Just Hammond: Difference between revisions
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This article features just intervals created by the mechanical tonegenerator of the classical Hammond B-3 Organ model. | This article features just intervals created by the mechanical tonegenerator of the classical Hammond B-3 Organ model. | ||
==Design of the Hammond B-3’s Tonegenerator== | ==Design of the Hammond B-3’s Tonegenerator== | ||
Since 1935 the Hammond Organ Company’s goal was to market electromechanical organs<ref>Webressource https://en.wikipedia.org/wiki/Hammond_organ (retrieved December 2019)</ref> with 12-tone equally tempered (12edo) tuning. The mechanical tonegenerator of the Hammond B-3 Organ is based on a set of | Since 1935 the Hammond Organ Company’s goal was to market electromechanical organs<ref>Webressource https://en.wikipedia.org/wiki/Hammond_organ (retrieved December 2019)</ref> with 12-tone equally tempered (12edo) tuning. The mechanical tonegenerator of the Hammond B-3 Organ is based on a set of twelve ''different pairings'' of gearwheels that make (12*4) ''driven'' shafts turn. The corresponding ''driving'' gearwheels are mounted on a common shaft and turn all at the same rotational speed ''n<sub>1</sub>''. Certain gears reduce, others increase rotational speed.<ref>Detailed photos of a similar M-1 tonegenerator are provided by https://modularsynthesis.com/hammond/m3/m3.htm (retrieved December 2019)</ref> | ||
For every chromatic pitch class four driven shafts are installed. Pure octaves are generated by dedicated ''tonewheels'' (with 2, 4, 8, 16, 32, 64 or 128 high and low points on their edges) that rotate with the driven shafts. Each high point on a tone wheel is called a ''tooth''. When the gears are in motion, magnetic pickups react to the tonewheels’ passing teeth and generate an electric signal that can be amplified and transmitted to a loudspeaker. | |||
For each pair of gearwheels the ratio of rotational speed ''n<sub>2</sub>''/''n<sub>1</sub>'' is determined by the inverse ratio of the gearwheels’ integer teeth numbers Z<sub>1</sub> and Z<sub>2</sub>: | For each pair of gearwheels the ratio of rotational speed ''n<sub>2</sub>''/''n<sub>1</sub>'' is determined by the inverse ratio of the gearwheels’ integer teeth numbers Z<sub>1</sub> and Z<sub>2</sub>: | ||
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<math>\frac{Z_1}{Z_2}=\frac{n_2}{n_1}</math> | <math>\frac{Z_1}{Z_2}=\frac{n_2}{n_1}</math> | ||
To calculate the rotational speed ''n<sub>2</sub>'' of the driven | To calculate the rotational speed ''n<sub>2</sub>'' of the driven shafts we write | ||
<math>n_2=\frac{Z_1}{Z_2}\cdot n_1</math> | <math>n_2=\frac{Z_1}{Z_2}\cdot n_1</math> | ||
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|- | |- | ||
| style="text-align: center;" | driving | | style="text-align: center;" | driving | ||
Z<sub>1</sub>[teeth] | |||
| style="text-align: center;" | driven | | style="text-align: center;" | driven | ||
Z<sub>2</sub>[teeth] | |||
| colspan="2" style="text-align: center;" | <br>Fraction | | colspan="2" style="text-align: center;" | <br>Fraction | ||
| style="text-align: center;" | Ratio | | style="text-align: center;" | Ratio | ||
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== Just Intervals == | == Just Intervals == | ||
When we associate ''“ratios of the gearwheels’ integer teeth numbers”'' with ''“frequency ratios between partials”'' we realize an intrinsic ''just interval'' determined by integer teeth numbers within such mechanical gear - even without turning the shafts! Although the Hammond Organ pretends to generate a 12edo scale, the instrument in fact creates a | When we associate ''“ratios of the gearwheels’ integer teeth numbers”'' with ''“frequency ratios between partials”'' we realize an intrinsic ''just interval'' determined by integer teeth numbers within such mechanical gear - even without turning the shafts! Although the Hammond Organ pretends to generate a 12edo scale, the instrument in fact creates a ''103-prime-limit'' just scale. | ||
== Tuning == | == Tuning == | ||
The whole set of frequency ''ratios'' is fixed by the design of the gear mechanism. The driving shaft’s (A) rotational speed ''n<sub>1</sub>'' determines the instrument’s (master-)tuning. Rotating at exactly 1200 rpm ( | The whole set of frequency ''ratios'' is fixed by the design of the gear mechanism. The driving shaft’s (A) rotational speed ''n<sub>1</sub>'' determines the instrument’s (master-) tuning. Rotating at exactly 1200 rpm (20 rev./sec), the pitch of note A equals precisely 27.500 Hz or one of its doublings. Therefore the instrument aligns note A with a concert pitch of 440.0 Hz. | ||
<math>f_\text{A}=20.0/\text{s}\cdot\frac{88}{64}\cdot(2^4)=440.0/\text{s} = 440.0 \text{ Hz}</math> | <math>f_\text{A}=20.0/\text{s}\cdot\frac{88}{64}\cdot(2^4)=440.0/\text{s} = 440.0 \text{ Hz}</math> | ||
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| style="text-align: center;" | | | style="text-align: center;" | | ||
8.6<br><br> | 8.6<br><br> | ||
Decimal printed<br> | |||
for orientation only | for orientation only | ||
|} | |} | ||
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(LCM)<br><br> | (LCM)<br><br> | ||
| style="text-align: center;" | | | style="text-align: center;" | | ||
45.1<br> | 45.1<br> | ||
Decimal printed<br> | Decimal printed<br> | ||
for orientation only | for orientation<br> | ||
only | |||
|} | |} | ||
== Discussion | == Discussion == | ||
No doubt - the evidence that a cluster of 12 simultaneously ringing semitones from a Hammond Organ is allocated around the 45<sup>th</sup> octave of the harmonic series is of limited practical | No doubt - the evidence that a cluster of 12 simultaneously ringing semitones from a Hammond Organ is allocated around the 45<sup>th</sup> octave of the harmonic series is of limited practical value. The intervals' ''far-up placement'' is mainly caused by Laurens Hammond’s implementation of various prime numbers (11, 13, 23, 37, 41, 73) in different gearwheel pairings. | ||
* Respective high-order partials are very densely spaced (in the range of ''pico-cents)'' and intervals between successive partials up there are too narrow for musical applications by far | |||
* Due to its construction the tonegenerator selects only twelve from 17.6 trillion varieties in the 45<sup>th</sup> octave where… | |||
** the partial number associated with the LCM, which is located exactly 8/11 below pitch class A, is not addressed because there is no gear with transmission ratio 1.000 | |||
** no pure octave above a virtual root (1/1; partial# (2<sup>44</sup>)) is playable, which would ring -624.997 cents way down from pitchclass A | |||
== General Applicability == | |||
The method of prime factorization to find the [[Least common multiple|LCM]] can be applied to arbitrary '''intervals, chords or scales built from rational intervals''' to identify their position in the harmonic series. Simply replace the gear-ratios by just intervals of interest. | |||
==References== | ==References== | ||
<references /> | <references /> | ||
== See also… == | == See also… == | ||
- Dismantling the tonegenarator of a scrapped H-Series Hammond Organ [8:47 min]<br> | |||
https://www.youtube.com/watch?v=7Qqmr6IiFLE <br> | |||
<nowiki>- An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min] </nowiki><br>https://www.youtube.com/watch?v=5CG81_Y8SvY | |||
* @ 4:48 min: ''“…these sounds are in there”'' | |||
* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”'' | |||
[[Category:Lists of intervals]] | |||
[[Category:Instruments]] | |||
[[Category:Organ]] | |||