0.25 and 0.4999 is an Ultra Minor (sweet).
(iv) Value between 0.125 and 0.2499 is "tolerable" but not sweet.
(v) Value less than 0.125 is "intolerable".

So if all the notes in a scale are within 6.776 cents (256/255) of any of the notes listed above (except the last note,17/16) then the scale is "tonally good". If a note occurs that isn't within 6.776 cents of any of the notes listed above then the scale is "tonally bad".

The last note in the list above, 17/16 has a value less than 0.25 but it is within 6.776 cents accuracy of 16/15. In fact it is right on the cusp. The distance between 16/15 and 17/16 is exactly my old friend 6.775877 cents (256/255). Interesting coincidence.

I have one more stipulation for a good scale. Most "tonally good" scales will contain other melodic intervals (that don't contain the tonic, 1/1) that are not "sweet (i) to (iii)" but are "tolerable (iv)" and/or "intolerable (v)". As long as all the notes that go with the tonic are "sweet" it doesn't seem to matter much if a few other intervals in the scale that don't contain the tonic, 1/1, are only "tolerable". The narrowest "true" interval that is "tolerable" is 32/31 with a value of 0.127 (just above the cutoff point of 0.125). Any interval within 6.776 cents of 32/31 should be tolerable.

32/31 is 54.964428 cents. Going down by 6.775877 cents we get 48.18855 cents. So any interval narrower than 48.18855 cents is "intolerable". So a scale that contains an interval narrower than 48.18855 cents is "intolerable". I know some will argue that if you don't play a pair of notes that are less than 48.18855 cents apart consecutively then you can "get away with it" but I disagree.

End of Part I. Part II will deal with Harmony in Scales.

John O'Sullivan

http://www.johnsmusic7.com

   The following intervals I found considerably more dissonant than their listed positions and  definitely in the "intolerable" range (most of them turn out to be toward the higher ends of 11 and 13 limit).

13/8
13/10
20/11
16/11
19/11
17/11
16/13

  Again, you put 13/9 as "ultra-minor"...which seems to hint to me that it could be used as a decent alternative type of fifth.  Even though John I recall you said the only valid "good"/consonant type of fifth is 3/2.
  Interestingly enough I also find 14/9 considerably more consonant than 13/9, again at odds with your list.

   For the most part, though, your list seems pretty accurate.

--- On Thu, 2/24/11, john777music
<jfos777@...> wrote:

From: john777music <jfos777@...>
Subject: [tuning] An Alternative Method for Building Scales
To: tuning@yahoogroups.com
Date: Thursday, February 24, 2011, 12:47 PM

An Alternative Method for Building Scales

Part I: Melody in Scales

No commas here, no generators, no circles or chains of intervals.

After 14 years of number crunching I arrived at two likely formulas for quantifying the strength/sweetness of melodic and harmonic intervals using sine wave tones (no overtones) only and not tones with a regular harmonic series.

Starting with Harmony I got...

(2 + 1/x - 1/y - diss(x,y) )/2

The diss(x,y) is a function.

If y/x <= 0.9375 the diss(x,y) is simply y/x

If y/x > 0.9375 the diss(x,y) is (1 - y/x)*15

x should be greater than y and both should be less than 256. Beyond 256 you get some intervals that the formula says are bad but some of these are within 6.776 cents (256/255) of a *good* interval and by virtue of this are in themselves also good.

Testing the above formula it was consistent with sine wave tones but not consistent with complex tones. Intervals that should have been stronger sounded weaker and vice versa. So I wrote a program using the formula above that paired the first 32 partials (harmonics) of each note in an interval with the first 1024 partials of both notes avoiding doing any pairing twice. Why not pair *all* 1024 partials of both notes? I'm not sure but it doesn't work. Perhaps it's because as you progress along the harmonic series the 1/x + 1/y part of the formula approaches zero (e.g. 1/256 + 1/512 = 0.0059) but the dissonance value does not diminish significantly (e.g. 256/512 = 0.5).

Right or wrong, the program I wrote (called IntCalc7) seems consistent when testing harmony intervals whose notes have a regular harmonic series/timbre (i.e. the frequencies of the partials are approximately x, 2x, 3x, 4x etc and the amplitudes of the partials are approximately y, y/2, y/3, y/4 etc).

What about melody? The formula I worked out was 2/x + 2/y. Why not

1/x + 1/y? This is purely cosmetic: using 2/x + 2/y any melodic interval with a value of 1.0 or higher sounds like a Major progression and any melodic interval with a value less than 1.0 sounds like a Minor progression.

I had a suspicion that the 2/x + 2/y formula might be good for complex tones as well as sine wave tones and indeed it is. I wrote two programs that used the 2/x + 2/y formula. The first program paired the first 32 partials of each note with the other 1024 partials and the second paired *all* of the first 1024 partials of both notes. The results were different in magnitude but in all three cases the proportions were pretty much exactly the same. In other words the 2/x + 2/y formula for melody is equally good for both sine wave tones *and* complex tones with a `regular' timbre.

For me this is relevant to scale construction. Say you want a scale exactly one octave wide. The tonic is 1/1 and the octave is 2/1. Now you select a few notes (I stick with eleven but you can choose any number) in between 1/1 and 2/1 so as to build a scale. It seems to me that (if 1/1 is the tonic), every note in the scale should "go" with 1/1 according to a *good* melodic interval within 6.776 cents (256/255) accuracy. What are the good notes that *go* with 1/1? Here they are in order of strength (use fixed width font)...

Interval, Strength, Width

1/1 4.000 0.0000c

2/1 3.000 1200.0000c

3/2 1.6667 701.9550c

4/3 1.1667 498.0450c

5/3 1.0667 884.3587

5/4 0.9000 386.3137

7/4 0.7857 968.8259

6/5 0.7333 315.6413

7/5 0.6857 582.5122

8/5 0.6500 813.6863

9/5 0.6222 1017.5963

7/6 0.6190 266.8709

8/7 0.5357 231.1741

11/6 0.5152 1049.3629

9/7 0.5079 435.0841

10/7 0.4857 617.4878

9/8 0.4722 203.9100

11/7 0.4675 782.4920

12/7 0.4524 933.1291

13/7 0.4396 1071.7018

11/8 0.4318 551.3179

10/9 0.4222 182.4037

11/9 0.4040 347.4079

13/8 0.4038 840.5277

15/8 0.3833 1088.2687

11/10 0.3818 165.0042

13/9 0.3761 636.6177

14/9 0.3651 764.9159

13/10 0.3538 454.2140

12/11 0.3485 150.6371

16/9 0.3472 996.0900

17/9 0.3399 1101.0454

13/11 0.3357 289.2097

14/11 0.3247 417.5080

13/12 0.3205 138.5727

17/10 0.3176 918.6417

15/11 0.3152 536.9508

16/11 0.3068 648.6821

19/10 0.3053 1111.1993

17/11 0.2995 753.6375

14/13 0.2967 128.2982

18/11 0.2929 852.5921

15/13 0.2872 247.7411

19/11 0.2871 946.1951

17/12 0.2843 603.0004

20/11 0.2818 1034.9958

16/13 0.2788 359.4723

21/11 0.2771 1119.4630

15/14 0.2762 119.4428

19/12 0.2719 795.5580

17/13 0.2715 464.4278

18/13 0.2650 563.3823

17/14 0.2605 336.1295

19/13 0.2591 656.9854

16/15 0.2583 111.7313

20/13 0.2538 745.7861

23/12 0.2536 1126.3193

17/15 0.2510 216.6867

--------------------------------

17/16 0.2426 104.9554

Anyone want to test these melodic intervals against their values?

I propose 5 categories of Melodic intervals...

(i) Value 1.0 or higher is Major (sweet).

(ii) Value between 0.5 and 0.9999 is a Blue Minor (sweet).

(iii) Value between 0.25 and 0.4999 is an Ultra Minor (sweet).

(iv) Value between 0.125 and 0.2499 is "tolerable" but not sweet.

(v) Value less than 0.125 is "intolerable".

So if all the notes in a scale are within 6.776 cents (256/255) of any of the notes listed above (except the last note,17/16) then the scale is "tonally good". If a note occurs that isn't within 6.776 cents of any of the notes listed above then the scale is "tonally bad".

The last note in the list above, 17/16 has a value less than 0.25 but it is within 6.776 cents accuracy of 16/15. In fact it is right on the cusp. The distance between 16/15 and 17/16 is exactly my old friend 6.775877 cents (256/255). Interesting coincidence.

I have one more stipulation for a good scale. Most "tonally good" scales will contain other melodic intervals (that don't contain the tonic, 1/1) that are not "sweet (i) to (iii)" but are "tolerable (iv)" and/or "intolerable (v)". As long as all the notes that go with the tonic are "sweet" it doesn't seem to matter much if a few other intervals in the scale that don't contain the tonic, 1/1, are only "tolerable". The narrowest "true" interval that is "tolerable" is 32/31 with a value of 0.127 (just above the cutoff point of 0.125). Any interval within 6.776 cents of 32/31 should be tolerable.

32/31 is 54.964428 cents. Going down by 6.775877 cents we get 48.18855 cents. So any interval narrower than 48.18855 cents is "intolerable". So a scale that contains an interval narrower than 48.18855 cents is "intolerable". I know some will argue that if you don't play a pair of notes that are less than 48.18855 cents apart consecutively then you can "get away with it" but I disagree.

End of Part I. Part II will deal with Harmony in Scales.

John O'Sullivan

http://www.johnsmusic7.com

Michael,

these are *melodic* intervals which have different values to their harmony counterparts.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
>    The following intervals I found considerably more dissonant than their listed positions and  definitely in the "intolerable" range (most of them turn out to be toward the higher ends of 11 and 13 limit).
>
> 13/8
> 13/10
> 20/11
> 16/11
> 19/11
> 17/11
> 16/13
>
>   Again, you put 13/9 as "ultra-minor"...which seems to hint to me that it could be used as a decent alternative type of fifth.  Even though John I recall you said the only valid "good"/consonant type of fifth is 3/2.
>   Interestingly enough I also find 14/9 considerably more consonant than 13/9, again at odds with your list.
>
>
>    For the most part, though, your list seems pretty accurate.
>
> --- On Thu, 2/24/11, john777music
> <jfos777@...> wrote:
>
> From: john777music <jfos777@...>
> Subject: [tuning] An Alternative Method for Building Scales
> To: tuning@yahoogroups.com
> Date: Thursday, February 24, 2011, 12:47 PM
>
>
>
>
>
>
>
>  
>
>
>
>
>
>
>
>
>
> An Alternative Method for Building Scales
>
>
>
> Part I: Melody in Scales
>
>
>
> No commas here, no generators, no circles or chains of intervals.
>
>
>
> After 14 years of number crunching I arrived at two likely formulas for quantifying the strength/sweetness of melodic and harmonic intervals using sine wave tones (no overtones) only and not tones with a regular harmonic series.
>
>
>
> Starting with Harmony I got...
>
>
>
> (2 + 1/x - 1/y - diss(x,y) )/2
>
>
>
> The diss(x,y) is a function.
>
> If y/x <= 0.9375 the diss(x,y) is simply y/x
>
> If y/x > 0.9375 the diss(x,y) is (1 - y/x)*15
>
> x should be greater than y and both should be less than 256. Beyond 256 you get some intervals that the formula says are bad but some of these are within 6.776 cents (256/255) of a *good* interval and by virtue of this are in themselves also good.
>
>
>
> Testing the above formula it was consistent with sine wave tones but not consistent with complex tones. Intervals that should have been stronger sounded weaker and vice versa. So I wrote a program using the formula above that paired the first 32 partials (harmonics) of each note in an interval with the first 1024 partials of both notes avoiding doing any pairing twice. Why not pair *all* 1024 partials of both notes? I'm not sure but it doesn't work. Perhaps it's because as you progress along the harmonic series the 1/x + 1/y part of the formula approaches zero (e.g. 1/256 + 1/512 = 0.0059) but the dissonance value does not diminish significantly (e.g. 256/512 = 0.5).
>
>
>
> Right or wrong, the program I wrote (called IntCalc7) seems consistent when testing harmony intervals whose notes have a regular harmonic series/timbre (i.e. the frequencies of the partials are approximately x, 2x, 3x, 4x etc and the amplitudes of the partials are approximately y, y/2, y/3, y/4 etc).
>
>
>
> What about melody? The formula I worked out was 2/x + 2/y. Why not
>
> 1/x + 1/y? This is purely cosmetic: using 2/x + 2/y any melodic interval with a value of 1.0 or higher sounds like a Major progression and any melodic interval with a value less than 1.0 sounds like a Minor progression.
>
>
>
> I had a suspicion that the 2/x + 2/y formula might be good for complex tones as well as sine wave tones and indeed it is. I wrote two programs that used the 2/x + 2/y formula. The first program paired the first 32 partials of each note with the other 1024 partials and the second paired *all* of the first 1024 partials of both notes. The results were different in magnitude but in all three cases the proportions were pretty much exactly the same. In other words the 2/x + 2/y formula for melody is equally good for both sine wave tones *and* complex tones with a `regular' timbre.
>
>
>
> For me this is relevant to scale construction. Say you want a scale exactly one octave wide. The tonic is 1/1 and the octave is 2/1. Now you select a few notes (I stick with eleven but you can choose any number) in between 1/1 and 2/1 so as to build a scale. It seems to me that (if 1/1 is the tonic), every note in the scale should "go" with 1/1 according to a *good* melodic interval within 6.776 cents (256/255) accuracy. What are the good notes that *go* with 1/1? Here they are in order of strength (use fixed width font)...
>
>
>
> Interval, Strength, Width
>
> 1/1 4.000 0.0000c
>
> 2/1 3.000 1200.0000c
>
> 3/2 1.6667 701.9550c
>
> 4/3 1.1667 498.0450c
>
> 5/3 1.0667 884.3587
>
> 5/4 0.9000 386.3137
>
> 7/4 0.7857 968.8259
>
> 6/5 0.7333 315.6413
>
> 7/5 0.6857 582.5122
>
> 8/5 0.6500 813.6863
>
> 9/5 0.6222 1017.5963
>
> 7/6 0.6190 266.8709
>
> 8/7 0.5357 231.1741
>
> 11/6 0.5152 1049.3629
>
> 9/7 0.5079 435.0841
>
> 10/7 0.4857 617.4878
>
> 9/8 0.4722 203.9100
>
> 11/7 0.4675 782.4920
>
> 12/7 0.4524 933.1291
>
> 13/7 0.4396 1071.7018
>
> 11/8 0.4318 551.3179
>
> 10/9 0.4222 182.4037
>
> 11/9 0.4040 347.4079
>
> 13/8 0.4038 840.5277
>
> 15/8 0.3833 1088.2687
>
> 11/10 0.3818 165.0042
>
> 13/9 0.3761 636.6177
>
> 14/9 0.3651 764.9159
>
> 13/10 0.3538 454.2140
>
> 12/11 0.3485 150.6371
>
> 16/9 0.3472 996.0900
>
> 17/9 0.3399 1101.0454
>
> 13/11 0.3357 289.2097
>
> 14/11 0.3247 417.5080
>
> 13/12 0.3205 138.5727
>
> 17/10 0.3176 918.6417
>
> 15/11 0.3152 536.9508
>
> 16/11 0.3068 648.6821
>
> 19/10 0.3053 1111.1993
>
> 17/11 0.2995 753.6375
>
> 14/13 0.2967 128.2982
>
> 18/11 0.2929 852.5921
>
> 15/13 0.2872 247.7411
>
> 19/11 0.2871 946.1951
>
> 17/12 0.2843 603.0004
>
> 20/11 0.2818 1034.9958
>
> 16/13 0.2788 359.4723
>
> 21/11 0.2771 1119.4630
>
> 15/14 0.2762 119.4428
>
> 19/12 0.2719 795.5580
>
> 17/13 0.2715 464.4278
>
> 18/13 0.2650 563.3823
>
> 17/14 0.2605 336.1295
>
> 19/13 0.2591 656.9854
>
> 16/15 0.2583 111.7313
>
> 20/13 0.2538 745.7861
>
> 23/12 0.2536 1126.3193
>
> 17/15 0.2510 216.6867
>
> --------------------------------
>
> 17/16 0.2426 104.9554
>
>
>
> Anyone want to test these melodic intervals against their values?
>
>
>
> I propose 5 categories of Melodic intervals...
>
>
>
> (i) Value 1.0 or higher is Major (sweet).
>
> (ii) Value between 0.5 and 0.9999 is a Blue Minor (sweet).
>
> (iii) Value between 0.25 and 0.4999 is an Ultra Minor (sweet).
>
> (iv) Value between 0.125 and 0.2499 is "tolerable" but not sweet.
>
> (v) Value less than 0.125 is "intolerable".
>
>
>
> So if all the notes in a scale are within 6.776 cents (256/255) of any of the notes listed above (except the last note,17/16) then the scale is "tonally good". If a note occurs that isn't within 6.776 cents of any of the notes listed above then the scale is "tonally bad".
>
>
>
> The last note in the list above, 17/16 has a value less than 0.25 but it is within 6.776 cents accuracy of 16/15. In fact it is right on the cusp. The distance between 16/15 and 17/16 is exactly my old friend 6.775877 cents (256/255). Interesting coincidence.
>
>
>
> I have one more stipulation for a good scale. Most "tonally good" scales will contain other melodic intervals (that don't contain the tonic, 1/1) that are not "sweet (i) to (iii)" but are "tolerable (iv)" and/or "intolerable (v)". As long as all the notes that go with the tonic are "sweet" it doesn't seem to matter much if a few other intervals in the scale that don't contain the tonic, 1/1, are only "tolerable". The narrowest "true" interval that is "tolerable" is 32/31 with a value of 0.127 (just above the cutoff point of 0.125). Any interval within 6.776 cents of 32/31 should be tolerable.
>
>
>
> 32/31 is 54.964428 cents. Going down by 6.775877 cents we get 48.18855 cents. So any interval narrower than 48.18855 cents is "intolerable". So a scale that contains an interval narrower than 48.18855 cents is "intolerable". I know some will argue that if you don't play a pair of notes that are less than 48.18855 cents apart consecutively then you can "get away with it" but I disagree.
>
>
>
> End of Part I. Part II will deal with Harmony in Scales.
>
>
>
> John O'Sullivan
>
>
>
> http://www.johnsmusic7.com
>

Michael, is it intentional that your changes fall not far from quarter tones
between chromatic degrees?

359.5 16/13
454.2 13/10
(.. 11/8?)
648.7 16/11
753.6 17/11
840.5 13/8
946.2 19/11
1035.0 20/11

Here's a little QBASIC program I wrote to test John's metrics:

_______________________________________________

'CONSON.BAS
'Evaluates John O'Sullivan consonance heuristics for pure sinusoids
'The values are scaled by 10 for easier readability
'Dan Nielsen

CONST consonanceType = 2 'Values: 1=melodic 2=harmonic
CONST eps = .0001 'Acceptable error in checking whether a value is an
integer
CONST domain = 15 'Checks ratios from 1/1 to 2*domain/domain

DEFDBL A-Z
SCREEN 12
PALETTE 0, 8 + 8 * 256 + 8 * 65536
FOR i = 1 TO 15
PALETTE i, 3 + 4 * i
NEXT
DIM res(1300) AS DOUBLE
DIM resq(1300) AS INTEGER
DIM resr(1300) AS INTEGER
i = 0
FOR r = 1 TO domain
FOR q = r TO 2 * r
'If q/r is not a reduced fraction, skip it
FOR s = 2 TO domain
f1 = q / s
f2 = r / s
WHILE f1 >= 1
f1 = f1 - 1
WEND
WHILE f2 >= 1
f2 = f2 - 1
WEND
IF ABS(f1) < eps AND ABS(f2) < eps GOTO skip
NEXT s
i = i + 1
x = 1 / (2 * q)
y = 1 / (2 * r)
IF (consonanceType = 1) THEN
h = 10 * 4 * (x + y)
ELSE
cond = (x / y) > 15 / 16
h = 10 * (x - y - .5 * x / y + 1 - cond * (8 * x / y - 7.5))
END IF
res(i) = h
resq(i) = q
resr(i) = r
LINE (10 * q, 10 * r)-(10 * q + 9, 10 * r + 9), h, BF
IF cond THEN LINE (10 * q, 10 * r)-(10 * q + 9, 10 * r + 9), 0, B,
&HFF00
skip:
NEXT q
NEXT r

'Sort results
FOR x = 1 TO 1300 - 1
FOR y = 1300 TO x + 1 STEP -1
IF res(y) > res(x) THEN
SWAP res(y), res(x)
SWAP resq(y), resq(x)
SWAP resr(y), resr(x)
END IF
NEXT y
NEXT x

LOCATE 1, 1
PRINT "Major"
j = 0
DO
CLS
LOCATE 1, 1
FOR i = 1 TO 25
k = j + i
IF k > 1 THEN
IF res(k) < 1.25 GOTO coda
IF res(k) < 2.5 AND res(k - 1) >= 2.5 THEN PRINT "Ultra minor";
STRING$(80 - 11, "-")
IF res(k) < 5 AND res(k - 1) >= 5 THEN PRINT "Blue minor"; STRING$(80
- 10, "-")
END IF
PRINT "#"; k, resq(k), resr(k), res(k)
NEXT
j = j + 25
SLEEP
LOOP

coda:
END

Daniel>"Michael, is it intentional that your changes fall not far from quarter tones between chromatic degrees?

359.5  16/13454.2  13/10(.. 11/8?)648.7  16/11753.6  17/11840.5  13/8946.2  19/111035.0  20/11"
  
      Funny, it's not intentional at all.  In fact I view 11/9, 9/7, 22/15, 14/9, 11/6...which are also fairly near quarter tones, as fairly consonant .

   What I view as unique is that, around the quarter tone areas, there are many dyads that appear dissonant within some odd 10-15 cents or so of (IE rather close to) ratios that appear fairly consonant.  To perhaps over simplify it: areas around quarter tones seem very delicate...either you get them just right or they sound dissonant.

    Interesting program to recreate John's melodic dissonance formula, by the way.  It's amusing to know they are so many programmers on-list (you, John, Gene, myself, others?).

--- On Fri, 2/25/11, Daniel Nielsen <nielsed@...> wrote:

From: Daniel Nielsen <nielsed@...>
Subject: Re: [tuning] An Alternative Method for Building Scales
To: tuning@yahoogroups.com
Date: Friday, February 25, 2011, 11:18 AM

Michael, is it intentional that your changes fall not far from quarter tones between chromatic degrees?

359.5  16/13454.2  13/10(.. 11/8?)648.7  16/11753.6  17/11840.5  13/8946.2  19/111035.0  20/11
Here's a little QBASIC program I wrote to test John's metrics:

_______________________________________________
'CONSON.BAS'Evaluates John O'Sullivan consonance heuristics for pure sinusoids'The values are scaled by 10 for easier readability
'Dan Nielsen
CONST consonanceType = 2 'Values: 1=melodic 2=harmonicCONST eps = .0001 'Acceptable error in checking whether a value is an integerCONST domain = 15 'Checks ratios from 1/1 to 2*domain/domain

DEFDBL A-ZSCREEN 12PALETTE 0, 8 + 8 * 256 + 8 * 65536FOR i = 1 TO 15PALETTE i, 3 + 4 * iNEXTDIM res(1300) AS DOUBLEDIM resq(1300) AS INTEGER
DIM resr(1300) AS INTEGERi = 0FOR r = 1 TO domain  FOR q = r TO 2 * r    'If q/r is not a reduced fraction, skip it    FOR s = 2 TO domain      f1 = q / s
      f2 = r / s      WHILE f1 >= 1        f1 = f1 - 1      WEND      WHILE f2 >= 1        f2 = f2 - 1      WEND      IF ABS(f1) < eps AND ABS(f2) < eps GOTO skip
    NEXT s    i = i + 1    x = 1 / (2 * q)    y = 1 / (2 * r)    IF (consonanceType = 1) THEN      h = 10 * 4 * (x + y)    ELSE      cond = (x / y) > 15 / 16
      h = 10 * (x - y - .5 * x / y + 1 - cond * (8 * x / y - 7.5))    END IF    res(i) = h    resq(i) = q    resr(i) = r    LINE (10 * q, 10 * r)-(10 * q + 9, 10 * r + 9), h, BF
    IF cond THEN LINE (10 * q, 10 * r)-(10 * q + 9, 10 * r + 9), 0, B, &HFF00skip:  NEXT qNEXT r
'Sort resultsFOR x = 1 TO 1300 - 1
   FOR y = 1300 TO x + 1 STEP -1    IF res(y) > res(x) THEN      SWAP res(y), res(x)      SWAP resq(y), resq(x)      SWAP resr(y), resr(x)    END IF  NEXT y
NEXT x
LOCATE 1, 1PRINT "Major"j = 0DO  CLS  LOCATE 1, 1  FOR i = 1 TO 25    k = j + i    IF k > 1 THEN
      IF res(k) < 1.25 GOTO coda      IF res(k) < 2.5 AND res(k - 1) >= 2.5 THEN PRINT "Ultra minor"; STRING$(80 - 11, "-")      IF res(k) < 5 AND res(k - 1) >= 5 THEN PRINT "Blue minor"; STRING$(80 - 10, "-")
    END IF    PRINT "#"; k, resq(k), resr(k), res(k)  NEXT  j = j + 25  SLEEPLOOP
coda:END

On Fri, Feb 25, 2011 at 2:00 PM, Michael <djtrancendance@...> wrote:

>
>
> Funny, it's not intentional at all. In fact I view 11/9, 9/7, 22/15,
> 14/9, 11/6...which are also fairly near quarter tones, as fairly consonant .
>
> What I view as unique is that, around the quarter tone areas, there are
> many dyads that appear dissonant within some odd 10-15 cents or so of (IE
> rather close to) ratios that appear fairly consonant. To perhaps over
> simplify it: areas around quarter tones seem very delicate...either you get
> them just right or they sound dissonant.
>
> Interesting program to recreate John's melodic dissonance formula, by
> the way. It's amusing to know they are so many programmers on-list (you,
> John, Gene, myself, others?).
>
>

Thanks, I'm very glad you liked the little program. I fixed it up a little
more, compiled it, and posted it to Files.

Ah, I wonder what function might be able to make the changes you suggested -
it would seem something that is centered at the fifth and has a period of
about a semitone. If that's right, it might be some pretty simple 1D
function: maybe something like a Gabor function, an increment Gabor
function, the log of a Gabor or some exponentiation, or some similar version
of sinc(). I think one thing that leads me to Gabor here is that the retina
responds to light along some similar lines, although it may not be related.

An Alternative Method for Building Scales

Part I b: More on Melody in Scales

This is a continuation of message number 96503.

No commas here, no generators, no circles or chains of intervals.

Here's my Blue Just scale...

1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.

Which note is the tonic or `most resolved' sounding note?

Using my formula for melody, 2/x + 2/y (see msg 96503) pair the first note (1/1) with the other 12 notes and add the 12 results. Next pair the second note (15/14) with the other 12 notes and add the 12 results. Next pair the third note (9/8) with the other 12, and so on. The ratios should be simplified (e.g. 16/12 simplifies to 4/3) before calculation. The formula above is good only for "just" intervals where x and y are less than 256.

The note with the highest value is the tonic (in this case 1/1).

So here 1/1 is stronger than 2/1. This is good for melodies or chord progressions that go up from 1/1 and back down to 1/1 again, but not so good for melodies/progressions that go down from 2/1 and back up to 2/1 again. Most music `lands' on 1/1 but there are exceptions. E.g. Sweet Jane by The Velvet Underground starts on a high E (the tonic), goes down to B, down to A, up to Db Minor, down to B and finally up to the high E again. Here the tonic is (relatively) 2/1 and not 1/1.

So it would seem that Blue Just is better for "up/down" music but not so good for "down/up" music.

One way to have 1/1 and 2/1 as tonics of equal strength is to build a symmetric scale. Here's a likely candidate (Michael S. correctly suggested that 10/9 and 9/5 should be better than 9/8 and 16/9)...

1/1, 16/15, 10/9, 6/5, 5/4, 4/3, sqrt(2), 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.

The notes going up from 1/1 are the exact mirror image of the notes going down from 2/1. So both 1/1 and 2/1 are equally good as tonics. Note that a sqrt(2) interval does not work in harmony (chords) but it is acceptable in melody. It is only 3 cents away from 17/12 which for me is a `sweet' (but minor) melodic interval (see msg 96503). Using the 2/x + 2/y formula, 2/17 + 2/12 = 0.2843. Any melodic interval with a value higher than 0.25 should be good.

I tempered the symmetric scale above (I call it the Mirror Scale) so as to yield as many good harmony intervals as possible while maintaining the symmetry and the resulting scale has only 83 good harmony intervals an octave or less wide. Blue Temperament has 97 good harmony intervals. Here are the good (according to my taste) harmony intervals...

9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7, 2/1.

I found symmetry another way. By chance I stumbled upon a scale where the notes going up from 1/1 were the exact mirror image of the notes going down from 3/2. Blue Just has this property. Here it is again...

1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.

Like I said, the notes going up from 1/1 are the exact mirror image of the notes going down from 3/2. So we have two usable tonics of equal strength, 1/1 and 3/2. 1/1 is better for "up/down" music and 3/2 is better for "down/up" music (see above). We now have the best of both worlds.

Blue Just has exactly 72 "good" harmony intervals an octave or less wide (including the octave and excluding the unison) over twelve keys. That's 72 out of a possible 144 (12 x 12) intervals, a "hit rate" of exactly 50%. These are all perfectly in tune. There's also a 13/7 interval lurking in there somewhere, not perfectly in tune but within 6.776 cents accuracy.

My chosen method for building scales is to find the best possible "just" scale (which should be "tonally good", see msg 96503) and then temper each note by not more than 6.776 cents (256/255) to squeeze as many good harmony intervals (an octave or less wide) out of it as possible. It may not be the best approach but there's nothing wrong with it and it's viable.

I'm pretty sure that Blue Just is the most versatile "symmetric" (two tonics of equal strength, in this case 1/1 and 3/2), 12 keys per octave, just scale. I tempered this scale (preserving the symmetry) to get Blue Temperament. Here it is...

0.0
121.6
200.7
313.5
388.4
501.2
580.4
702.0
816.9
889.4
1012.5
1085.1
1200.0

Blue Temperament has 97 good harmony intervals (within 6.776 cents accuracy) out of a possible 144, a "hit rate" of just over 67%.

Can anyone beat this? Mike sent me a few scales and the most versatile (Kornerup's Golden Meantone) has only 94 good harmony intervals.

My tempering algorithm does not use chains or circles of intervals. The program works out every possible combination of frequencies and analyses each combination. Each frequency starts with "original just freq - 6.776 cents" and ends with "original just freq + 6.776 cents" with a step size of 0.28233 cents. Because of the symmetry (pairs of notes and not single notes are tempered equally) I only need 5 nested "for loops" (the program takes over five hours to run) and not 11 nested "for loops" (which would take years to run).

You can hear five compositions by Chris Vaisvil that use both Blue Just and Blue Temperament on my web site...

http://www.johnsmusic7.com

John.

I'm enjoying these posts, John. The melodic formula is easy enough to
understand in general, at least, because it not a piecewise function, and it
can be expressed simply using Tenney height; for q/r, consonance=(q+r)/T.H.

The harmonic formula is more difficult for me to interpret until you explain
further, because I don't have your book.

You probably noticed my rambling post concerning the similarity of these to
basic formulas in optics. Letting q and r be focal lengths, the melodic
formula is the effective focal length of 2 thin lenses in contact, and the
harmonic formula is much like the effective focal length of a telephoto lens
with a certain distance of separation between lenses, except that in
analogizing the case of your formula, this distance between lenses would
often be negative (impossible).

I'm interested in finding out what motivations or data you used to construct
the harmonic version, and also why you decided to use 2 functions to define
diss(). Would it be sensible to interpolate these functions?

(2 + 1/x + 1/y - diss(x,y))/2

Here's briefly how I arrived at my formula.

Listening to harmony and melody intervals (x/y) it seems that, in general, the smaller the numbers in the ratio, the sweeter the interval sounds. Based on this I assumed that there must be a mathematical formula for quantifying the harmony and melody values of intervals. Four wild guesses and likely candidates were: 1/xy, 1/(x+y), 1/x + 1/y and 1/x (note x>=y and x and y are less than 256). It seems reasonable that one of these formulas might be correct as the smaller x and y the greater the resulting value.

With the 1/x formula the y doesn't feature. Just as a chain is only as strong as its weakest link perhaps the strength value of an interval is only as strong as the weaker of either 1/x or 1/y. With 5/3 the 5 has a value of 0.2 (1/5) and the 3 has a value of 0.3333 (1/3). 0.2 is less than 0.3333 so the strength value here would be 0.2.

I started with melody because it seemed simpler than harmony (beating is a factor in harmony but is irrelevant in melody). I made three lists of notes (going up from 1/1) that I would pair with 1/1. I arranged the notes in decreasing order of strength (when paired with 1/1) in each list. One list corresponded to 1/x, another to 1(x+y) and another to 1/x + 1/y. I didn't bother making a list for 1/xy as I considered it unlikely but I checked it later, found one clear inconsistency and so ruled it out.

One list indicated that interval `a' should be stronger than interval `b' and another list would indicate that the opposite was true. Only one of the three lists seemed consistent with what my ears were telling me and that was the 1/x + 1/y list. I used sine wave tones for the testing but it seems now that 1/x + 1/y is consistent for complex tones (with a `regular' harmonic series) as well.

I guessed that 1/x + 1/y should be a part of a `harmony formula' but beating (AKA dissonance) must be a factor also. Using sine waves I tested 14/13, 15/14, 16/15, 17/16, 18/17 etc. It seemed to me that either 16/15 or 17/16 or thereabouts was the most dissonant interval. In the end I chose 16/15 . My first formula for dissonance was simply y/x. 9999/10000 would have a dissonance value of 0.9999 but it's almost a unison and doesn't sound dissonant so for me dissonance peaks at 16/15 or 0.9375.

Where y/x <= 0.9375 dissonance = y/x

Where y/x > 0.9375 dissonance = (1 - y/x)*15

So my harmony formula is the consonance value minus the dissonance value...

1/x + 1/y - diss(x,y).

In hindsight the 1/x + 1/y part of the formula seems to have to do with periodicity.

Assume a single note has a strength value of 1.0 . It seems to me that any harmony interval that sounds stronger than a single note (value 1.0) is a Major interval and any harmony interval that sounds weaker than a single note is a Minor interval.

Using sine wave tones for the testing the 3/2 interval has a value of 1/3 + 1/2 - 2/3 = 0.166667 which is less than 1.0 and should therefore be Minor but it clearly sounds Major. I made another educated guess and added 2 to the formula (two notes with a strength value of 1 each) to get...

2 + 1/x + 1/y - diss(x,y)

This seemed to work better after testing. Finally I divided the whole lot by 2 purely because when you do this any interval with a value of 1.0 or higher indeed sounds Major and any interval with a value less than 1.0 sounds Minor (using sine wave tones only). So the final formula is...

(2 + 1/x + 1/y - diss(x,y))/2

I spent a long time testing it and it seems consistent so far. Note that it only works for sine wave tones and not complex tones. I used the formula in a program that worked out the harmony values of complex tones by pairing the first 32 partials of each note with the other 1024 partials of each note and again, after much testing, the program seems consistent.

John.

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
>
> I'm enjoying these posts, John. The melodic formula is easy enough to
> understand in general, at least, because it not a piecewise function, and it
> can be expressed simply using Tenney height; for q/r, consonance=(q+r)/T.H.
>
> The harmonic formula is more difficult for me to interpret until you explain
> further, because I don't have your book.
>
> You probably noticed my rambling post concerning the similarity of these to
> basic formulas in optics. Letting q and r be focal lengths, the melodic
> formula is the effective focal length of 2 thin lenses in contact, and the
> harmonic formula is much like the effective focal length of a telephoto lens
> with a certain distance of separation between lenses, except that in
> analogizing the case of your formula, this distance between lenses would
> often be negative (impossible).
>
> I'm interested in finding out what motivations or data you used to construct
> the harmonic version, and also why you decided to use 2 functions to define
> diss(). Would it be sensible to interpolate these functions?
>

John,

Not feeling good here. The little I read remind me of counterpoint rules in
theory class. Have you compared to this:
http://en.wikipedia.org/wiki/Counterpoint

all the best,

Chris

On Fri, Mar 4, 2011 at 5:15 PM, john777music <jfos777@...> wrote:

>
>
> (2 + 1/x + 1/y - diss(x,y))/2
>
> Here's briefly how I arrived at my formula.
>
> Listening to harmony and melody intervals (x/y) it seems that, in general,
> the smaller the numbers in the ratio, the sweeter the interval sounds. Based
> on this I assumed that there must be a mathematical formula for quantifying
> the harmony and melody values of intervals. Four wild guesses and likely
> candidates were: 1/xy, 1/(x+y), 1/x + 1/y and 1/x (note x>=y and x and y are
> less than 256). It seems reasonable that one of these formulas might be
> correct as the smaller x and y the greater the resulting value.
>
> With the 1/x formula the y doesn't feature. Just as a chain is only as
> strong as its weakest link perhaps the strength value of an interval is only
> as strong as the weaker of either 1/x or 1/y. With 5/3 the 5 has a value of
> 0.2 (1/5) and the 3 has a value of 0.3333 (1/3). 0.2 is less than 0.3333 so
> the strength value here would be 0.2.
>
> I started with melody because it seemed simpler than harmony (beating is a
> factor in harmony but is irrelevant in melody). I made three lists of notes
> (going up from 1/1) that I would pair with 1/1. I arranged the notes in
> decreasing order of strength (when paired with 1/1) in each list. One list
> corresponded to 1/x, another to 1(x+y) and another to 1/x + 1/y. I didn't
> bother making a list for 1/xy as I considered it unlikely but I checked it
> later, found one clear inconsistency and so ruled it out.
>
> One list indicated that interval `a' should be stronger than interval `b'
> and another list would indicate that the opposite was true. Only one of the
> three lists seemed consistent with what my ears were telling me and that was
> the 1/x + 1/y list. I used sine wave tones for the testing but it seems now
> that 1/x + 1/y is consistent for complex tones (with a `regular' harmonic
> series) as well.
>
> I guessed that 1/x + 1/y should be a part of a `harmony formula' but
> beating (AKA dissonance) must be a factor also. Using sine waves I tested
> 14/13, 15/14, 16/15, 17/16, 18/17 etc. It seemed to me that either 16/15 or
> 17/16 or thereabouts was the most dissonant interval. In the end I chose
> 16/15 . My first formula for dissonance was simply y/x. 9999/10000 would
> have a dissonance value of 0.9999 but it's almost a unison and doesn't sound
> dissonant so for me dissonance peaks at 16/15 or 0.9375.
>
> Where y/x <= 0.9375 dissonance = y/x
>
> Where y/x > 0.9375 dissonance = (1 - y/x)*15
>
> So my harmony formula is the consonance value minus the dissonance value...
>
> 1/x + 1/y - diss(x,y).
>
> In hindsight the 1/x + 1/y part of the formula seems to have to do with
> periodicity.
>
> Assume a single note has a strength value of 1.0 . It seems to me that any
> harmony interval that sounds stronger than a single note (value 1.0) is a
> Major interval and any harmony interval that sounds weaker than a single
> note is a Minor interval.
>
> Using sine wave tones for the testing the 3/2 interval has a value of 1/3 +
> 1/2 - 2/3 = 0.166667 which is less than 1.0 and should therefore be Minor
> but it clearly sounds Major. I made another educated guess and added 2 to
> the formula (two notes with a strength value of 1 each) to get...
>
> 2 + 1/x + 1/y - diss(x,y)
>
> This seemed to work better after testing. Finally I divided the whole lot
> by 2 purely because when you do this any interval with a value of 1.0 or
> higher indeed sounds Major and any interval with a value less than 1.0
> sounds Minor (using sine wave tones only). So the final formula is...
>
> (2 + 1/x + 1/y - diss(x,y))/2
>
> I spent a long time testing it and it seems consistent so far. Note that it
> only works for sine wave tones and not complex tones. I used the formula in
> a program that worked out the harmony values of complex tones by pairing the
> first 32 partials of each note with the other 1024 partials of each note and
> again, after much testing, the program seems consistent.
>
> John.
>
>
> --- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
> >
> > I'm enjoying these posts, John. The melodic formula is easy enough to
> > understand in general, at least, because it not a piecewise function, and
> it
> > can be expressed simply using Tenney height; for q/r,
> consonance=(q+r)/T.H.
> >
> > The harmonic formula is more difficult for me to interpret until you
> explain
> > further, because I don't have your book.
> >
> > You probably noticed my rambling post concerning the similarity of these
> to
> > basic formulas in optics. Letting q and r be focal lengths, the melodic
> > formula is the effective focal length of 2 thin lenses in contact, and
> the
> > harmonic formula is much like the effective focal length of a telephoto
> lens
> > with a certain distance of separation between lenses, except that in
> > analogizing the case of your formula, this distance between lenses would
> > often be negative (impossible).
> >
> > I'm interested in finding out what motivations or data you used to
> construct
> > the harmonic version, and also why you decided to use 2 functions to
> define
> > diss(). Would it be sensible to interpolate these functions?
> >
>
>
>

This is a different formula than the one you originally stated. On the
bright side for myself, it is much easier to analogize with an optical
system. On the downside, it means the previous wrangling was for naught.

Anyway, thank you for the description. Knowing where this came from helps to
understand how it may be correctly used, and which mathematical forms,
variations, and approximations may plausibly render results in a similar
spirit.

Let me restate your metric, John:

c(x,y) = (2 + 1/x + 1/y - diss(x,y))/2
where y/x <= 0.9375, diss(x,y) = y/x
where y/x > 0.9375, diss(x,y) = (1 - y/x)*15

I'll just note (for y>=1; y<=x<=2*y; x/y reduced) that

0 <= diss(x,y) <= 15/16

and

0 < 1/x + 1/y <= 2

and it seems (after some testing) also that

0 < 1/x + 1/y - diss(x,y) <= 2

Let's add 2 to this last expression:

2 < 2 + 1/x + 1/y - diss(x,y) <= 4

Now divide by 2:

1 < (2 + 1/x + 1/y - diss(x,y)) / 2 <= 2

It seems that any possible dyad is classified as major, unless I'm doing
something wrong here.

Have you considered evaluating the base-2 logs
1+lb(1/x + 1/y)
1+lb(1/x + 1/y - diss(x,y))
(so that major is the condition >=0)?

Since 0<1/x<=1 and 0<1/y<=1, we might also consider the parameters something
like probabilities.

The binary entropy function of such is

H(1/x) = -(1/x)lb(1/x) + -((x-1)/x)lb((x-1)/x) = ( lb(x) + (x-1)
(lb(x)-lb(x-1))
) / x

and the mutual information (which is always nonnegative) resembles your
consonance function:

I(1/x,1/y) = H(1/x) + H(1/y) - H(1/x,1/y) =

( lb(x) + (x-1) (lb(x)-lb(x-1)) ) / x + ( lb(y) + (y-1) (lb(y)-lb(y-1)) ) /
y - H(1/x,1/y)

We might define the joint entropy H(1/x,1/y) in the above as something like

H(1/x,1/y) = H(1/xy) = ( lb(x) + lb(y) + (xy-1) (lb(x)+lb(y)-lb(xy-1)) ) /
xy

It might be interesting to evaluate consonance by that shared information
quantity (of course, there's a lot of talk on this list about harmonic
entropy, but I have no idea if that's a similar concept or not).

_______________________________

One final note. You may want to consider

1/x + 1/y - diss(x,y) = (x + y - d(x,y)) / xy

since this expression explicitly shows Tenney height, and d(x,y) is distance
in the two-lens equation.

Dan,

I made a typo in message number 96503. I wrote...

(2 + 1/x - 1/y - diss(x,y))/2

when I should have written...

(2 + 1/x + 1/y - diss(x,y))/2

John.

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
>
> This is a different formula than the one you originally stated. On the
> bright side for myself, it is much easier to analogize with an optical
> system. On the downside, it means the previous wrangling was for naught.
>
> Anyway, thank you for the description. Knowing where this came from helps to
> understand how it may be correctly used, and which mathematical forms,
> variations, and approximations may plausibly render results in a similar
> spirit.
>

Dan,

you said...

....and it seems (after some testing) also that

....0 < 1/x + 1/y - diss(x,y) <= 2

This can't be right. If x/y is 161/150 you get

0.0062 + 0.0067 - 0.9317 = -0.9188 a negative number.

The following should be more accurate...

-0.9375 < 1/x + 1/y - diss(x,y) <= 2

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
>
> Let me restate your metric, John:
>
> c(x,y) = (2 + 1/x + 1/y - diss(x,y))/2
> where y/x <= 0.9375, diss(x,y) = y/x
> where y/x > 0.9375, diss(x,y) = (1 - y/x)*15
>
> I'll just note (for y>=1; y<=x<=2*y; x/y reduced) that
>
> 0 <= diss(x,y) <= 15/16
>
> and
>
> 0 < 1/x + 1/y <= 2
>
> and it seems (after some testing) also that
>
> 0 < 1/x + 1/y - diss(x,y) <= 2
>
> Let's add 2 to this last expression:
>
> 2 < 2 + 1/x + 1/y - diss(x,y) <= 4
>
> Now divide by 2:
>
> 1 < (2 + 1/x + 1/y - diss(x,y)) / 2 <= 2
>
> It seems that any possible dyad is classified as major, unless I'm doing
> something wrong here.
>
> Have you considered evaluating the base-2 logs
> 1+lb(1/x + 1/y)
> 1+lb(1/x + 1/y - diss(x,y))
> (so that major is the condition >=0)?
>
> Since 0<1/x<=1 and 0<1/y<=1, we might also consider the parameters something
> like probabilities.
>
> The binary entropy function of such is
>
> H(1/x) = -(1/x)lb(1/x) + -((x-1)/x)lb((x-1)/x) = ( lb(x) + (x-1)
> (lb(x)-lb(x-1))
> ) / x
>
> and the mutual information (which is always nonnegative) resembles your
> consonance function:
>
> I(1/x,1/y) = H(1/x) + H(1/y) - H(1/x,1/y) =
>
> ( lb(x) + (x-1) (lb(x)-lb(x-1)) ) / x + ( lb(y) + (y-1) (lb(y)-lb(y-1)) ) /
> y - H(1/x,1/y)
>
> We might define the joint entropy H(1/x,1/y) in the above as something like
>
> H(1/x,1/y) = H(1/xy) = ( lb(x) + lb(y) + (xy-1) (lb(x)+lb(y)-lb(xy-1)) ) /
> xy
>
> It might be interesting to evaluate consonance by that shared information
> quantity (of course, there's a lot of talk on this list about harmonic
> entropy, but I have no idea if that's a similar concept or not).
>
> _______________________________
>
> One final note. You may want to consider
>
> 1/x + 1/y - diss(x,y) = (x + y - d(x,y)) / xy
>
> since this expression explicitly shows Tenney height, and d(x,y) is distance
> in the two-lens equation.
>

>
>
> This can't be right. If x/y is 161/150 you get
>
> 0.0062 + 0.0067 - 0.9317 = -0.9188 a negative number.
>
> The following should be more accurate...
>
> -0.9375 < 1/x + 1/y - diss(x,y) <= 2
>
>
Yeah, you're right, John, thanks (was a bit rushed on that last night :/).

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> (2 + 1/x + 1/y - diss(x,y))/2
>
> Here's briefly how I arrived at my formula.
>
> Listening to harmony and melody intervals (x/y) it seems that, in general, the smaller the numbers in the ratio, the sweeter the interval sounds. Based on this I assumed that there must be a mathematical formula for quantifying the harmony and melody values of intervals. Four wild guesses and likely candidates were: 1/xy, 1/(x+y), 1/x + 1/y and 1/x (note x>=y and x and y are less than 256). It seems reasonable that one of these formulas might be correct as the smaller x and y the greater the resulting value.
>
> With the 1/x formula the y doesn't feature. Just as a chain is only as strong as its weakest link perhaps the strength value of an interval is only as strong as the weaker of either 1/x or 1/y. With 5/3 the 5 has a value of 0.2 (1/5) and the 3 has a value of 0.3333 (1/3). 0.2 is less than 0.3333 so the strength value here would be 0.2.
>
> I started with melody because it seemed simpler than harmony (beating is a factor in harmony but is irrelevant in melody). I made three lists of notes (going up from 1/1) that I would pair with 1/1. I arranged the notes in decreasing order of strength (when paired with 1/1) in each list. One list corresponded to 1/x, another to 1(x+y) and another to 1/x + 1/y. I didn't bother making a list for 1/xy as I considered it unlikely but I checked it later, found one clear inconsistency and so ruled it out.

John, I'm still in the process of reading your book and will submit a review once I finish it. I'm currently having to put in overtime at work, which will continue for at least the next two weeks, which has not left me very much free time, and I want to give your book the amount of attention it deserves.

One thing mentioned in the book is repeated above: You didn't tell what is the "one clear inconsistency" with 1/xy (Tenney height) that you found. What is it?

--George

George,

I can't remember the particular inconsistency but the 1/x + 1/y formula indicated that interval 'a' should be stronger than interval 'b' and the 1/xy formula indicated that the opposite was true. The first formula seemed consistent to my ears and the second formula did not.

If you want you could test these yourself. List about two dozen simple just intervals and arrange them in decreasing order of strength according to each formula, one list for each formula. Compare the lists and there will be disagreement between them as regards the 'order' of the intervals.

John.

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > (2 + 1/x + 1/y - diss(x,y))/2
> >
> > Here's briefly how I arrived at my formula.
> >
> > Listening to harmony and melody intervals (x/y) it seems that, in general, the smaller the numbers in the ratio, the sweeter the interval sounds. Based on this I assumed that there must be a mathematical formula for quantifying the harmony and melody values of intervals. Four wild guesses and likely candidates were: 1/xy, 1/(x+y), 1/x + 1/y and 1/x (note x>=y and x and y are less than 256). It seems reasonable that one of these formulas might be correct as the smaller x and y the greater the resulting value.
> >
> > With the 1/x formula the y doesn't feature. Just as a chain is only as strong as its weakest link perhaps the strength value of an interval is only as strong as the weaker of either 1/x or 1/y. With 5/3 the 5 has a value of 0.2 (1/5) and the 3 has a value of 0.3333 (1/3). 0.2 is less than 0.3333 so the strength value here would be 0.2.
> >
> > I started with melody because it seemed simpler than harmony (beating is a factor in harmony but is irrelevant in melody). I made three lists of notes (going up from 1/1) that I would pair with 1/1. I arranged the notes in decreasing order of strength (when paired with 1/1) in each list. One list corresponded to 1/x, another to 1(x+y) and another to 1/x + 1/y. I didn't bother making a list for 1/xy as I considered it unlikely but I checked it later, found one clear inconsistency and so ruled it out.
>
> John, I'm still in the process of reading your book and will submit a review once I finish it. I'm currently having to put in overtime at work, which will continue for at least the next two weeks, which has not left me very much free time, and I want to give your book the amount of attention it deserves.
>
> One thing mentioned in the book is repeated above: You didn't tell what is the "one clear inconsistency" with 1/xy (Tenney height) that you found. What is it?
>
> --George
>