Just Hammond: Difference between revisions
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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 high prime limit just scale. | 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 ''high 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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== 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 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] https://www.youtube.com/watch?v=7Qqmr6IiFLE <nowiki>- An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min] </nowiki>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”'' | |||