Just Hammond: Difference between revisions

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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 ''n1'' 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.

== 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~~

== Mapping the Hammond’s Rational Intervals to the Harmonic Series ==

To find out, where the rational intervals played on a Hammond Organ occur in the harmonic series we

* 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

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== Numbering Octaves ==

We apply the scheme ~~used~~ from the article ''[[First Five Octaves of the Harmonic Series]]'' and number the octaves as follows:

We apply the scheme 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

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* The '''4th''' octave starts at partial #8 (= 23) 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 nth octave contains 2n-1 pitches''”. [4]

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 nth octave contains 2n-1 pitches''”.[4]

@@ Line 195: / Line 195: @@
 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.
-== Mapping the Hammond Organ’s Rational Intervals to the Harmonic Series ==
+<math>f_A=20.0/sec\cdot\frac{88}{64}\cdot(2^4)=440.0/sec = 440.0 Hz</math>
-To find out, where the rational intervals played on a Hammond occur in the harmonic series, we have to
+== Mapping the Hammond’s Rational Intervals to the Harmonic Series ==
+To find out, where the rational intervals played on a Hammond Organ occur in the harmonic series we
 * 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
@@ Line 203: / Line 205: @@
 == Numbering Octaves ==
-We apply the scheme used from the article ''[[First Five Octaves of the Harmonic Series]]'' and number the octaves as follows:
+We apply the scheme 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
@@ Line 210: / Line 212: @@
 * 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>
+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>