Frequency modulation synthesis (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Frequency modulation synthesis" in English language version.

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6yamaha.com

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3books.google.com

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1stanford.edu

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archive.org

Dr. Hubert Howe (1960s). Buchla Electronic Music System: Users Manual written for CBS Musical Instruments (Buchla 100 Owner's Manual). Educational Research Department, CBS Musical Instruments, Columbia Broadcasting System. p. 7. At this point we may consider various additional signal modifications that we may wish to make to the series of tones produced by the above example. For instance, if we would like to add frequency modulation to the tones, it is necessary to patch another audio signal into the jack connected by a line to the middle dial on the Model 158 Dual Sine-Sawtooth Oscillator. ...
Atten Strange (1974). Programming and Metaprogramming in the Electro-Organism - An Operating Directive for the Music Easel. Buchla and Associates.

books.google.com

McGuire, Sam; Matějů, Zbyněk (2020-12-28). The Art of Digital Orchestration. CRC Press. ISBN 978-1-000-28699-1.
Curtis Roads (1996). The computer music tutorial. MIT Press. p. 226. ISBN 0-262-68082-3. Retrieved 2011-06-05.
Holmes, Thom (2008). "Early Computer Music". Electronic and experimental music: technology, music, and culture (3rd ed.). Taylor & Francis. pp. 257–8. ISBN 978-0-415-95781-6. Retrieved 2011-06-04.

clavia.se

Rob Hordijk. "FM synthesis on Modular". Nord Modular & Micro Modular V3.03 tips & tricks. Clavia DMI AB. Archived from the original on 2007-04-07. Retrieved 2013-03-23.

cnx.org

Doering, Ed. "Frequency Modulation Mathematics". Retrieved 2013-04-11.

elektron.se

Digitone product page

javelinart.com

Zollinger, W. Thor (Dec 2017). "FM_Synthesis_of_Real_Instruments" (PDF). Archived (PDF) from the original on 2017-09-25.

korg.com

Volca FM product page

mixonline.com

"1978 New England Digital Synclavier". Mix. Penton Media. September 1, 2006.

musicradar.com

"The top 10 classic synth presets (and where you can hear them)". MusicRadar. Retrieved October 19, 2018.

patents.google.com

"Patent US4018121 - Method of synthesizing a musical sound - Google Patents". Retrieved 2017-04-30.

psu.edu

math.psu.edu

The above expression is transformed using trigonometric addition formulas
${\begin{aligned}\sin(x\pm y)&=\sin x\cos y\pm \cos x\sin y\end{aligned}}$
and a lemma of Bessel function
${\begin{aligned}\cos(\beta \sin \theta )&=J_{0}(\beta )+2\sum _{n=1}^{\infty }J_{2n}(\beta )\cos(2n\theta )\\\sin(\beta \sin \theta )&=2\sum _{n=0}^{\infty }J_{2n+1}(\beta )\sin((2n+1)\theta )\end{aligned}}$

(Source: Kreh 2012)
as following:
${\begin{aligned}&\sin \left(\theta _{c}+\beta \,\sin(\theta _{m})\right)\\&\ =\ \sin(\theta _{c})\cos(\beta \sin(\theta _{m}))+\cos(\theta _{c})\sin(\beta \sin(\theta _{m}))\\&\ =\ \sin(\theta _{c})\left[J_{0}(\beta )+2\sum _{n=1}^{\infty }J_{2n}(\beta )\cos(2n\theta _{m})\right]+\cos(\theta _{c})\left[2\sum _{n=0}^{\infty }J_{2n+1}(\beta )\sin((2n+1)\theta _{m})\right]\\&\ =\ J_{0}(\beta )\sin(\theta _{c})+J_{1}(\beta )2\cos(\theta _{c})\sin(\theta _{m})+J_{2}(\beta )2\sin(\theta _{c})\cos(2\theta _{m})+J_{3}(\beta )2\cos(\theta _{c})\sin(3\theta _{m})+...\\&\ =\ J_{0}(\beta )\sin(\theta _{c})+\sum _{n=1}^{\infty }J_{n}(\beta )\left[\,\sin(\theta _{c}+n\theta _{m})\ +\ (-1)^{n}\sin(\theta _{c}-n\theta _{m})\,\right]\\&\ =\ \sum _{n=-\infty }^{\infty }J_{n}(\beta )\,\sin(\theta _{c}+n\theta _{m})\qquad (\because \ J_{-n}(x)=(-1)^{n}J_{n}(x))\end{aligned}}$
Kreh, Martin (2012), "Bessel Functions" (PDF), The Pennsylvania State University, pp. 5–6, archived from the original (PDF) on 2017-11-18, retrieved 2014-08-22

soundonsound.com

Gordon Reid (September 2001). "Sounds of the '80s Part 2: The Yamaha DX1 & Its Successors (Retro)". Sound on Sound. Archived from the original on 17 September 2011. Retrieved 2011-06-29.
Poyser, Debbie; Johnson, Derek (1998). "Yamaha FS1R - FM Synthesis / Formant-shaping Tone Generator". Sound on Sound. No. December 1998.

stanford.edu

news.stanford.edu

Stanford University News Service (06/07/94), Music synthesis approaches sound quality of real instruments

umich.edu

web.eecs.umich.edu

Chowning 1973, pp. 1–2 Chowning, J. (1973). "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation" (PDF). Journal of the Audio Engineering Society. 21 (7).

uspto.gov

patft.uspto.gov

"U.S. Patent 4018121 Apr 1977". patft.uspto.gov. Retrieved 2017-04-30.

web.archive.org

The above expression is transformed using trigonometric addition formulas
${\begin{aligned}\sin(x\pm y)&=\sin x\cos y\pm \cos x\sin y\end{aligned}}$
and a lemma of Bessel function
${\begin{aligned}\cos(\beta \sin \theta )&=J_{0}(\beta )+2\sum _{n=1}^{\infty }J_{2n}(\beta )\cos(2n\theta )\\\sin(\beta \sin \theta )&=2\sum _{n=0}^{\infty }J_{2n+1}(\beta )\sin((2n+1)\theta )\end{aligned}}$

(Source: Kreh 2012)
as following:
${\begin{aligned}&\sin \left(\theta _{c}+\beta \,\sin(\theta _{m})\right)\\&\ =\ \sin(\theta _{c})\cos(\beta \sin(\theta _{m}))+\cos(\theta _{c})\sin(\beta \sin(\theta _{m}))\\&\ =\ \sin(\theta _{c})\left[J_{0}(\beta )+2\sum _{n=1}^{\infty }J_{2n}(\beta )\cos(2n\theta _{m})\right]+\cos(\theta _{c})\left[2\sum _{n=0}^{\infty }J_{2n+1}(\beta )\sin((2n+1)\theta _{m})\right]\\&\ =\ J_{0}(\beta )\sin(\theta _{c})+J_{1}(\beta )2\cos(\theta _{c})\sin(\theta _{m})+J_{2}(\beta )2\sin(\theta _{c})\cos(2\theta _{m})+J_{3}(\beta )2\cos(\theta _{c})\sin(3\theta _{m})+...\\&\ =\ J_{0}(\beta )\sin(\theta _{c})+\sum _{n=1}^{\infty }J_{n}(\beta )\left[\,\sin(\theta _{c}+n\theta _{m})\ +\ (-1)^{n}\sin(\theta _{c}-n\theta _{m})\,\right]\\&\ =\ \sum _{n=-\infty }^{\infty }J_{n}(\beta )\,\sin(\theta _{c}+n\theta _{m})\qquad (\because \ J_{-n}(x)=(-1)^{n}J_{n}(x))\end{aligned}}$
Kreh, Martin (2012), "Bessel Functions" (PDF), The Pennsylvania State University, pp. 5–6, archived from the original (PDF) on 2017-11-18, retrieved 2014-08-22
"[Chapter 2] FM Tone Generators and the Dawn of Home Music Production". Yamaha Synth 40th Anniversary - History. Yamaha Corporation. 2014. Archived from the original on 2017-05-11.
Rob Hordijk. "FM synthesis on Modular". Nord Modular & Micro Modular V3.03 tips & tricks. Clavia DMI AB. Archived from the original on 2007-04-07. Retrieved 2013-03-23.
Gordon Reid (September 2001). "Sounds of the '80s Part 2: The Yamaha DX1 & Its Successors (Retro)". Sound on Sound. Archived from the original on 17 September 2011. Retrieved 2011-06-29.
Zollinger, W. Thor (Dec 2017). "FM_Synthesis_of_Real_Instruments" (PDF). Archived (PDF) from the original on 2017-09-25.

yamaha.com

usa.yamaha.com

"[Chapter 2] FM Tone Generators and the Dawn of Home Music Production". Yamaha Synth 40th Anniversary - History. Yamaha Corporation. 2014. Archived from the original on 2017-05-11.
Yamaha Montage Product Features Page
Yamaha MODX Product Features Page
MODX8+, MODX7+, and MODX6+ Features
MONTAGE M Synthesizer

jp.yamaha.com

"Yamaha SY99 spec". Yamaha Corporation (in Japanese).