Just Hammond - Revision history

Holger Stoltenberg: replaced "high prime limit" by "103-prime-limit"

2025-01-08T16:48:12Z

replaced "high prime limit" by "103-prime-limit"

← Older revision		Revision as of 16:48, 8 January 2025
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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 ''103-prime-limit'' just scale.

	== Tuning ==		== Tuning ==

BudjarnLambeth: Categorised this uncategorised page

2023-04-27T00:27:37Z

Categorised this uncategorised page

← Older revision		Revision as of 00:27, 27 April 2023
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	* @ 4:48 min: ''“…these sounds are in there”''		* @ 4:48 min: ''“…these sounds are in there”''
	* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''		* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''
			[[Category:Lists of intervals]]
			[[Category:Instruments]]
			[[Category:Organ]]

Holger Stoltenberg: number of driven shafts corrected

2020-01-21T16:57:07Z

number of driven shafts corrected

Holger Stoltenberg: _{in table; "See also…"-section formatted}

2020-01-13T13:08:05Z

<sub> in table; "See also…"-section formatted

← Older revision		Revision as of 13:08, 13 January 2020
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	\|-		\|-
	\| style="text-align: center;" \| driving		\| style="text-align: center;" \| driving
	Z1[teeth]		Z<sub>1</sub>[teeth]
	\| style="text-align: center;" \| driven		\| style="text-align: center;" \| driven
	Z2[teeth]		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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	== 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 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.		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		* 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

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	== 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		- 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”''		* @ 4:48 min: ''“…these sounds are in there”''
	* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''		* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''

Holger Stoltenberg: finalized sections "Discussion" and "General Applicability"

2020-01-10T17:26:05Z

finalized sections "Discussion" and "General Applicability"

← Older revision		Revision as of 17:26, 10 January 2020
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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 (~~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 (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 ~~and General Applicability~~ ==		== 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 ~~use~~. ~~Nevertheless~~ the method of prime factorization 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.		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		- 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
	* An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min] https://www.youtube.com/watch?v=5CG81_Y8SvY		* @ 4:48 min: ''“…these sounds are in there”''
	** @ 4:48 min: ''“…these sounds are in there”''		* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''
	** @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''

Holger Stoltenberg at 16:10, 6 January 2020

2020-01-06T16:10:35Z

← Older revision		Revision as of 16:10, 6 January 2020
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	\| style="text-align: center;" \|		\| style="text-align: center;" \|
	8.6<br><br>		8.6<br><br>
	(Decimal printed<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>~~<br>~~		45.1<br>

	Decimal printed<br>		Decimal printed<br>
	for orientation only		for orientation<br>
			only
	\|}		\|}

Holger Stoltenberg at 11:28, 6 January 2020

2020-01-06T11:28:03Z

← Older revision		Revision as of 11:28, 6 January 2020
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	~~The~~ 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 12 pairs of gearwheels that make twelve ''driven'' shafts turn. The corresponding twelve ''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>		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 12 pairs of gearwheels that make twelve ''driven'' shafts turn. The corresponding twelve ''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>
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	The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11<sup>th</sup> and the 12<sup>th</sup> octave.		The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11<sup>th</sup> and the 12<sup>th</sup> octave.

	==== Example 3: Mapping all of the ~~tonegenarator~~'s pitchclasses ====		==== Example 3: Mapping all of the tonegenerator's pitchclasses ====
	The full set of the Hammond Organ’s intervals resides surprisingly far up in the Harmonic Series:		The full set of the Hammond Organ’s intervals resides surprisingly far up in the Harmonic Series:
	* The '''44th''' '''octave starts''' at partial #(2<sup>43</sup>), just below the set of partials determined by the Hammond Organ’s tonegenerator		* The '''44th''' '''octave starts''' at partial #(2<sup>43</sup>), just below the set of partials determined by the Hammond Organ’s tonegenerator
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	== Discussion and General Applicability ==		== Discussion and General Applicability ==
	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 ~~relevance~~. Nevertheless the method of prime factorization 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.		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 use. Nevertheless the method of prime factorization 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]		* Dismantling the tonegenarator of a scrapped H-Series Hammond Organ [8:47 min] https://www.youtube.com/watch?v=7Qqmr6IiFLE
	https://www.youtube.com/watch?v=7Qqmr6IiFLE ~~<br>~~		* An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min] https://www.youtube.com/watch?v=5CG81_Y8SvY
	An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min]		** @ 4:48 min: ''“…these sounds are in there”''
	* @ 4:48 min: ''“…these sounds are in there”''		** @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''
	* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''
	~~https://www.youtube.com/watch?v=5CG81_Y8SvY~~

Holger Stoltenberg at 11:03, 6 January 2020

2020-01-06T11:03:21Z

← Older revision		Revision as of 11:03, 6 January 2020
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	== See also… ==		== See also… ==
	Dismantling the tonegenarator of a scrapped H-Series Hammond Organ [8:47 min]		Dismantling the tonegenarator of a scrapped H-Series Hammond Organ [8:47 min]
	https://www.youtube.com/watch?v=7Qqmr6IiFLE		https://www.youtube.com/watch?v=7Qqmr6IiFLE <br>
	An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min]		An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min]
	* @ 4:48 min: ''“…these sounds are in there”''		* @ 4:48 min: ''“…these sounds are in there”''
	* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''		* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''
	https://www.youtube.com/watch?v=5CG81_Y8SvY		https://www.youtube.com/watch?v=5CG81_Y8SvY

Holger Stoltenberg: Table 4 finalized, sections "Discussion" and "See also..." added

2020-01-06T10:56:47Z

Table 4 finalized, sections "Discussion" and "See also..." added

← Older revision		Revision as of 10:56, 6 January 2020
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	~~[THIS PAGE IS A WORK IN PROGRESS...]~~

	The article features just intervals created by the mechanical tonegenerator of the classical Hammond B-3 Organ model.		The 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==
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	==== Example 1: Mapping a single interval====		==== Example 1: Mapping a single interval====

	In this example we map the combination of a Hammond Organ’s note E and a higher note A ~~(a fourth up)~~ to the harmonic series.		In this example we map the combination of a Hammond Organ’s note E and a higher note A to the harmonic series.

	<u>Table 2</u>: ~~The~~ fourth E-A		<u>Table 2</u>: Mapping the fourth E-A

	{\| class="wikitable"		{\| class="wikitable"
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	==== Example 2: Mapping a chord ====		==== Example 2: Mapping a chord ====
	Adding an upper fifth (note B), the second example illustrates how to map the sus4-chord E-A-B to the harmonic series.		Adding an upper fifth (note B), the second example illustrates how to map the resulting sus4-chord E-A-B to the harmonic series.

	<u>Table 3</u>: sus4-chord E-A-B		<u>Table 3</u>: sus4-chord E-A-B
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	The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11<sup>th</sup> and the 12<sup>th</sup> octave.		The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11<sup>th</sup> and the 12<sup>th</sup> octave.

	==== Example 3: ~~Hammond - full scale~~ ====		==== Example 3: Mapping all of the tonegenarator's pitchclasses ====
	Table 4: ~~Full~~ set of ~~pitchclasses~~		The full set of the Hammond Organ’s intervals resides surprisingly far up in the Harmonic Series:
			* The '''44th''' '''octave starts''' at partial #(2<sup>43</sup>), just below the set of partials determined by the Hammond Organ’s tonegenerator
			* The '''45th''' '''octave''' starts '''right within the derived set of partials''' and starts at partial #(2<sup>44</sup>)
			<u>Table 4</u>: The full set of intervals' position in the Harmonic Series
	{\| class="wikitable"		{\| class="wikitable"
	! rowspan="3" style="text-align: center;" \| <br><br><br>		! rowspan="3" style="text-align: center;" \| <br><br><br>
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	\| style="text-align: center;" \| (C)		\| style="text-align: center;" \| (C)
	\| style="text-align: center;" \| (D)		\| style="text-align: center;" \| (D)
	\| ~~rowspan~~="2" ~~colspan~~="16" style="text-align: center; background-color:#cbcefb;" \|		\| colspan="16" rowspan="2" style="text-align: center; background-color:#cbcefb;" \|
	... of Column (D)<br><br><br><br>		... of Column (D)<br><br><br><br>
	\| rowspan="2" style="text-align: center;" \|		\| rowspan="2" style="text-align: center;" \|
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	\|}		\|}

	[...~~WILL BE CONTINUED]~~		== Discussion and General Applicability ==
			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 relevance. Nevertheless the method of prime factorization 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… ==
			Dismantling the tonegenarator of a scrapped H-Series Hammond Organ [8:47 min]
			https://www.youtube.com/watch?v=7Qqmr6IiFLE
			An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min]
			* @ 4:48 min: ''“…these sounds are in there”''
			* @ 5:40 min: ''“16 foot, biggest pipes, the deepest sounds – they come from the foot”''
			https://www.youtube.com/watch?v=5CG81_Y8SvY

Holger Stoltenberg at 09:53, 6 January 2020

2020-01-06T09:53:38Z

Show changes

← Older revision		Revision as of 16:57, 21 January 2020
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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 ~~12 pairs~~ of gearwheels that make ~~twelve~~ ''driven'' shafts turn. The corresponding ~~twelve~~ ''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>		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>

	~~Every~~ chromatic pitch class ~~has a separate~~ driven ~~shaft~~. 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 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 ~~shaft~~ we write		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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	== 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 (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 (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>