Harmonics Relative Strength

Please excuse the number of lines in this post - many of them are blank.
By sending this once it will be possible to discuss parts of the
structure (such as a single octave) in future.

The following shows the strongest harmonics according to my calculations
from 48 to 69120 and shows the relative strength by the size of the
line. At various parts various scales can be found. At all parts there
is the greatest possible degree of simple whole number ratios between
strong harmonics. The vertical scale is as near to a cents scale as is
possible with the coarse resolution and the need to separate close
figures. Some interesting parts are:

48 - 96 is a clear just major scale plus two flats.
720 - 1440 is a sort of minor scale.
17280 - 34560 is a sort of blues scale.

In the region 360 - 720 or 576 - 1152 there is a sort of transition zone
between major and minor and this might be interesting for some composers
to explore the use of. This is the zone where a major third shift
happens in the key (i.e a 5 ratio exists between the strongest
harmonics).

Of course in the upper region the large numbers will generally be able
to be reduced by large factors (e.g. 12, 24, 60, 144 etc) to give more
familiar ratios.

The whole structure has many subtleties such as the gradual change of
relative strength of the 80/81 ratios as we go up octaves. The switch
over happens for octaves of 80/81 between 320/(324) and (640)/648.
Similar things happen for ratios of 63/64 and 35/36 at various places.
Music that explores these parts will clearly need to have both the notes
in these pairs and to be sure which one it is using on each occasions to
avoid the drift that Paul refers to.

Both otonal and utonal relationships are visible and the emphasis
changes slightly at different parts. The very strongest harmonics are
12, 24, 288, 2880 and 343560 and ratios will work upwards and downwards
from these ones especially. That is why the region 1440 to 2880 is a
minor scale region as these values like to be fractions of 34560.

I believe that the popular lattice diagrams are mostly contained in this
structure (when divided by some large factor), although those involving
primes larger than 7 will be mostly beyond the end of the part I have
printed here.

The upper regions contain some scales which have probably gone
unexplored until now and so I am hoping that some composers will have a
look at them and see what happens. It may be best to work from the
bottom upwards in finding a place where the scale is not too unfamiliar
to begin with.

--------------- 48

----- 54

----- 56

---------- 60

------ 64

------------- 72

------- 80

------ 84

----- 90

------------- 96

------- 108

---- 112

----------- 120

--- 126
---- 128

--- 132

--- 140

------------- 144

------ 160

------ 168

-------- 180

----------- 192

--- 200

--------- 216

--- 224

----------- 240

----- 252
--- 256

--- 264
--- 270

--- 280

------------- 288

---- 300

----- 320
---- 324

------- 336

---------- 360

--------- 384

--- 400

---- 420

---------- 432

--- 448

----------- 480

------ 504

--- 528
----- 540

--- 560

------------ 576

----- 600

---- 640
----- 648

------- 672

------------ 720

--- 756
------- 768

--- 792
--- 800

----- 840

----------- 864

--- 900

---------- 960

------- 1008

--- 1056
-------- 1080

--- 1120

----------- 1152

----- 1200

---- 1260
--- 1280
------- 1296

------ 1344

------------- 1440

----- 1512
------ 1536

---- 1584
--- 1620

------ 1680

----------- 1728

----- 1800

---------- 1920
--- 1944

-------- 2016

--- 2112
---------- 2160

--- 2240

---------- 2304

------ 2400

------ 2520
--- 2560
-------- 2592
--- 2640
------ 2688
--- 2700

-------------- 2880

------ 3024
----- 3072

---- 3168
------ 3240

------- 3360

----------- 3456

------- 3600

--- 3780
--------- 3840
---- 3888
--- 3960
-------- 4032

--- 4200
--- 4224
------------ 4320

--- 4536
-------- 4608

--- 4752
------ 4800

-------- 5040

--------- 5184
--- 5280
----- 5376
----- 5400

-------------- 5760

------- 6048
---- 6144

---- 6336
-------- 6480

------- 6720

---------- 6912

-------- 7200

----- 7560
-------- 7680
----- 7776
---- 7920
-------- 8064

--- 8400

-------------- 8640

----- 9072
------- 9216

---- 9504
------ 9600
---- 9720

--------- 10080

--------- 10368
---- 10560
---- 10752
------- 10800

------------- 11520

-------- 12096
--- 12288

--- 12600
---- 12672
---------- 12960

------- 13440

---------- 13824
--- 14112
--------- 14400

-------- 15120
------- 15360
------ 15552
----- 15840
-------- 16128
--- 16200

---- 16800

--------------- 17280

--- 18000
------ 18144
------ 18432

---- 19008
------ 19200
------ 19440

----------- 20160

---------- 20736
---- 21120
---- 21504
--------- 21600

--- 22176
---- 22680
------------ 23040
--- 23328
---- 23760
--- 24000
--------- 24192

----- 25200
---- 25344
------------ 25920

------- 26880
--- 27216
--------- 27648
--- 28224
--- 28512
---------- 28800

---------- 30240
------ 30720
------- 31104
----- 31680
-------- 32256
------ 32400

----- 33600

---------------- 34560
--- 35280
---- 36000
-------- 36288
----- 36864

--- 37800
----- 38016
------ 38400
-------- 38880

------------ 40320

---------- 41472
--- 42240
--- 42336
--- 43008
----------- 43200

--- 44352
------ 45360
------------ 46080
---- 46656
--- 47040
----- 47520
--- 48000
--------- 48384

------- 50400
---- 50688
-------------- 51840

------- 53760
--- 54000
----- 54432
-------- 55296
--- 55440
---- 56448
---- 57024
---------- 57600
---- 58320

------------ 60480
----- 61440
-------- 62208
------ 63360
------- 64512
-------- 64800

--- 66528
----- 67200
--- 68040
---------------- 69120

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
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