Grackle

Grackle from Entropic Loop is a three-operator phase modulation voice with a unique take on a familiar architecture. Phase modulation, popularly known as FM synthesis, gained popularity in the 1980s with the rise of more affordable digital synthesizers such as the Yamaha DX-7 after being developed by researcher John Chowning while at Stanford University in the late 1960s. While Grackle uses the fundamental techniques found in FM synthesizers, its implementation is unique; unlocking a new dimension in digital synthesis with sounds that stretch from the familiar to deep uncharted territory. Although it has a distinctly digital sound, Grackle's interface is immediate and playable in a way that almost feels analog.

Phase Modulation Synthesis

The principles that underlie the record player give a rough approximation of the relationship between frequency and phase applied to phase modulation. Imagine if you record a sine wave onto what is called a locked groove record, i.e. a record that continuously spins on a loop without moving inward. Locked grooves are typically placed at the end of the side of a record and just contain silence. However, they are sometimes used to create infinite loops, and in our hypothetical example, a single sine wave.

The speed that the record player spins determines the pitch of the sound, if it spins faster, then the sound is higher, if it spins slower, then the sound is lower. The position of the stylus on the vinyl record is the phase of the sine wave in this example. If you rapidly change the position of the record in relation to the stylus such as scratching the record, the pure sine wave tone changes in character.

Phase modulation synthesis uses one or more sine wave oscillators to change the phase of another sine wave oscillator (the modulation sine wave oscillators in this example are our turntablists). This transforms the simple sine wave into a more complex waveform with more harmonics, but retains the fundamental pitch of the original sine wave (since it is still rotating at the same speed.)

Interface

Grackle features two modulator oscillators (mod1 and mod2), a carrier oscillator, and an envelope generator. Grackle’s main controls are broken up into sliders, knobs, and switches. The sliders are the main controls, and the knobs are attenuators. Grackle defaults to a base note of C, and the frequency slider changes the octave. Holding the shift button provides fine tuning to change the C to a different root note. The switches select the algorithm, interval divisor, and the envelope mode. The algorithm and interval switches both have CV inputs that add or subtract to the switch position. Grackle’s main output is the carrier output, but also features the result of the modulator oscillators before they phase modulate the carrier oscillator. Additionally, the envelope used for the VCA has a dedicated output.

The knobs on Grackle are all black except for the two depth CV attenuators, which are pink. There is a line on the panel from the envelope output to the depth CV input, which indicates that the envelope is normalled to these CV inputs. Inserting a cable overrides this routing. This preconfigured routing was selected because we found ourselves manually patching up this configuration often while testing Grackle. It adds a bit of movement to the sound and works especially well for percussive sound. All the knobs on Grackle are attenuators which control the level of the incoming CV, much in the same way as a volume control does. Grackle responds to both negative and positive voltages, but the ability to completely turn off modulation is useful in quickly altering sounds on the fly. Grackle also features a shift button, used to allow access to additional features not available with the front panel controls alone.

There are four kinds of input types on Grackle; attenuated, unattenuated, AC-coupled, and a gate input. The attenuated inputs are all of the CV inputs that have attenuators on the inputs (interval1 + interval2, mod1 depth + mod2 depth, growth, and decay); with the attenuators at max, they have an input range of +/-5.5V. The unattenuated inputs are the interval selection, algorithm selection, and the 1v/oct input. The interval and algorithm CV inputs have a range of +/-5V. A jumper on the rear of Grackle determines the range of the 1V/oct input. By default, the range is -3V to +7V, but switching the jumper makes it +/-5V. The single AC-coupled input is the actualFM input (a play on the fact that phase modulation synthesis is often referred to as frequency modulation synthesis), tied to the audio input on the Daisy microcontroller. The Daisy is a microcontroller platform intended specially for music, including synthesizers and audio effects. These AC-coupled inputs are typically intended for audio effects, such as delays, reverbs, or distortions. This input is processed at the speed of the microcontroller, 48kHz, giving it a much higher resolution. If you take the output of the mod oscillators and feed it back into the actualFM input, it creates a unique form of feedback, since the mod oscillator’s frequency is derived from the carrier oscillator’s frequency. The gate input is tied to Grackle’s envelope, and affects it in different ways depending on the envelope mode.

Algorithms

 The algorithms determine how the modulation oscillators affect the timbre of the carrier oscillator. Each mode imparts a unique sonic character to Grackle’s final output. The three modes are Add (addition), Mul (multiply), and Cas (Cascade).  Add mode uses addition to combine the two mod oscillators, which mixes them together. Cascade utilizes two stages of phase modulation. First mod1 modulates the phase of mod2, and then mod2 modulates the phase of the carrier oscillator.

In both add and cascade mode, the depth sliders and CV inputs simply control the amplitude of the two mod oscillators. Multiply mode is a little different. The depth2 slider and CV input controls the amplitude of both oscillators which are multiplied together. The depth1 slider and CV input applies wave folding to the multiplied oscillators. In both cascade and multiply mode, the carrier oscillator will not be modulated if depth2 is not engaged.

Below is a representation of Grackle’s three algorithms using the online graphing calculator Desmos. You can adjust the sliders to get an idea of how the three algorithms behave. The top graph is the add mode, the middle graph is multiply, and the bottom graph is cascade.

The controls are in order

1. mod1 interval slider

2. mod1 depth slider

3. mod2 interval slider

4. mod2 depth slider

5. interval denominator

6. carrier oscillator pitch

Intervals

Typically, FM synthesizers use the term ratio when describing the relationship between the carrier and modulator oscillators. The pitch of Grackle’s mod oscillators are derived from the frequency of the carrier oscillator multiplied by fraction set by the combination of the interval sliders, interval CV input, and interval switch. The switch selects the bottom part of the fraction (the denominator), or how much the top number (the numerator) is divided by. The options available, as outlined on the panel are 3, 5, and 8. The interval sliders set the top part of the fraction, which is an integer between 1 and 32. That fraction is then multiplied by the frequency of the carrier oscillator’s frequency to derive the modulation frequency. To explain where this technique comes from and how it influences the sound of Grackle, it’s important to review the harmonic series and just-intonation intervals.

There are three aspects of a tonal sound: pitch, loudness, and timbre. Pitch is a function of how humans perceive frequency content.  The number of times an event happens per second determines its frequency, measured in hertz or cycles per second. When we talk about A440, we’re talking about a sound that expands and compresses air particles 440 times per second. The faster the frequency of a sound, the higher its perceived pitch. The distance from one pitch to another is an interval. An octave corresponds to a doubling in frequency, so an octave above A440 is A880.

Timbre is a perceptual characteristic of sound that allows our ear to distinguish two instruments playing the same note. The fundamental frequency is what we perceive as a sound’s pitch, typically the lowest frequency in a sound and often the loudest. However, related vibrations often exist above this fundamental frequency, called overtones. A sine wave is the simplest waveform, and only contains the fundamental frequency with no overtones.  Harmonics are whole number multiples of the fundamental frequency, which are commonly described by the harmonic series. Partials are the overtones that occupy frequency ranges between the harmonics. Although they are less common; subharmonics, or harmonics present below the fundamental frequency do exist in some sounds. When we talk about a sound being more or less inharmonic, we’re talking about the amount of partials versus overtones. The second harmonic is twice the frequency of the root note (an octave), the third harmonic is three times the root note, which is a perfect fifth above the octave. The tuning system used in Eurorack is 1V/oct, where the increase of a volt increases the frequency of an oscillator by an octave, doubling its frequency.

The vast majority of western music for the past several hundred years uses what is called 12 tone equal temperament (12-tet).  This system splits an octave into twelve equally spaced notes, and the difference between each of those twelve notes is a semitone or 100 cents (but not a dollar). One of the biggest advantages of 12-tet is the ability to play a wide range of scales and modulate between them without needing to retune your instrument, or in some cases an entire orchestra. The mathematical ratio of a semitone is the 12th root of 2, or approximately 1.05946. On a guitar, two adjacent frets are spaced by a semitone. Just intonation represents intervals as whole number ratios, which more closely approximate the harmonic series.

One of the most common musical intervals is a perfect fifth. In 12-tet, a perfect 5th spans seven semitones. In just intonation, a perfect fifth is represented as a ratio of 3/2, which is equal to 1.5. Due to a semitone being equal to the 12th root of 2, an interval of seven semitones equals approximately 1.498307, a relatively small difference, but a difference nonetheless. 12-tet and just intonation represent an octave the same way, a ratio of 2/1 or double the frequency.

Let’s focus on the main panel denominators, 3, 5, and 8; and ignore alternative modes accessed with the shift function (see the shift section for more details). Using material properties as metaphor is a common technique for describing the timbral characteristics of a sound. We like to think of each set of intervals as having a its own unifying set of timbral qualities, which we would describe respectively as "glass," "metal," and "enamel." We hope this gives a rough idea of the characteristics of the modes, both in terms of the features of the overtones and the frequency range available in each.

Below is a table with the decimal equivalents to the interval fractions from 1-16 for the 3, 5, and 8 interval modes. The column to the right indicates when the fraction corresponds to a common just intonation musical interval. (note: while a perfect fifth is commonly written as 3/2, this is the same ratio as 12/8, and 10/8 is the same ratio as 5/4, or a major third). With the slider at 16/8, the ratio is 2, or an octave above the carrier frequency. This chart is limited to the first 16 slider positions because there is a pattern that repeats every 3, 5, and 8 values, and we typically look at intervals that span less than an octave.

Although the majority of the intervals are unique to an interval switch setting, there are a few that correspond to intervals in more than one. Slider setting 5 is equal to a major sixth (5/3), and an octave (5/5). Slider setting 8 is equal to both a minor sixth (8/5) and an octave (8/8). 9 equals three octaves (9/3), a whole tone (9/5), and a minor seventh (9/8). 12 equals both four octaves (12/3) and is equal to a perfect fifth (12/8, which can be reduced to 3/2). 15 is equal to both five octaves (15/3), three octaves (15/5), and a major seventh (15/8).

Envelope

The envelope has three modes, triggered, asymmetric slew, and cycling. The modes are modified by growth and decay, which determine the time it takes for the envelope to reach its highest point (growth is often labeled as attack on envelopes) and the time for it to return to its lowest point(often labeled decay or release). Triggered mode causes the envelope to fire off once based on the slider positions. In this mode the length of the input signal does not affect the envelope time. In asymmetric slew mode, the length of the input gate affects the length of the envelope. If the gate input is not active for the time allotted to the growth portion of the envelope, it will be cut short. If the gate extends beyond the growth portion, the envelope will remain fully open for as long as the gate input is high and then decay once the gate input goes low. The shape of the envelope in cycling mode is the same as in triggered mode, but the envelope will repeatedly trigger for as long as the input is high. With no jack inserted into the gate input, a high voltage is normalled into the gate input. In asymmetric slew mode, the envelope is constantly held open, producing a drone. In cycling mode, the envelope will continually cycle. The high voltage is turned off in triggered mode. The envelope has a logarithmic rise and exponential decay, mimicking many percussive sounds found in nature.

Shift Functions

Fine Tuning: Shift + Freq Slider

By default, Grackle is tuned to a C, holding the shift button and adjusting the frequency slider allows you to retune the base frequency of the oscillator. This control has a range of an octave from C to a C an octave above it.

Interval Unlock: Shift + Interval Slider

Holding the shift button and moving either one of the interval sliders its full travel length will put it into “unlocked mode.” In standard operation, the mod oscillators and CV inputs are quantized to integer (whole number) steps with the slider selecting a whole number between 1 and 32 for the numerator of the interval fraction. Positive CV values affect the numerator, but if negative voltages cause the numerator to drop below one, the CV increases the denominator (the bottom part of the fraction). In unlocked mode, the denominator is still applied,, but the numerator slider and CV input is not locked to integers and instead includes real numbers -- continuous values that include digits after the decimal such as 1.75 or 3.141… Additionally, the unlocked mode supports thru-zero frequency modulation, meaning negative values cause the mod oscillator to invert its phase and “run backwards.”

Updating the Firmware

Updating the firmware on Grackle is fairly straight forward. You will need a micro-USB cable and Google Chrome.

1. Download the firmware

2. Connect the Daisy microcontroller on the rear of your Grackle to your computer with a micro-USB cable

3. Using Chrome, go to Electro Smith’s Daisy Web Programmer and follow the instructions on the page, or proceed as follows:

4. Hold down the BOOT button and then press and release the RESET button. The BOOT button is closer to the corner of the Daisy microcontroller, while the RESET button is located next to it.

5. Click the connect button at the top of the page

6. Select ‘DFU in FS Mode’

7. Ignore the Flash Blink button and the menu dropdowns and go to the part that says “Or select a file from your computer,” drag and drop the file or hit the browse button to locate it on your computer.

8. Click program, wait for the progress bars to finish up and you’re good to go

Calibration

Your Grackle has already been calibrated, so this process should be unnecessary, but if for whatever reason you want to recalibrate your Grackle, please follow these instructions carefully. Failure to execute these steps properly may cause your Grackle to not track voltages, especially 1V/oct inputs correctly.

1. Power down your case.

2. Remove all cables from Grackle, both inputs and outputs

3. Turn all of the attenuator knobs all the way up

4. Press and hold the shift button, power on your case with the shift button still pressed down, wait for the LEDs to finish blinking, and release the shift button. Failure to hold the shift button down until the LEDs have completely stopped blinking will leave the module in manual calibration mode.

5. Your Grackle should be fully calibrated now.