Yamaha DX7 - FM & Timbre in a digital world

 I have recently been exploring the Yamaha DX7 and thinking about how it compares with standard analog subtractive synthesizers....... and contrasting Analog FM with Digital FM

There is something very special about the DX7 and how it produces sounds.
The Yamaha DX7 is of course digital and for many, the DX7 = FM synthesis.

However, FM isn't the sole domain of digital. FM existed in the analog world before the DX range was produced. Below is a pic of a Buchla oscillator... the 158. You can see that Don was very much into wave shaping & FM. This module dates from around 1968/69.

The pic below is of a Buchla 208 with its analog FM, dating from 1973.

Notice the word "timbre".

So in a non FM analog subtractive synth you usually start with harmonically rich wave forms like saw, square, etc. and you use filters to smooth the sound out a bit (technically we are subtracting overtones).
In analog FM synthesis, you start with a wave form and you then use other waves to distort it.... to create new tones & harmonics.

The best way to see this is with a spectrum analyser. This will show the additional harmonics created by the FM process.

Timbre

We come across this term again and again. It is also called tone or colour. What exactly is it?
It's one of the 3 main parameters of sound. ...which are volume, pitch & timbre.

The Oxford dictionary says that
"timbre" describes those characteristics which allow the ear to distinguish sounds which have the same pitch and loudness.

For example a piano, a flute and a guitar may all play the same (frequency) note, at the same (amplitude) loudness, but we can still tell them apart. This is due to their different timbres.

The timbre of a sound depends on its wave form. Though the frequency & amplitude may stay the same what varies are the number of overtones, or harmonics, that are present, their frequencies, and their relative intensities. 

To elaborate further, the timbre of a tuning fork is clear and pure because the sound it produces is almost without overtones. It vibrates at 440Hz. This is known as it's fundamental frequency.
You can use this fork to tune our piano, flute or guitar to 440 Hz (Middle A), but each instrument will sound different because they have unique overtones which are related to the fundamental.

Harmonic          Frequency
First                   440 Hz
Second               880 Hz
Third                  1320Hz
Fourth                1760Hz
Fifth                   2200Hz

To calculate the next harmonic, you of course just add 440.
  These relationships appear to be very important when using the Yamaha DX FM system.
This is where concept of Frequency Modulation Ratios is impt. 
Harmonic overtones are created based on the ratio between the carrier frequency & the modulator frequency. Musically pleasing overtones are based on simple fractional ratios like 1:2, 1:3, 1:5, etc
Complex fractional ratios like 1.73 produce complex waveforms when combined with operators set to other ratios.
 
Below is a list of the ratios used in the DX21

 

What's important to remember is that when you hear any sound (in nature or music) you are hearing lots of different frequencies all mixed together.... fundamentals, harmonics of the fundamentals, 
dis-harmonics / en-harmonics of the fundamental, etc etc..... this is timbre.

More harmonics = more "musical" sounds
More dis-harmonics = more "non-musical" sounds

Digital synths generally produce sounds in two ways ... FM or additive
Additive digital synths like the Fairlight and the Synclavier can specify a fundamental frequency and add overtones (of harmonics & disharmonics) to create new sounds.  The Fairlight allows sampling where it will analyse a inputted sound, and recreate the sound using its additive synthesis ability.

Though the DX7 can do some additive, this is not it's main way of making new sounds.
It's method uses Digital FM
It's all accomplished in a virtual world consisting (not of waveforms distorting one another),. but of programs in a computer calculating what would have happened if these waveforms had really existed and were distorting one another.

The core of the DX7 is the operator.
Its not a physicial thing like a string on a guitar or an oscillating electrical signal.
It's a bit of software.
You can picture it something like this:

The Oscillator is the sound source.
It generates numbers.

The operator also has a digital amplifier and EG
The amplifier's job is to increase or decrease those numbers exiting the oscillator.
It may be changing volume (if the operator is a carrier).
The EG will send numbers (issuing software instructions) to effect the amplifier.... its changing volume over time in this example.

Note That the operators oscillator has two inputs:
1. pitch data
2. modulation data

When the output of one operator is patched into the input of another operator, the result is a complex waveform. This is the essence of FM synthesis.
By varying the ratio of the modulator & carrier frequencies, and by varying the envelope of the modulator, you can create some very complex waveforms.

They use DACs to convert those numbers to analog voltages which we can hear.

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