Glass Organ: Tuning Part 2

Layout and Tuning as of March 2001

I noticed when I converted the ratios of my scale to cents that I had a 182 cent step between the 1/1 and the 10/9. After playing the organ for a while, I found the consonances plentiful and sweet, yet harsh dissonances were available as well - the clash of two notes separated by less than a half step is always jarring. Yet I began to yearn for a different quality of consonance/dissonance. The sound I was after was a gradual extension of a consonance to the point of dissonance. An effect similar to modulation.

First I analyzed the scale and found that certain intervals occurred statistically often between scale steps: 21/20 and 35/18. I built two organs with that configuration but was never really happy with it. Recently, I added a third note and I think I have just what I want. Here's how I decided on the final notes.

28/15
G# - 19

9/5
G + 18

16/9
G - 4

7/4
G - 31

12/7
F# + 33

5/3
F# - 16

40/21
G# + 15

2/1
A

8/5
F + 14

14/9
F - 35

3/2
E + 2

10/7
D# + 17

36/35
A + 49

35/18

9/7
C# + 35

4/3
D - 2

7/5
D# - 17

1/1
A

5/4
C# - 14

21/20
A# - 15

9/8
B + 4

8/7
B + 31

7/6
C - 33

6/5
C + 16

10/9
B - 18

15/14
A# + 19

The key to choosing good filler notes for my scale was that ratios with common factors in their numerators or denominators are generally consonant together. Since 3, 6, 9, and 12 have a common factor of 3, my basic scale was 3-heavy. Meaning that there were many more ratios with a factor of 3 in them than a factor of 2, 5, or 7. This is because 9 is not prime (3 × 3 = 9). I wanted to fill the gaps in my scale with notes that had factors of 2, 5, or 7, but not 3. To find the notes I wanted, I extended the tonality diamond a good bit, and looked for some higher non-prime numbers. Eventually, I found some by multiplying the factors I wanted. 7 × 5 = 35, 7 × 3 = 21, 4 × 5 = 20, 3 × 5 = 15.

Big Tonality Diamond

UNDER CONSTRUCTION

13 prime-limit, with reducing ratios removed.

	2	3	5	7	9	11	13	15	21	25	27	33	35
2	1/1 A	3/2 E + 2	5/4 C# - 14	7/4 G - 31	9/8 B + 4	11/8 D# - 49	13/8	15/8	21/16 D - 29	25/16	27/16	33/32 A# - 27	35/32 B - 45
3	4/3 D - 2		5/3 F# - 16	7/6 C - 33		11/6	13/12 A# + 39			25/24 A# - 29			35/24 E - 47
5	8/5 F + 14	6/5 C + 16		7/5 D# - 17	9/5 G + 18	11/10 B - 35	13/10 D - 46		21/20 A# - 15		27/20 D + 20	33/20
7	8/7 B + 31	12/7 F# + 33	10/7 D# + 17		9/7 C# + 35	11/7	13/7	15/14 A# + 19		25/14	27/14	33/28 C - 16
9	16/9 G - 4		10/9 B - 18	14/9 F - 35		11/9 C + 47	13/9 D# + 37			25/18 D# - 31			35/18
11	16/11 D# + 49	12/11 B - 49	20/11	14/11 C# + 18	18/11		13/11 C - 11	15/11 D + 37	21/11	25/22 B + 21	27/22 C# - 45		35/22
13	16/13 C# - 40	24/13	20/13	14/13 A# + 28	18/13 D# - 37	22/13		15/13 B + 48	21/13	25/13	27/26 A# - 35	33/26 C# + 13	35/26 D + 15
15	16/15 A# + 12			28/15		22/15 E - 37	26/15
21	32/21 E + 29		40/21			22/21 A# - 19	26/21 C# - 30			25/21 C + 2
25	32/25 C# + 27	48/25		28/25 B - 14	36/25 D# + 31	44/25	26/25 A# - 32		42/25		27/25 A# + 33	33/25 D - 19
27	32/27 C - 6		40/27 E - 20	28/27 A# - 37		44/27	52/27			50/27			35/27 C# + 49
33	64/33		40/33 C + 33	56/33			52/33			50/33 E + 19			35/33 A# + 2
35	64/35	48/35 D + 47			36/35 A + 49	44/35 C# - 4	52/35 E - 15				54/35	66/35

Layout and Tuning as of May 2001

The glasses get filthy if water is left in them and emptying them is time consuming. I wanted to be able to dump the whole instrument out over a bathtub. That instrument proved too big to fit through doorways and difficult to lift. Also, I wasn't using 36/35 or 35/18 very much so I removed them and shrunk the organ.