Just Hammond: Difference between revisions
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Table 1: Pairings of Gearwheels<ref>Gearing details were taken from http://www.goodeveca.net/RotorOrgan/ToneWheelSpec.html (retrieved Dec 29, 2019) | Table 1: Pairings of Gearwheels<ref>Gearing details were taken from http://www.goodeveca.net/RotorOrgan/ToneWheelSpec.html (retrieved Dec 29, 2019) | ||
The German Wikipedia provides the same technical information (in German): https://de.wikipedia.org/wiki/Hammondorgel#Tonerzeugung (retrieved Dec 29, 2019) | The German Wikipedia provides the same technical information (in German): https://de.wikipedia.org/wiki/Hammondorgel#Tonerzeugung (retrieved Dec 29, 2019) | ||
The ''HammondWiki'' publishes a second, alternative set of gear ratios with slightly deviating pitch class “E”. Certain other pitch classes are shifted by pure octaves. http://www.dairiki.org/HammondWiki/GearRatio (retrieved Dec 29, 2019)</ref> / Ratios and Intervals | The ''HammondWiki'' publishes a second, alternative set of gear ratios with slightly deviating pitch class “E”. Certain other pitch classes are shifted by pure octaves. http://www.dairiki.org/HammondWiki/GearRatio (retrieved Dec 29, 2019)</ref> / Ratios and Intervals | ||
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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 (which equals 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. | 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 (which equals 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_A=20.0/sec\cdot\frac{88}{64}\cdot(2^4)=440.0/sec = 440.0 Hz</math> | == Mapping the Hammond Organ’s Rational Intervals to the Harmonic Series == | ||
To find out, where the rational intervals played on a Hammond occur in the harmonic series, we have to | |||
* cancel the fractions of gear-ratios specified by Hammond and | |||
* calculate the ''[[Least common multiple|least common multiple (LCM)]]'' of the denominators of "''intervals of interest"'' by prime factorization | |||
* With this specific LCM we recalculate the numerators of the intervals. The resulting numerators correspond to the partial numbers we are looking for. | |||
Before we proceed, we have to agree on a numbering scheme for octaves in the harmonic series. | |||
== Numbering Octaves == | |||
We apply the scheme used from the article ''[[First Five Octaves of the Harmonic Series]]'' and number the octaves as follows: | |||
* Integer octave numbering starts with '''#1''' for the range between the 1st and < 2nd partial | |||
* The '''2nd''' octave starts at partial #2 (= 2<sup>1</sup>) and covers partials 2 and 3 | |||
* The '''3rd''' octave starts at partial #4 (= 2<sup>2</sup>) and covers partials 4, 5, 6 and 7 | |||
* The '''4th''' octave starts at partial #8 (= 2<sup>3</sup>) and covers partials 8, 9, 10, 11, 12, 13, 14 and 15. | |||
* ... | |||
<math>f_A=20.0/sec\cdot\frac{88}{64}\cdot(2^4)=440.0/sec = 440.0 Hz</math>This numbering scheme is consistent with the scheme used by Bill Sethares<ref>Sethares, William A. ''Tuning Timbre Spectrum Scale.'' London: Springer Verlag , 1999.</ref> : “''In general, the n<sup>th</sup> octave contains 2<sup>n-1</sup> pitches''”. <sup>[4]</sup> | |||