JP80x0 midi to coefficient mapping tables

Pitch

Midi	Pitch
0	1555
1	1647
2	1745
3	1849
4	1959
5	2075
6	2198
7	2329
8	2468
9	2615
10	2770
11	2935
12	3110
13	3295
14	3491
15	3698
16	3918
17	4151
18	4397
19	4658
20	4936
21	5230
22	5541
23	5871
24	6220
25	6590
26	6982
27	7396
28	7836
29	8302
30	8794
31	9317
32	9873
33	10460
34	11082
35	11742
36	12441
37	13181
38	13965
39	14792
40	15672
41	16604
42	17589
43	18635
44	19747
45	20921
46	22164
47	23485
48	24883
49	26362
50	27930
51	29585
52	31345
53	33209
54	35179
55	37270
56	39494
57	41842
58	44329
59	46970
60	49767
61	52725
62	55860
63	59171
64	62691
65	66418
66	70358
67	74541
68	78988
69	83684
70	88659
71	93941
72	99534
73	105450
74	111720
75	118342
76	125382
77	132836
78	140716
79	149082
80	157976
81	167368
82	177318
83	187882
84	199068
85	210900
86	223440
87	236684
88	250764
89	265672
90	281432
91	298164
92	315952
93	334736
94	354636
95	375764
96	398136
97	421800
98	446880
99	473368
100	501528
101	531344
102	562864
103	596328
104	631904
105	669472
106	709272
107	751528
108	796272
109	843600
110	893760
111	946736
112	1003056
113	1062688
114	1125728
115	1192656
116	1263808
117	1338944

 

Detune:

Midi	uC
0	1
1	1
2	2
3	2
4	3
5	3
6	4
7	4
8	5
9	5
10	6
11	6
12	7
13	7
14	8
15	8
16	9
17	9
18	10
19	10
20	11
21	11
22	12
23	12
24	13
25	13
26	14
27	14
28	15
29	15
30	16
31	16
32	17
33	17
34	18
35	18
36	19
37	19
38	20
39	20
40	21
41	21
42	22
43	22
44	23
45	23
46	24
47	24
48	25
49	25
50	26
51	26
52	27
53	27
54	28
55	28
56	29
57	29
58	30
59	30
60	31
61	31
62	32
63	32
64	33
65	34
66	35
67	36
68	37
69	38
70	39
71	40
72	41
73	42
74	43
75	44
76	45
77	46
78	47
79	48
80	49
81	51
82	53
83	55
84	57
85	59
86	61
87	63
88	65
89	67
90	69
91	71
92	73
93	75
94	77
95	79
96	81
97	83
98	85
99	87
100	89
101	91
102	93
103	95
104	97
105	99
106	101
107	103
108	105
109	107
110	109
111	111
112	113
113	115
114	117
115	119
116	121
117	123
118	125
119	127
120	129
121	137
122	145
123	153
124	169
125	193
126	225
127	321

 

Mix

Midi	Mix
0	102400
1	118784
2	135168
3	151552
4	167936
5	184320
6	200704
7	217088
8	233472
9	249856
10	266240
11	282624
12	299008
13	315392
14	331776
15	348160
16	364544
17	380928
18	397312
19	413696
20	430080
21	446464
22	462848
23	479232
24	495616
25	512000
26	528384
27	544768
28	561152
29	577536
30	593920
31	610304
32	626688
33	643072
34	659456
35	675840
36	692224
37	708608
38	724992
39	741376
40	757760
41	774144
42	790528
43	806912
44	823296
45	839680
46	856064
47	872448
48	888832
49	905216
50	921600
51	937984
52	954368
53	970752
54	987136
55	1003520
56	1019904
57	1036288
58	1052672
59	1069056
60	1085440
61	1101824
62	1118208
63	1134592
64	1150976
65	1167360
66	1183744
67	1200128
68	1216512
69	1232896
70	1249280
71	1265664
72	1282048
73	1298432
74	1314816
75	1331200
76	1347584
77	1363968
78	1380352
79	1396736
80	1413120
81	1429504
82	1445888
83	1462272
84	1478656
85	1495040
86	1511424
87	1527808
88	1544192
89	1560576
90	1576960
91	1593344
92	1609728
93	1626112
94	1642496
95	1658880
96	1675264
97	1691648
98	1708032
99	1724416
100	1740800
101	1757184
102	1773568
103	1789952
104	1806336
105	1822720
106	1839104
107	1855488
108	1871872
109	1888256
110	1904640
111	1921024
112	1937408
113	1953792
114	1970176
115	1986560
116	2002944
117	2019328
118	2035712
119	2052096
120	2068480
121	2084864
122	2101248
123	2117632
124	2134016
125	2150400
126	2166784
127	2183168

 

Mix and oscillator amplitudes, more research

I just can't let this one go. The code appears to add a linear amount of the side oscillators to the sum, but Adam Szabo says differently - the center oscillator is attenuated and the outer ones follow a curved gain.

I just had a happy accident. I am trying to find the contribution of each oscillator to the total if one normalize the sum - saying the total should always be 1.

However, I only added one single outer oscillator - but the output was very interesting:

Here it is compared to the graph in the article:

 

Here, the value of oscillator 1 is 1 / (1 + mix), whereas the plot for the others is mix / (1 + mix). The plots are quite similar! It really makes me want to understand this even more!

But - if I assume that ALL 6 outer oscillators should be included in the normalization, everything breaks down, so clearly that's not correct.

Going back to the article, we have this graph (Figure 9):

Note that the amplitudes of 5, 6 and 7 are higher than 1, 2, 3.

In the text, Szabo says that 2, 3, 5, 6 and 7 are removed, leaving 1 and 4, as illustrated in the first graph.

I don't know if he DID measure those too, but there is a chance that they don't follow the exact same curve. We'll see if we can figure that one out.

Also, let's renumber the spikes in the plot to match the order in the detune_table:

[0, 318, -318, 1020, -1029, 1760, -1800], let's call them A-G to keep them separated

That gives us:

A = 4, B = 5, C = 3, D = 6, E = 2, F = 7, G = 1

In case it matters, what is called 1 here is actually the last element added in the summing in the ESP code. 

 

Now, I'm not entirely sure how to interpret Figure 9 in terms of "max wave amplitude". The spectrum has peaks of a certain width, not just a single frequency, and there are no units on the Y axis. If the scale is linear and one assumes that the amplitude of each wave in the result is actually propotional to the max value of the center oscillator, we get the numbers in the table (each pure frequency is a sine wave, so the highest peak would correspond to the root frequency of the saw waves, wouldn't it)?

It looks like that's what Szabo means that they are, so let's accept that.

If so, the sum of all amplitudes is definitely not 1. Could the perceived total "loudness" be equal if one uses dB instead of a linear scale? The total energy or something?  

And in any case, how does one go from the sum += saw[i] * mix to this thing? 

My own measurements

I wanted to confirm my understanding of Szabo's graph, so I fired up JE-8086 and coded a spectrum analyzer in web audio. I fed the audio from JE-8086 back to the input of my mac using the virtual microphone/input "VB-cable".

I used a linear Y-axis and a logarithmic X-axis. Then, with detuning at max, for every line on the mix pot (11 in total) I screenshot'ed the spectrum. Finally, I went through every graph, measuring the height in pixels.

Here is the first and last spectrum plots:

The measurements were of course wildly inaccurate, and the spacing between the mix values not quite even as I couldn't see the slider value in the display (and also, the slider resolution wasn't high enough). 

Here are the values:

And, more importantly, the plot of the values relative to max value of the center oscillator:

 

This is indeed very cool. It does confirm most of what Szabo described - the center oscillator is fairly linear and the others are definitely curved, and the center oscillator ends at an amplitude lower than the outer ones. There are some small differences though:

- I don't get the feeling that the outer oscillators actually go DOWN in amplitude at the end. The 7th oscillator appears to go slightly  down again, but I think it's more likely a measuring error. 

- The higher pitch / right hand oscillators have a higher amplitude than the lower ones. This matches what can be seen in Figure 9 in Szabo. The top oscillator is even higher though, and that does not match. I am still not sure if this is an artifact of the spectrum analyzer or if it is real. It could be an artifact of approximate multiplication or running average or something. 

- Something else to note: Both in mine and Szabo's spectrum analyzers, the minimum amplitudes for the outer oscillators is not zero. If the summing is actually saw[i] * spread, there should be no trace of the oscillator if spread is 0. Very strange. Also - why do they call it spread and not mix?