Visualizing Harmonics

by
Benjamin Tubb
1/3/97
[rev. 11/4/01]
brtubb@pdmusic.org

Hammond Organ Drawbar Conversions

The convention of using Pipe Organ stop footages has been applied to the following chart where 8' is equivalent to sounding the pitch of the key played, 16' would play an octave below, 4' would play one octave above, etc.

Converting Hammond drawbar values to harmonic series - three "base" examples:
Drawbar #s			 Base 4'		8'		16'
1 = 16'								1	
2 = 5 1/3	'							3
3 = 8'						1		2
4 = 4'				1		2		4
5 = 2 2/3'						3		6
6 = 2'				2		4		8
7 = 1 3/5'						5		10
8 = 1 1/3'				3		6		12
9 = 1'				4		8		16
10 = 1 1/7' and 8/9'				7,9		14,18
11 = 4/5' and 2/3'				10,12		20,24

Drawbar #s where A=10, B=11, converted to harmonic placeholders
Base 16' -> 12 3456 789 AB = 1324050607080A090A0B000B [24 harmonics]
Base  8' -> 00 3456 789 AB = 345678A9AB0B [12 harmonics]
Base  4' -> 00 0406 080 A0 = 468A [4 harmonics]

Drawbar values may range from 0 to 9. All "normal" settings will at least include one of the three "base" drawbars: 1, 3, or 4. If they do not then Base 16' will be assumed since all drawbars can be defined in relation to it.

Function Definitions

A 16 Harmonics Plot Function

A function to produce a 2D plot for the first 16 harmonics of any fundamental.

\(harmonicsPlot[H1_, H2_, H3_, H4_, H5_, H6_, H7_, H8_, H9_, H10_, H11_, 
        H12_, H13_, H14_, H15_, H16_] := 
	
      Plot[H1*Sin[x] + H2*Sin[2*x] + H3*Sin[3*x] + H4*Sin[4*x] + 
          H5*Sin[5*x] + H6*Sin[6*x] + H7*Sin[7*x] + H8*Sin[8*x] + 
          H9*Sin[9*x] + H10*Sin[10*x] + H11*Sin[11*x] + H12*Sin[12*x] + 
          H13*Sin[13*x] + H14*Sin[14*x] + H15*Sin[15*x] + H16*Sin[16*x], {x, 
          0, Pi}]\)

A 24 Harmonics Organ Drawbar Plot Function

A function for a 2D plot which automatically converts an 11 drawbar setting of the form "12 3456 789 AB" (with the included spaces) into the proper correspondence for coefficients of harmonics based on 16' as the fundamental.
Again: Base 16' -> 12 3456 789 AB = 1324050607080A090A0B000B [24 harmonics]

\(\(drawbarHarmonics[Drawbars_String] := 
      Module[{h1, h2, h3, h4, h6, h8, h10, h12, h14, h16, h18, h20, h24}, 
	
        h1 := ToExpression[StringTake[Drawbars, {1}]]; 
	
        h2 := ToExpression[StringTake[Drawbars, {4}]]; 
	
        h3 := ToExpression[StringTake[Drawbars, {2}]]; 
	
        h4 := ToExpression[StringTake[Drawbars, {5}]]; 
	
        h6 := ToExpression[StringTake[Drawbars, {6}]]; 
	
        h8 := ToExpression[StringTake[Drawbars, {7}]]; 
	
        h10 := ToExpression[StringTake[Drawbars, {9}]]; 
	
        h12 := ToExpression[StringTake[Drawbars, {10}]]; 
	
        h14 := ToExpression[StringTake[Drawbars, {13}]]; 
	
        h16 := ToExpression[StringTake[Drawbars, {11}]]; 
	h18 := h14; \ \ 
        h20 := ToExpression[StringTake[Drawbars, {13}]]; 
	h24\  := h20; 

        Plot[h1*Sin[x] + h2*Sin[x*2] + h3*Sin[x*3] + h4*Sin[x*4] + 
            h6*Sin[x*6] + h8*Sin[x*8] + h10*Sin[x*10] + h12*Sin[x*12] + 
            h14*Sin[x*14] + h16*Sin[x*16] + h18*Sin[x*18] + h20*Sin[x*20] + 
            h24*Sin[x*24], {x, 0, Pi}]]
	\)\)

The Foundation Stops (including Diapasons)

Diapason 8'

drawbarHarmonics["00 6744 010 10"];

The Flute Stops

Bourdon (big-scale) 8'

drawbarHarmonics["00 7010 000 00"];

The String Stops

Aoline 8'

drawbarHarmonics["00 1133 111 11"];

'Cello 8'

drawbarHarmonics["00 3462 240 11"];

The Reed Stops

Bassoon 8'

drawbarHarmonics["00 0360 321 10"];

The Percussion Stops

Banjo 8'

drawbarHarmonics["00 1356 662 00"];

Bells 8'

drawbarHarmonics["00 2003 002 00"];

The Full Organs

Full Great

drawbarHarmonics["52 8888 542 00"];

Full Accompanimental

drawbarHarmonics["31 6666 543 00"];

Full Choir

drawbarHarmonics["52 7777 753 00"];

Novelty-Type Combinations

These are represented by non-family oriented harmonic settings including those that don't even have a "reference" to a fundamental defined such as "02 0052 073 76".

No.  1

drawbarHarmonics["55 0000 000 00"];

No. 35

drawbarHarmonics["00 0000 505 00"];

No. 70

drawbarHarmonics["00 8800 003 00"];

Reference

Irwin, Stevens, "Dictionary of Hammond Organ Stops: A Translation of Pipe-Organ Stops into Hammond Organ Number Arrangements - An Introduction to Playing the Hammond Organ", Schirmer Books, Revised Fourth Edition 1970.