# Almond Organ using Fundamental Modules

Just made a short series of videos exploring some additive synthesis techniques with the fundamental modules.

Saw some users asing questions about the tuning and setting of harmonics that the hammond uses. I actually don’t know much about the specific design of the hammond, but if it behaves according to the laws of acoustics … then it’s useful to think about (at least) two different aspects that combine make the instrumental characteristics:

A choir (of registers) will be in tune to some temperament. In the case of the fundamental modules this temperament happens to be 12TET.

Discrete registers are set or built according to the principle of the harmonic series. So, that means that when employing certain registers (duodecime, duotessaron, etc…) there will be some very slight deviation from equal tempered tuning.

You can find exact tuning data for hammonds on the internet. It’s close to even tempered. I myself can’t hear the difference, but I’m sure someone can

``````Obviously the haromonics are just g(x)
where x is the partial number and g is the fundamental.

to calculate 12TET

**fn = f1 * pow(q, (n-1))**

| n   | number of the wanted tone                     |
|-----|-----------------------------------------------|
| fn  | frequency of the wanted tone                  |
| f1  | frequency of the reference tone               |
| q   | 2 ^ ( 1 / 12 ) when the wanted tone is higher |
| q   | ½ ^ (1 / 12 ) when the wanted tone is lower   |
| n-1 | means the number of the wanted tone -1        |

You can see the deviations in the table below
a4 = 442Hz as a reference
a2 = 221Hz is the fundamental for the different registers
so g(1) = a2 = 221Hz

``````
Midi no midi name ET-freq tone number(f) Pure-freq Partial No.
21 A0 27.625 49
22 Bf0 29.2676679816755 48
23 B0 31.0080140845464 47
24 C1 32.8518465519502 46
25 Cs1 34.8053190033459 45
26 D1 36.8749509714472 44
27 Ds1 39.0676496605567 43
28 E1 41.3907329987183 42
29 F1 43.8519540606215 41
30 Fs1 46.4595269427677 40
31 G1 49.2221541772538 39
32 Gs1 52.1490557756636 38
33 A1 55.25 37
34 Bf1 58.5353359633511 36
35 B1 62.0160281690928 35
36 C2 65.7036931039003 34
37 Cs2 69.6106380066918 33
38 D2 73.7499019428944 32
39 Ds2 78.1352993211135 31
40 E2 82.7814659974367 30
41 F2 87.703908121243 29
42 Fs2 92.9190538855355 28
43 G2 98.4443083545075 27
44 Gs2 104.298111551327 26
45 A2 110.5 25
46 Bf2 117.070671926702 24
47 B2 124.032056338186 23
48 C3 131.407386207801 22
49 Cs3 139.221276013383 21
50 D3 147.499803885789 20
51 Ds3 156.270598642227 19
52 E3 165.562931994873 18
53 F3 175.407816242486 17
54 Fs3 185.838107771071 16
55 G3 196.888616709015 15
56 Gs3 208.596223102654 14
57 A3 221 13 221 1
58 Bf3 234.141343853404 12
59 B3 248.064112676371 11
60 C4 262.814772415601 10
61 Cs4 278.442552026767 9
62 D4 294.999607771578 8
63 Ds4 312.541197284454 7
64 E4 331.125863989747 6
65 F4 350.815632484972 5
66 Fs4 371.676215542142 4
67 G4 393.77723341803 3
68 Gs4 417.192446205309 2
69 A4 442 1 442 2
70 Bf4 468.282687706809 2
71 B4 496.128225352743 3
72 C5 525.629544831203 4
73 Cs5 556.885104053534 5
74 D5 589.999215543155 6
75 Ds5 625.082394568908 7
76 E5 662.251727979493 8 663 3
77 F5 701.631264969944 9
78 Fs5 743.352431084284 10
79 G5 787.55446683606 11
80 Gs5 834.384892410617 12
81 A5 884 13 884 4
82 Bf5 936.565375413617 14
83 B5 992.256450705486 15
84 C6 1051.25908966241 16
85 Cs6 1113.77020810707 17
86 D6 1179.99843108631 18
87 Ds6 1250.16478913782 19
88 E6 1324.50345595899 20 1326 6
89 F6 1403.26252993989 21
90 Fs6 1486.70486216857 22
91 G6 1575.10893367212 23
92 Gs6 1668.76978482123 24
93 A6 1768 25 1768 8
94 Bf6 1873.13075082723 26
95 B6 1984.51290141097 27
96 C7 2102.51817932481 28
97 Cs7 2227.54041621414 29 2210 10
98 D7 2359.99686217262 30
99 Ds7 2500.32957827563 31
100 E7 2649.00691191797 32 2652 12
101 F7 2806.52505987978 33
102 Fs7 2973.40972433713 34
103 G7 3150.21786734424 35
104 Gs7 3337.53956964247 36
105 A7 3536 37
106 Bf7 3746.26150165447 38
107 B7 3969.02580282194 39
108 C8 4205.03635864962 40
1 Like

I’m confused here. Are you explaining in detail even temperament? Because a Hammond is not strictly even tempered, right? Confused…

From what I just read it’s as close as they could accomodate with mechanical cogs that needed to have a whole number of teeth. When discussed further, it seems like it only makes one cent of difference if I read correctly. More the discussion is the even temperament - if input in to Cheby is even-tempered will the harmonics out be even tempered? I may be asking a completely dumb question.

Yes, they tried to get it very close, and it is very close.

Not a dumb question at all. And, no, the outputs of Chebyshev are not even tempered at all. Waveshapers (of which Chebyshev is one) can only generate perfect integer multiples of what you put into them. So the outputs are exactly 1x, 2x, 3x, 4x, 5x, etc… Some of those perfect harmonics are pretty far from even tempered.

Which begs the question - a) are the harmonics in the Hammond even tempered and b) can you make an even tempered Chebyshev?

The Chebyshev functions will not do that. I did make a fake Hammond once with 128 sin generators tuned exactly to the Hammond pitches. It was ok.

okay, apologies - I haven’t investigated in detail the mechanics behind the hammond and how the tonwheel system is working, I also can’t ever remember actually playing a hammond. So, my almond organ is very much a design built to illustrate a number of things: