Digitone Guide to FM Synthesis

  1. Basics of sound

  2. Fundamental Frequency

  3. Digital FM Synthesis

  4. Modulator and Carrier

  5. Side Bands

  6. FM Basic Waveforms

  7. Model a Digital Clarinet on Digitone

  8. Designing Sounds on the Digitone

  9. How to use Visual Analyser with the Digitone

  10. Model Instruments on the Digitone with Visual Analyser 

  11. Suggested Sound Design Tutorials

  12.  History

  13.  Math

  14.  Sources

 

If a violin and a viola both play an A note, they sound completely different, even though they are playing the same note! Different harmonics are what cause the same musical note (the same frequency) played on different instruments to sound unique.

Oscillator – A device for generating waveforms. A string on a violin is a great example of an oscillator. 

Sine wave – Sine waves are representations of a single frequency with no harmonics.To the human ear, a sound that is made of more than one sine wave will have perceptible harmonics

Amplitude Domain 

A simple method of depicting sound waveforms is to draw them in the form of a graph of air pressure (amplitude) versus time. This is called a time domain representation. When the curved line is near the bottom of the graph, the air pressure is lower, and when the curve is near the top of the graph, the air pressure has increased. 

Frequency Domain

 A standard way is to plot each frequency (pitch) as a line along an x-axis. The height of each line indicates the strength (or amplitude) of each frequency component. The purest signal is a sine waveform, so named because it can be calculated using the trigonometric formulae for the sine of an angle. A pure sine wave represents just one frequency component, or one line in a spectrum.   

 

Sound is created by changes in air pressure.

 

Watch this to learn more: https://www.youtube.com/watch?v=npGMhaOR7Rw

 

 

 

 

Timbre and Complex Waves

 

Sine Waves are perfect waves that don’t occur in nature. These are pure sounds of a single frequency that are usually generated by a computer, or some other tone generator. The sound wave created by a cello, is much more inconsistent than the sine wave. That is because, instead of just having a single frequency, like the sine wave, it has multiple. If a sine wave were to generate the note A2, or 220 Hz, only one frequency would play. But if a Cello plays the note A2, it has the pitches: 220Hz, 440Hz, 660Hz, and 880Hz, all at different amplitudes, or volumes. The volumes of these frequencies all vary and change with time as the instrument is played. This gives the sound a movement, that a sine wave just doesn’t have. The dominant note, A2, is the fundamental frequency. It is the pitch that we hear.  All sounds that we hear on a daily basis are made up of a variety of sine waves, these are known as complex waves. The note that we identify is known as the Fundamental. The other frequencies that are in the wave, are known as Harmonics. These frequencies all work together to form the unique sound of an instrument, known as its timbre. Fundamental + Harmonics = Timbre. 

 

Watch this to learn more: https://www.youtube.com/watch?v=nlv5bylQDsE

 

Examples of different timbres:

https://www.youtube.com/watch?v=VRAXK4QKJ1Q

 

Production of sound:

 https://www.khanacademy.org/science/ap-physics-1/ap-mechanical-waves-and-sound/introduction-to-sound-waves-ap/v/production-of-sound

Sound Properties:  

https://www.khanacademy.org/science/ap-physics-1/ap-mechanical-waves-and-sound/introduction-to-sound-waves-ap/v/sound-properties-amplitude-period-frequency-wavelength

 

 

 

2. Fundamental Frequency

Joseph Fourier 

It was a scientist, Joseph Fourier, who in the early nineteenth century gave a mathematical basis to the idea that certain complex sounds, such as that from a violin or trumpet, are just the sum of a collection of simple tones(sine waves), at frequencies which are whole number multiples of the fundamental frequency. 

Source Page 11 ( http://www.burnkit2600.com/manuals/fm_theory_and_applications.pdf )

Img source(https://wolfe4e.sinauer.com/wa10.02.html)

The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal tones present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Together they form the harmonic series. Overtones that are perfect integer multiples of the fundamental are called harmonics

Any frequency(pitch) component can be called a partial.

The Fundamental Frequency + Harmonics = Timbre

Harmonics, or more precisely, harmonic partials, are partials whose frequencies are multiples of the fundamental.

Harmonics are positive integer multiples of the fundamental. For example, if the fundamental frequency is 10 Hz (also known as the first harmonic) then the second harmonic will be 20Hz (10 * 2 = 20 Hz), the third harmonic will be 30 Hz (10* 3 = 30 Hz), and so on.

Above Img source ( http://www.muzines.co.uk/images_mag/articles/emm/EMM_81_08_harmonics_2_large.jpg )

3. Digital FM Synthesis

If you have a Digitone, watch the video below before continuing! 

Digitone Guide to FM synthesis (https://youtu.be/FWIiY1twioc?list=TLPQMzEwMzIwMjArGn6zhQv9dw

 

Frequency Modulation – Frequency Modulation is a very well known digital synthesis method. However, FM is not one technique, but a family of methods.

Source Page 224(https://books.google.com/books?id=nZ-TetwzVcIC&&pg=PA232#v=onepage&q&f=false)

Frequency Modulation and Phase Modulation:

FM and the closely related technique called phase modulation (PM) represent two virtually identical cases of the same type of angle modulation.(Black 1953, pp.28-30)

 The amplitudes of the partials generated by the two methods exhibit slight differences, but in musical practice there is no great distinction between PM and FM, particularly in the case of time-varying spectra.  

Source Page 224(https://books.google.com/books?id=nZ-TetwzVcIC&&pg=PA232#v=onepage&q&f=false)

John Chowning at Stanford University was the first to explore the musical potential of digital FM synthesis. Prior to this, most digital sounds had been produced by a fixed-waveform, fixed spectrum techniques. Hence, Chowning sought a way to generate synthetic sounds that had the animated spectra characteristic of natural sounds. 

Chowning says: 

“I found that with two simple sinusoids I could generate a whole range of complex sounds which done by other means demanded much more powerful and extensive tools. If you want to have a sound that has, say 50 harmonics, you have to have 50 oscillators. And I was using just two oscillators to get something that was very similar.”

(Chowning 1973)

The two oscillators that Chowning used to create such a complex sound were not normal oscillators. He named them operators.(Keep in mind that the Digitone has 4 of these, A, C, B1, and B2) 

Operators- are oscillators that have two functions. They can function as a Carrier or a Modulator. It is important to note, in some cases an Operator can be both a Carrier and a Modulator.

 

Carrier Frequency– The frequency of the oscillator which is being modulated. It outputs sound

Modulator Frequency – The frequency of the oscillator which modulates the Carrier. The modulator does not output sound. The frequency of the Modulator determines what the Carriers harmonics will be.

4. Modulator and Carrier

Critical relationship between "M" & "C"

Let's look at the one Modulator & one Carrier set-up.

MODULATOR ——> CARRIER ——-> sound output

The carrier frequency "C" and the modulator frequency "M" will together determine which harmonics will exist in the harmonic spectrum. The harmonic spectrum is a graphic representation of frequencies where “ (f)c ”  is the fundamental frequency and the other harmonics are just multiples of the fundamental.

Source (http://www.kratzer.at/DXFMSynthesis.htm)

(img source n/a)

5. Side Bands

What is happening is that the energy of the modulation is transformed into "Sidebands" (the series of harmonics on both sides of the Carrier).

(img source n/a)

Img source (https://web.eecs.umich.edu/~fessler/course/100/misc/chowning-73-tso.pdf)

The appearance of Sidebands is always in pairs on each side of "C". These Sideband pairs are ranked by their "order" of separation from "C" (eg 1st pair is "M" distance apart from "C", 2nd pair is 2x"M" distance apart from "C"… etc).

Now, it is important to note the following:-

|||||||||

1357911131517

C-4MC-3MC-2MC-M Carrier C+M    C+2MC+3MC+4M

If "C" was detuned down to 8.5, then the whole harmonic spectrum would be shifted down by 0.5! So detuning "C" shifts the entire spectrum.

If "M" was detuned down to 1.5, then the Sidebands would move in closer and be separated by 1.5!

 So detuning "M" compresses or expands the Sideband separation.

Source ( http://yala.freeservers.com/2fmsynth.htm#2Mod )

Img source ( https://web.eecs.umich.edu/~fessler/course/100/misc/chowning-73-tso.pdf )

See the above image. I = Intensity. As you increase I, the energy from the carrier is spread out to the side bands.

When “I” = 0 there is no modulation. As “I” increases from zero, energy is stolen from the carrier and distributed among an increasing number of sidebands! 

The modulation of any carrier in any way produces sidebands. 

Source (https://www.electronics-notes.com/articles/radio/modulation/frequency-modulation-fm-sidebands-bandwidth.php)

The FM sidebands are dependent on both the level of deviation  and the frequency of the modulation.

Frequency Deviation – maximum difference between a modulator frequency and the carrier frequency

Source ( https://www.revolvy.com/page/Frequency-deviation )

In fact the total frequency modulation spectrum consists of the carrier plus an infinite number of sidebands spreading out on either side of the carrier at integral multiples of the modulating frequency. The values for the levels of the sidebands rise and fall with varying values of deviation and modulating frequency. The parameters for the FM sidebands are determined by a formula using Bessel functions of the first kind.

Negative Frequencies

Some frequencies will be negative. We sometimes say the negative frequencies “wrap around” (zero) to become positive. 

That is why when observing the Frequency Domain , we do not see any negative frequencies:

Above Img Source (http://sites.music.columbia.edu/cmc/MusicAndComputers/chapter3/03_03.php)

The main caveat here is that when frequencies wrap around and add to positive frequencies of the same magnitude, the components may not add in phase. The complexity of all this tends to give FM signals a complex behavior as the index of modulation increases, adding more and more components, both positive and negative.

Source ( http://yala.freeservers.com/2fmsynth.htm#2Mod )

Harmonic Ratio

The human ear is very sensitive to harmonic vs. inharmonic spectra. Perceptually,

harmonic spectra are very distinctive because they give a strong sense of

pitch. The harmonic ratio [Truax 1977] is the ratio of the modulating frequency

to the carrier frequency, such that H =M/C

 

Source ( http://www.cs.cmu.edu/~music/icm-online/readings/fm-synthesis/index.html )

Inharmonic Frequencies: The model of Fourier analysis provides for the inclusion of inharmonic partials, which are partials whose frequencies are not whole number ratios of the fundamental (such as 1.1 or 2.14179)

Source ( https://en.wikipedia.org/wiki/Overtone )

6. FM Basic Waveforms

When the Frequency Ratio between Modulator and Carrier is 1:1, that is, they have the same value, this will create a Sawtooth Wave (Sawtooth describes a wave that contains all harmonics).

If the Modulator:Carrier ratio is 2:1 – this results in generating every other harmonic (odd numbered multiples of the Fundamental). The Square wave will include all the odd number harmonics 3.00, 5.00, 7.00, 9.00 etc. This sound we describe as hollow, woody, reedy. 

If Ratio of the Modulator is set to 3.00. You will hear a very tight, narrow waveform (Pulse wave). And as you can guess, the 4:1 and 5:1 will give ever narrower results – and tighter narrower pulse waves.

Source ( https://yamahasynth.com/yamaha-synth-rss/fm-basic-waveforms )

Basic waveforms on the Digitone:

Watch the video below before continuing! 

Digitone Guide to FM synthesis (https://youtu.be/FWIiY1twioc?list=TLPQMzEwMzIwMjArGn6zhQv9dw)

A sawtooth wave sounds harsh and clear. Its spectrum contains both even and odd harmonics of the fundamental frequency. Each partial in a sawtooth wave is a positive integer multiple of the Fundamental Frequency. The multiplier resulted in both even and odd harmonics existing. 

Source( https://quizlet.com/77401222/music-technology-chapter-3-flash-cards/ )

Img Source ( http://www.personal.psu.edu/meb26/INART55/oscillators.html )

You can easily create a Sawtooth wave on the Digitone! If you are on Algo 1, set your MIX to the X channel! Our C operator has a line directly connecting it to X, making it our carrier. Make sure your C operator (the carrier) and your A operator (the modulator) are both set to a value of 1.00…

On the Digitone, there are 4 operators, C, A, B1, and B2. X and Y represent our sound outputs. If there is a line directly connecting an operator to X or Y, that means it is outputting sound. Any operator that outputs sound, is defined as a Carrier. If an operator does not have a line, directly connecting it to the X or Y sound outputs, then it is not outputting sound, it is going to modulate that sound.

Checking the oscilloscope below, all we see now is a sine wave, because we have not added any amplitude to our modulator. (A)

 Switch to the SYN2 page. Set the LEV for your A operator to a value of 50. This brings our sidebands, or harmonics into existence!

Now the oscilloscope displays a sawtooth wave. 

Img Source ( http://www.personal.psu.edu/meb26/INART55/oscillators.html )

A square wave is constructed from only odd harmonics. The odd harmonics were a result of a multiplier that was applied to the fundamental frequency. 

Source( https://quizlet.com/77401222/music-technology-chapter-3-flash-cards/ )

On the Digitone, a square wave is made from a 1:2 ratio. Change your (A) oscillator to a value of 2.00.

For now, we only see a sine wave, because we need to bring the harmonics into existence through applying amplitude to our modulator (A)

Switch to your SYN2 page and set to a value of 50.

Below is the resulting square wave displayed on an oscilloscope. 

7. Clarinet sound on the Digitone

A M:C of 2:1 could be used to create a digital clarinet.

Source (https://yamahasynth.com/yamaha-synth-rss/fm-basic-waveforms) 

Switch to your SYN2 page and set your LEV to 35. It could be anywhere from 30-40, depending on the tone you would like.

But this will only create the correct frequencies for the clarinet! An envelope needs to be applied to the overall amplitude of the Clarinet to create a more realistic sound. The next step is to set up the way the synthesizer plays each note. ADSR stands for Attack-Decay-Sustain-Release. The Attack rate sets how fast the synth gets from silence up to full volume for the note. A fast attack means it hits the note hard. A medium attack means it raises the volume of the note slower and eases into the note smoothly. A slow attack rate means it’s going to take several seconds to get the note 

 

Img Source Page 9 ( http://www.javelinart.com/FM_Synthesis_of_Real_Instruments.pdf )

In reality, a Clarinet has a more rounded envelope (see figure 5) which we approximate with a rectangular shaped envelope (see figure 6). Switch over to your AMP page. We set the Attack to medium 37. It could be anywhere from 26-40 depending on how you want the note to sound.  Set the Decay rate to 32, the Sustain rate to 127, and the Release to 21. This gives us an envelope like the one below. It isn’t exactly like the real thing above, but it’s close enough.

While the M:C of 2:1 could be used to recreate a Clarinet. The M:C of 4:1 could be used to recreate an Oboe. Both "reedy" sounds but one more nasal than the other.

8. Designing sounds on the Digitone:

Note: the Digitone has 4 oscillators! C, A, B1, and B2

If you haven’t already, be sure to watch the video below before continuing! 

Digitone Guide to FM synthesis (https://www.youtube.com/watch?v=Wf203qcju5w

Oscillator – A device for generating waveforms. A string on a violin is a great example of an oscillator. The Digitone has 4 oscillators. 

Sine wave – Sine waves are representations of a single frequency with no harmonics.To the human ear, a sound that is made of more than one sine wave will have perceptible harmonics; The default waveform on the Digitone is a sine wave. 

Operators- are oscillators that have two functions. They can function as a Carrier or a Modulator. It is important to note, in some cases an Operator can be both a Carrier and a Modulator.

 

Carrier Frequency– The frequency of the oscillator which is being modulated. It outputs sound

Modulator Frequency – The frequency of the oscillator which modulates the Carrier. The modulator does not output sound. The frequency of the Modulator determines what the Carriers harmonics will be.

On the Digitone, the oscillator’s frequency value is represented by number values assigned to each operator. A value of 1.00 = a C4 note. 

 

Listed below are the oscillator’s available frequency ratios on the Digitone, and what note they represent on our chromatic scale:

 

C – 0.25/0.50/1.00/2.00/4.00/16.00

Csharp/Dsharp – 4.25/8.50

D – 4.50/2.25/4.00

Dsharp/Eflat – 1.25/2.50/5.00/10.00

E – 1.25/2.50/5.00/10.00

F/Esharp – 2.75$(not exact!)

Fsharp – 5.50/11.00

G – 0.75/1.50/3.00/6.00/12.00

Gsharp/Aflat – 3.25/6.50$(not exact!)/13.00

A

Asharp/Bflat – 1.75$(not exact!)/3.50/7.00$(not exact!)/14.00

B – 3.75/ 7.50/15.00

 

A value of 1.00 is C4, so a value of 0.50 is going to be C3, and a value of 2.00 is going to be C5!  All of the 12 notes from the chromatic scale are available except for notes A and F. Notice some have $(not exact!) attached to them. These notes are note values that are not one of the 12 notes on the chromatic scale, which is kind of cool. There are a total of 4 of these notes. They are 1.75, 2.75, 6.50, and 7.00.

Enharmonic Detuned Frequencies

The frequency of an operator is dependent on 3 parameters (1) the Coarse Frequency, (2) the Fine Frequency, and (3) Detune. The Coarse Frequency is the main integer ratio of the base frequency (eg M:C = 2:1). We have also looked at Detune and its effects where (a) detuning the Carrier shifts the entire spectrum, and (b) detuning the Modulator changes the separation between the sidebands.

FM synths will have some form of Fine Frequency. The Fine Frequency allows the selection of non-integer multiples of the base frequency. Normally, this would yield something clangorous or enharmonic. This brings another dimension into FM sounds because a new set of overtones are introduced. Typically, this would be used for bell-type or percussion sounds. You can also obtain very unique and strange timbres too (a great source of experimentation).

Source pg. 1 (https://sites.google.com/site/yalaorg/audio-music-synthesis/fmsynth/fmsynthdx?tmpl=%2Fsystem%2Fapp%2Ftemplates%2Fprint%2F&showPrintDialog=1)

On the digitone you are given a Coarse Frequency that functions as both the Coarse Frequency and the Fine Frequency. The coarse frequency represents the whole number values, (1.00, 2.00) and the Fine Frequency represents the numbers in between (1.25, 1.50). You are also given a dual detuner. 

 

Detune is applied to both A and B2 at the same level and time. 

And finally, if you go to the second SYN1 page, you will see the option to individually detune your operators! This represents your fine tuner! It is how you reach the values 1.32, or 56.

What is cool about the Digitone, is that it comes packed with a ton of blueprints on how to make sounds. Do you want to know what algorhythm and settings are best to create a kick? Load up a preset of a kick, and take a look at the settings it has. Check out the envelope that created the kick like sound. Check out the operator values, and the level of modulation. Check out the LFO settings to see what parameters were being modulated. You can look at these presets and learn a lot from them!

A big part of designing sounds, is seeing what is happening. I would suggest for you to get a tool to see the frequency spectrum with. The tool that I am using is Visual Analyser. It works on PC only! It will not work on MAC computers. 

Here is the link: :http://www.sillanumsoft.org/download.htm 

Visual Analyser shows us both our waveform and frequency spectrum. Once you start using this tool, you will come back to it again and again when designing sounds. And now that Overbridge works with the Digitone, you will be able to bypass needing an audio interface. You are able to connect directly to the program via usb. This is extremely convenient and Elektron really deserves praise for making it so easy to do. 

9. How to use the Digitone with Visual Analyser:

  1. Plug your Digitone into the usb port 

  2. Open Overbridge Engine. 

  3. Open the Overbridge Digitone application 

  4. Open up Visual Analyser

  5. Go to "Settings" on Visual Analyser

  6. From the top tabs, pick Device

  7. Under Input device, choose the option Line In (Elektron Digitone)

  8. Hit OK to confirm

  9. In the top left corner, it the On button

10: Model Instruments on the Digitone with Visual Analyser 

How to model a flute on the Digitone: 

https://www.youtube.com/watch?v=HoKQIND3NBc

How to model an alto flute, bassoon, & clarinet: 

https://www.youtube.com/watch?v=CmeULzdOM6o

11. Suggested Sound Design Tutorials

I would suggest watching these tutorials with visual analyzer open, so that you can see and hear what is happening to the waveform. 

Ivar Tryti:

Digitone Bass Tutorial: https://www.youtube.com/watch?v=gfFCnILXr20

Digitone Lead & Pads Tutorial https://www.youtube.com/watch?v=42pJG6Gm5es

Red Means Recording:

Intro and bass patch: https://www.youtube.com/watch?v=T5Oo56G6Ft4

Making a pad: https://www.youtube.com/watch?v=T5Oo56G6Ft4

Making an Arpeggio: https://www.youtube.com/watch?v=m4XDFgTO35g

 

Making ambient with arpeggios: https://www.youtube.com/watch?v=Sdn9G54JeXA

 

Sound pool and drums: https://www.youtube.com/watch?v=Bu7JEXdDKhM

Oscillator Sink:

Digitone’s LFO features https://www.youtube.com/watch?v=bv4WdFxoAc0

12. History

FM broadcasting:  is a method of radio broadcasting using frequency modulation (FM) technology. Invented in 1933 by American engineer Edwin Armstrong, wide-band FM is used worldwide to provide high-fidelity sound over broadcast radio. FM broadcasting is capable of better sound quality than AM broadcasting, the chief competing radio broadcasting technology, so it is used for most music broadcasts. 

              AM and FM modulated signals for radio.

In the case of AM, modulation is done by altering the amplitude of the carrier wave with time, according to the original signal. In the case of FM, it is the frequency of the carrier wave that is varied. A radio receiver (a "radio") contains a demodulator that extracts the original program material from the broadcast wave.

FM has better rejection of static (RFI) than AM. This was shown in a dramatic demonstration by General Electric at its New York lab in 1940. The radio had both AM and FM receivers. With a million-volt arc as a source of interference behind it, the AM receiver produced only a roar of static, while the FM receiver clearly reproduced a music program from Armstrong's experimental FM transmitter in New Jersey.

Reception distance

VHF Radio waves do not travel far beyond the visual horizon, so reception distances for FM stations are usually limited to 30—40 miles (50—65 km) They can also be blocked by hills or buildings.

The knife edge effect can permit reception where there is no direct line of sight between broadcaster and receiver. The reception can vary considerably depending on the position. One example is the 

U?ka mountain range, which makes constant reception of Italian signals from Veneto and Marche possible in a good portion of Rijeka, Croatia, despite the distance being over 200km.[citation needed] 

Other radio propagation effects such as tropospheric ducting and Sporadic E can occasionally allow distant stations to be intermittently received, but cannot be relied on for commercial broadcast 

purposes.

This is still less than the range of AM radio waves, which because of their lower frequency can travel as ground waves or reflect off the ionosphere, so AM radio stations can be received at hundreds 

(sometimes thousands) of miles. This is a property of the carrier wave's typical frequency (and power), not its mode of modulation.

Reference ( https://en.wikipedia.org/wiki/FM_broadcasting )

Frequency modulation synthesis (or FM synthesis) is a form of sound synthesis where the timbre of a simple waveform (such as a square, triangle, or sawtooth) called the carrier, is changed by modulating its frequency with a modulator frequency that is also in the same or similar audio range, so that a more complex timbre results. The frequency of an oscillator is altered "in accordance with the amplitude of a modulating signal."

FM synthesis can create both harmonic and inharmonic sounds. For synthesizing harmonic sounds, the modulating signal must have a harmonic relationship to the original carrier signal. As the amount of frequency modulation increases, the sound grows progressively more complex. Through the use of modulators with frequencies that are non-integer multiples of the carrier signal (i.e. non-harmonic), atona land tonal bell-like and percussive sounds can easily be created.

FM synthesis using analog oscillators may result in pitch instability. However, FM synthesis can also be implemented digitally, the latter proving to be more 'reliable' and is currently seen as standard practice. Digital FM synthesis (implemented as phase modulation) was the basis of several musical instruments beginning as early as 1974. Yamaha built the first prototype digital synthesizer in 1974, based on FM synthesis, before commercially releasing the Yamaha GS-1 in 1980. Yamaha's groundbreaking DX7, released in 1983, brought FM to the forefront of synthesis in the mid-1980s.

The technique of the digital implementation of frequency modulation, which was developed by John Chowning (Chowning 1973, cited in Dodge & Jerse 1997, p. 115) at Stanford University in 1967–68, was patented in 1975. Prior to that, the FM synthesis algorithm was licensed to Japanese company Yamaha in 1973.

Reference (https://en.wikipedia.org/wiki/Frequency_modulation_synthesis)

John Chowning Quote:

“A new application of the well-known process of frequency modulation is shown to result in a surprising control of audio spectra. The technique provides a means of great simplicity to control the spectral components and their evolution in time. Such dynamic spectra are diverse in their subjective impressions and include sounds both known and unknown”

(http://people.ece.cornell.edu/land/courses/ece4760/Math/GCC644/FM_synth/Chowning.pdf)

FM synthesis was the basis of some of the early generations of digital synthesizers, most notably those from Yamaha, as well as New England Digital Corporation under license from Yamaha.[3] Yamaha's popular DX7 synthesizer, released in 1983, was ubiquitous throughout the 1980s. Several other models by Yamaha provided variations and evolutions of FM synthesis during that decade.

With the expiration of the Stanford University FM patent in 1995, digital FM synthesis can now be implemented freely by other manufacturers. The FM synthesis patent brought Stanford $20 million before it expired, making it (in 1994) "the second most lucrative licensing agreement in Stanford's history".[10] FM today is mostly found in software-based synths such as FM8 by Native Instruments or Sytrus by Image-Line, but it has also been incorporated into the synthesis repertoire of some modern digital synthesizers, usually coexisting as an option alongside other methods of synthesis such as subtractive, sample-based synthesis, additive synthesis, and other techniques. The degree of complexity of the FM in such hardware synths may vary from simple 2-operator FM, to the highly flexible 6-operator engines of the Korg Kronos and Alesis Fusion, to creation of FM in extensively modular engines such as those in the latest synthesisers by Kurzweil Music Systems.

Most recently, in 2016, Korg released the Korg Volca FM, an FM iteration of the Korg Volca series of compact, affordable desktop modules, and Yamaha released the Montage, which combines a 128-voice sample-based engine with a 128-voice FM engine. This iteration of FM is called FM-X, and features 8 operators; each operator has a choice of several basic wave forms, but each wave form has several parameters to adjust its spectrum. Elektron now in 2018 have the Digitone, an FM synth in a Elektrons renown sequence engine.

12. Math

Check the website below for a great guide on the math of FM Synthesis!

((http://www.cs.cmu.edu/~music/icm-online/readings/fm-synthesis/fm_synthesis.pdf)

The FM Formula

Resource pg(528)( https://web.eecs.umich.edu/~fessler/course/100/misc/chowning-73-tso.pdf )

When the carrier and the modulator are both sine waves, the formula for a frequency modulated signal FM at time t is as follows:

FMt= A x sin(Ct=[I x sin(Mt)])

where A is the peak amplitude of the carrier, Ct = 2pi x C, Mt =  2pi x M, and I is the index of modulation. 

The equation for a frequency modulated wave is 

 e= A sin(at + I sinBt)

e= the instantaneous amplitude of the modulated carrier

a= the carrier frequency

B= the modulating frequency

I= d/m = the modulation index, the ratio of the peak deviation to the modulating frequency.

d= peak deviation= the maximum difference between an FM modulated frequency and the nominal carrier frequency

As the modulation index increases from zero, the energy gets distributed from the carrier out to the sidebands. The amplitudes of the carrier and sideband components are determined by Bessel functions "of the first kind" and nth order, J(n)(I). 

The overall spectrum of the signal is dependent upon the not only the deviation, but also the level of deviation, i.e. the modulation index M. The total spectrum is an infinite series of discrete spectral components expressed by a complex formula using Bessel functions of the first kind.

 the total bandwidth  is approximately equal to twice the sum of the frequency deviation and the modulating frequency.

BW ~ 2(d+m)

Trigonometric expansion:

 e = A { J0 (l) sinat 

            +J1(I) [sin(a+ B)t-sin(a- B)]

            +J2(I) [sin(a + 2B)t + sin(a – 2B) ] 

            +J3(I) [sin(a+ 3B)t- sin(a– 3B)

            + ……………………………………………. ). (2)  

  It can be seen in (2) that the odd-order lower-side frequencies, sin(a-B), sin(a-3B), etc, are preceded by a negative sign, and that for an index greater than 2.5, the Bessel Functions (Fig. 2) will yield a negative scaling co- where efficient for some components . Ordinarily, these negative signs are ignored in plotting spectra, as in Fig. 1, since they simply indicate a phase inversion of the frequency component, -sin(0) = sin(-0). In the application of FM described below, this phase information is significant and must be considered in plotting spectra. 

By way of demonstration, Fig. 1e is plotted, but with the phase information included. The carrier and the first upper-side frequency are plotted with a downward bar representing the phase inversion resulting from the negative bessel coefficients. 

These negative components reflect around 0 Hz and mix with the components in the positive domain. The variety of frequency relations which result from this mix is vast and includes both harmonic and inharmonic spectra. 

A simple but very useful example of reflected side frequencies occurs if the ratio of the carrier to modulating frequencies is unity. For the values

c = 100 Hz

m = 100 Hz

I = 4 (I = d/m , or the modulation index, the ratio of the peak deviation to the modulating frequency) 

Bessel Functions

The amplitudes of the individual sideband components vary according to a class of mathematical functions called Bessel functions of the first kind and the nth order Jn(I), where the argument to the function is the modulation index I. The FM equation just given can be re expressed in an equivalent representation that incorporates the Bessel function terms directly:

FMt = Jn(I) x sin(2pi x {Fc = or – {n x fm}])t

Each n is an individual partial. So to calculate the amplitude of the third partial, we multiply the third Bessel function at point I, that is, J3(I), times two sine waves on either side of the carrier frequency. Odd-order lower-side frequency components are phase inverted. 

Resources:

http://www.javelinart.com/FM_Synthesis_of_Real_Instruments.pdf

https://www.teachmeaudio.com/recording/sound-reproduction/fundamental-harmonic-frequencies/ 

https://books.google.com/books?id=nZ-TetwzVcIC&&pg=PA232#v=onepage&q&f=false

 (  https://en.wikipedia.org/wiki/FM_broadcasting  )

https://en.wikipedia.org/wiki/Frequency_modulation_synthesis

http://www.kratzer.at/DXFMSynthesis.htm

https://www.electronics-notes.com/articles/radio/modulation/frequency-modulation-fm-sidebands-bandwidth.php

http://www.cs.cmu.edu/~music/icm-online/readings/fm-synthesis/fm_synthesis.pdf

http://yala.freeservers.com/2fmsynth.htm#2Mod

https://web.eecs.umich.edu/~fessler/course/100/misc/chowning-73-tso.pdf  

https://en.wikipedia.org/wiki/Frequency_modulation_synthesis

https://en.wikipedia.org/wiki/Overtone

https://www.cs.sfu.ca/~tamaras/freqMod/Effect_Phase_in.html

http://www.burnkit2600.com/manuals/fm_theory_and_applications.pdf

http://www.hibberts.co.uk/make.htm

https://method-behind-the-music.com/mechanics/physics/

https://blog.landr.com/adsr-envelopes-infographic/

 If you have any questions about where I got my information, please email me at ([email protected]).

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