Crows

My formula for a melodic interval is 2/x + 2/y where x and y are less than 256. I propose six categories of melodic intervals.

(i) Super Major...value >= 2.0
(ii) Blue Major...1.0 <= value < 2.0
(iii) Blue Minor...0.5 <= value < 1.0
(iv) Ultra Minor...0.25 <= value < 0.5
(v) Tolerable (but not sweet)...0.125 <= value < 0.25
(vi) Intolerable...value < 0.125

(i) to (iv) are sweet. It would be nice to have all possible intervals in a scale "sweet" but with, say, 12 notes per octaves it's impossible. You have to allow a few "tolerable" intervals.

It is possible however to build a scale where all the notes going up from the tonic (1/1) are "sweet" when paired with the tonic. It seems that as long as all the notes paired with the tonic are sweet it doesn't matter too much if some of the other intervals are only "tolerable". Obviously "intolerable intervals" should be avoided.

Here again is the list of sweet melodic intervals arranged in decreasing order of strength (use fixed width font).

In order...
Super Major
1/1 4 0.0000
2/1 3 1200.0000
------------------------
Blue Major
3/2 1.6667 701.9550
4/3 1.1667 498.0450
5/3 1.0667 884.3587
------------------------
Blue Minor
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
------------------------
Ultra Minor
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 (exactly 6.7758758 cents away from 16/15)

Anyone want to test these categories?

Any melodic interval within 6.7758758 cents (256/255) of any of the sweet intervals listed above (e.g. 17/16) should also be sweet enough though not perfectly in tune.

The narrowest (apart from the unison) just tolerable interval that has a value greater than or equal to 0.125 is 32/31 (54.964428 cents). 2/32 + 2/31 = 0.127
Any interval within 6.7758758 cents of 32/31 should also be tolerable. Going down from 54.964428 cents (32/31) by 6.7758758 cents we get 48.1885522 cents so any interval narrower than 48.1885522 cents (apart from the unison) should sound "intolerable". I tested a 47 cents interval and it sounded sour to me.

The unison is good so going up from 0 cents to 6.7758758 cents (256/255) should also be good. So any melodic interval between 6.6775876 cents and 48.1885522 cents should be intolerable (and therefore illegal).

I came up with a name for this "illegal" range: a "crow". The cawing of crows sounds harsh compared to the singing of other songbirds.

Definition...

A "crow" is any melodic interval between 6.6775876 cents and 48.1885522 cents wide which sounds unpleasant.

This is bad news for fans of 25 or higher EDOs where the gaps between consecutive notes are all crows.

John.

On Sat, Mar 12, 2011 at 3:41 PM, john777music <jfos777@...> wrote:
>
> Anyone want to test these categories?
>
> Any melodic interval within 6.7758758 cents (256/255) of any of the sweet intervals listed above (e.g. 17/16) should also be sweet enough though not perfectly in tune.

So what about 300 cents and 400 cents? They aren't usable as melodic intervals?

-Mike

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> My formula for a melodic interval is 2/x + 2/y where x and y are less than 256. I propose six categories of melodic intervals.

Some comments:

(1) I don't think names like "Super Major" or "Blue Minor" are going to work as names, bringing associations to mind you don't want.

(2) I think your goodness figure doesn't work as well as Tenney height. For instance, you rate 9/5 as better than 7/6, while Tenney height rates it worse, and I think TH is right. If you want to argue that 9/5 should win because it is lower limit, then what about 11/6, which you rate as better than 9/7?

(3) There's no problem with divisions of the octave with more than 24 notes to the octave, and anyway, following your logic, with more than 177 notes they should become good again.

I'll get back to you on this. I suspect and will check tomorrow that 300 cents and 400 cents are both within 6.776 cents of a "tolerable" interval (i.e. 2/x + 2/y is greater than or equal to 0.125 and less than 0.25). The "tolerable" intervals don't appear in my list because there are so many of them.

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 12, 2011 at 3:41 PM, john777music <jfos777@...> wrote:
> >
> > Anyone want to test these categories?
> >
> > Any melodic interval within 6.7758758 cents (256/255) of any of the sweet intervals listed above (e.g. 17/16) should also be sweet enough though not perfectly in tune.
>
> So what about 300 cents and 400 cents? They aren't usable as melodic intervals?
>
> -Mike
>

1. The names Super Major and Blue Minor are serviceable enough for my purposes.

2. I thought that Tenney Height applied to harmony intervals only and not melodic intervals. Harmony and Melody intervals have different values in my system. My melody formula gives 9/5 a value of 0.622 and gives 7/6 a value of 0.619 so if the formula is correct it would be pretty hard to decide which was better in a listening test, the values being so close to each other. Also my formula has nothing to do with odd or prime limits, these are not a factor and are irrelevant.

3. I suspected someone would point this out regarding higher than 177 EDOs. I knew about it but didn't mention it for brevity's sake. As I said, I tested a melodic interval that was 47 cents wide (a crow) and it definitely sounded sour to me. So if it's down to a matter of taste I'll go with my own taste.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > My formula for a melodic interval is 2/x + 2/y where x and y are less than 256. I propose six categories of melodic intervals.
>
> Some comments:
>
> (1) I don't think names like "Super Major" or "Blue Minor" are going to work as names, bringing associations to mind you don't want.
>
> (2) I think your goodness figure doesn't work as well as Tenney height. For instance, you rate 9/5 as better than 7/6, while Tenney height rates it worse, and I think TH is right. If you want to argue that 9/5 should win because it is lower limit, then what about 11/6, which you rate as better than 9/7?
>
> (3) There's no problem with divisions of the octave with more than 24 notes to the octave, and anyway, following your logic, with more than 177 notes they should become good again.
>

Mike,

300 cents is within 6.776 cents of 31/26 (304.5 cents) which has a value of (according to 2/x + 2/y) 0.1414.

400 cents is within 6.776 cents of 27/34 (399.1 cents) which has a value of 0.1329.

Any just melodic interval (where x and y are less than 256) with a value of 0.125 or higher is "tolerable".

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 12, 2011 at 3:41 PM, john777music <jfos777@...> wrote:
> >
> > Anyone want to test these categories?
> >
> > Any melodic interval within 6.7758758 cents (256/255) of any of the sweet intervals listed above (e.g. 17/16) should also be sweet enough though not perfectly in tune.
>
> So what about 300 cents and 400 cents? They aren't usable as melodic intervals?
>
> -Mike
>

Gene,

apologies for being abrupt in my last reply.

(1) What associations are you talking about?

(2) I tested 9/5 and 7/6 and 9/5 definitely sounded sweeter. Note that I am talking about melodic intervals, two notes played in sequence, not simultaneously. Going from 1/1 to 9/5 sounds sweeter to me than going from 1/1 to 7/6. You might want to test these again yourself. Did you have harmony, not melody, in mind when you posted your comment?

(3) I suspected that the worst melodic interval would be the midpoint of a "crow" (midway between 48.18855c and 6.7758758c) which is 27.5 cents approximately. I was surprised when I tested it that it didn't sound as terrible as I expected although it didn't sound especially good either. I was using a piano voice on my keyboard. Next I tried a few sustained voices (i.e. Church Organ and Chapel Organ) and the sourness was much more obvious. Take any tuning you like where all the notes going up from 1/1 are more than 100 cents higher than 1/1 with one exception: a note that is 27.5 cents above 1/1. Now play 1/1 paired with each note going up (play each pair in sequence) and the 27.5 cents note paired with 1/1 will always sound much worse (intolerable to my ear) in comparison to all the other notes paired with 1/1, whatever they may be (all above 100 cents). Can you find an exception to this rule?

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > My formula for a melodic interval is 2/x + 2/y where x and y are less than 256. I propose six categories of melodic intervals.
>
> Some comments:
>
> (1) I don't think names like "Super Major" or "Blue Minor" are going to work as names, bringing associations to mind you don't want.
>
> (2) I think your goodness figure doesn't work as well as Tenney height. For instance, you rate 9/5 as better than 7/6, while Tenney height rates it worse, and I think TH is right. If you want to argue that 9/5 should win because it is lower limit, then what about 11/6, which you rate as better than 9/7?
>
> (3) There's no problem with divisions of the octave with more than 24 notes to the octave, and anyway, following your logic, with more than 177 notes they should become good again.
>