KEDITOR PHYSICAL

What a sample has is the truth of one recording, every detail of the instrument and of the room. A model has only what you told it, so the art of modelling is knowing which few numbers matter. There are not many: the frequencies of the modes, their levels at the strike, how long each one rings, how the strike reaches them, and what happens when you play harder. The rest of this part is about those numbers.

A linear object struck once rings as a sum of decaying sine waves, one for each mode: y(t) = Σ aₙ · e^(−6.91 t / T60ₙ) · sin(2π fₙ t), where fₙ is the mode's frequency, T60ₙ the time it takes to fall 60 dB and aₙ how hard the strike sets it going. Each term is exactly what a two-pole resonant filter does when it is struck, so a modal synthesizer is an exciter feeding a bank of resonant filters in parallel, one a mode, their outputs summed.

In samples a resonator is two multiplications and two memories: yₖ = 2R cos θ · yₖ₋₁ − R² · yₖ₋₂ + xₖ, where θ = 2π f / fs sets its pitch and R = e^(−6.91 / (T60 · fs)) its ring. Modal synthesis is at its best on struck and plucked objects with inharmonic modes: bars, bells, plates, drums. Its cost grows with the modes, and a bell wants dozens.

A string's motion is two waves travelling in opposite directions, reflected and inverted at each end. A digital waveguide keeps them in two delay lines as long as the string, the sample rate over the note's frequency, and gathers all the string's losses and stiffness into one filter in the loop. Every harmonic comes out at once for the cost of a single delay line, which makes the waveguide the cheapest model of harmonic, sustained instruments: strings, bores, the vocal tract.

Karplus-Strong is the simplest case: a burst of noise recirculating through one delay line and a two-point average. The average takes a little more of each harmonic on every pass, the upper ones most, and the noise becomes a plucked string. Put a nonlinear element at one end, a model of a reed or of a player's lips, and the loop sustains itself: the only method in which the model itself keeps a note going.

The most literal methods cut the object into small pieces and compute every one. In a mass-spring model a string or a plate is a lattice of masses joined by springs, each mass pushed by its neighbours. Finite differences solve the wave equation itself on a grid in space and time: the next displacement of every point from its own last two and its neighbours'. They handle anything, stiff strings, collisions, cymbals, a piano's hammer, felt and soundboard, a room. They are also by far the most expensive: millions of operations a second for a single string.

The oldest and cheapest physical idea: a source rich in harmonics, shaped by a filter that stands for the resonator. The voice works this way, the vocal folds' buzz through the formants of the throat and mouth, and so, nearly, does a saxophone: a reed's tone, every harmonic in it, through the resonances of its bore and bell. The filter is linear and fixed, so it can be measured on a real instrument and copied. What source-filter cannot do by itself is the loop: the pitch is the source's, not the resonator's, so it never overblows or squeals.

Around them sit the hybrids: commuted synthesis, which folds a body's response into the excitation; a sampled attack over a modelled resonance, as in modelled pianos; and FM, which its inventor presented as a model of what we hear rather than of the instrument, though each of its operators can stand for a decaying mode.

A string under tension, fixed at both ends, has its modes at whole multiples of the first: its harmonics, each sin(nπx) in shape. Real strings are a little stiff and their upper modes run sharp, fₙ = n f₁ √(1 + B n²), which a piano's tuning has to live with.

A bar free at both ends, a marimba's or a glockenspiel's, bends instead of stretching, and its modes go as the squares of a series of numbers: 1 : 2.756 : 5.404 : 8.933. That is why a plain bar sounds like a clang. Marimba and vibraphone bars are thinned underneath until their second and third modes sit at 4 and 10 times the first, a xylophone's at 3; each hangs on cords at its first mode's nodes, 22.4 % from each end, so that the cords do not damp it.

A tine clamped at one end, a kalimba's or a music box's, goes 1 : 6.267 : 17.55: its second mode lies far above, a bright tick in the attack.

A string is a very thin bar. It resists bending a little, and the resistance matters more the shorter the wave: fₙ = n f₁ √(1 + B n²). B is tiny, about 0.0004 in the middle of a piano, and the upper partials still run many cents sharp of the whole multiples. A piano is tuned with its octaves stretched to live with it. Thick, short strings have the largest B, which is why the bass of a small piano clangs.

A circular drumhead's modes are Bessel functions, Jₘ(jₘₙ r), each named by two numbers: m nodal diameters and n nodal circles, the rim included. Their frequencies are the Bessel functions' zeros over the first: 1, 1.594, 2.136, 2.296, 2.653, 2.918, 3.156, 3.501, an inharmonic cluster heard as a thud with a pitch in it. Struck in the centre, only the modes with no diameter sound; drummers strike off centre for the rest.

Plates have modes in both directions; a flat plate's go roughly as (m + 2n)², the patterns Chladni made visible with sand. A steel pan's notes are panels of its surface hammered until their modes fall on harmonics, 1 : 2 : 3 : 4, which is why a pan sounds so clearly pitched.

A bell is a shell. Its modes bend the rim into 4, 6, 8 or more waves round its circumference, and the founder tunes the profile until five of them land on a chord: the hum an octave under the note, the prime on it, the tierce a minor third above, the quint a fifth and the nominal an octave above. The note you hear, the strike note, is none of them alone: the ear makes it out of upper partials that sit nearly at 2 : 3 : 4.

A bowl is a bell without the tuning. Its rim bends in 2, 3, 4 and 5 waves, at frequencies near m(m² − 1) / √(m² + 1): 1 : 2.8 : 5.4 : 8.8 for a thin, even rim, 1 : 2.71 : 5.06 : 8.0 in KEDITOR PHYSICAL's Singing Bowl.

The air in a tube resonates like a string: open at both ends, every harmonic; closed at one end, the odd ones. Tone holes let the high frequencies out of the tube, which gives a clarinet its cut-off near 1.5 kHz; a trumpet's bell lets its high frequencies out and, with the mouthpiece, pulls its modes into a nearly harmonic series.

A body is a resonator that is not meant to set the pitch: a harp's soundboard, a singer's throat and mouth. Its modes are many, broad and fixed in frequency whatever the note, so each note's harmonics are coloured differently. The voice's are its formants: a tenor's sung aah has them near 650, 1080 and 2650 Hz, and a change of vowel moves them while the note stays where it is.

An object much smaller than the wavelength mostly pushes the air aside without compressing it. How much of its movement becomes sound goes, for a source of radius a, as (ka)² / (1 + (ka)²), with k = 2πf / 343. At 100 Hz a 3 cm source is about 16 dB less efficient than a 20 cm one, and that is the whole reason a small object has no bass.

Instrument makers answer in two ways. A soundboard is a large, light surface driven by a small, stiff vibrator: a harp's, a piano's. A resonator is a second tuned system: a marimba's tube, closed at the bottom and a quarter of a wave long, sits under the bar and takes up its first mode and nothing else.

A tube that draws energy out of the bar also shortens its ring, a trade every builder makes. In a model the radiator is one more filter on the whole sound, or simply a mode's level: a marimba's first mode is loud because of its tube, and its level on the panel says so. The body bands of KEDITOR PHYSICAL's winds and voices are the same idea: fixed resonances in Hz that colour every note differently.

These are where the other methods earn their cost. On a K2700 the way round is to imitate what is heard: a filter that moves as a note dies, a second layer that opens with the force of the blowing. KEDITOR PHYSICAL's EDGE is one of those, a rasp heard only above a threshold of blowing, built in step 14 of Part four.

Mode 1 first. Everything else is heard against it. Set its level and its ring before you touch the others, and when a sound goes wrong come back to it.

Spend voices last. Every mode costs one voice for every note. If taking a mode away changes nothing you can hear, it was a voice you did not need.

Do not chase a level you could get elsewhere. A mode's LEVEL is its share of the strike. VOLUME is the program. LOW/HIGH tilts the keyboard. Each has its job, and the three together are how an object stays even from C2 to C6.

The Marimba is the reference: a bar thinned underneath until its upper modes sit at 4 and 10, rosewood's short rings and a yarn mallet that barely reaches the second one when played softly. The Xylophone is tuned to the twelfth, 1 : 3, and struck with a hard mallet: the same family, and a much drier, brighter sound.

The Vibraphone is the Marimba's ratios in aluminium: seven seconds asked, and dampers on the key. The Glockenspiel is a plain free bar of steel, its ratios the ones the physics gives, and it is the only one of the five to use them. The Wood Block shows how short a sound can be and still have a pitch: a fifth of a second at C4.

The Harp plucks a string at 15 % of its length, near the node of its seventh harmonic, and that one mode drops to −26 dB: the point decides the colour of a string more than anything else. Eight modes make it the most expensive plucked object, nine voices a note. The Kalimba is the cheapest instrument here: a clamped tine, two modes, three voices. Its second mode, far above the first, is the bright tick at the start of each note.

The Tenor's bands are the Alto's at three quarters of the frequency and the Baritone's are half of it, the ratios of their lengths. That is the most physical thing in the program: the body scales, the reed does not.

They are played as a wind is. WIND MAP gives the aftertouch to the breath, the pitch bend to the lips and the mod wheel to the vibrato. LEGATO keeps one voice and slides between notes. BREATH ties the brightness and the pitch to how hard the note is being blown. FLOW AIR makes the air ride the tone, in bursts at the note's own period.

Both are objects from the strike family with the mallet taken away. The Singing Bar is the Vibraphone's bar with its upper modes much weaker, since a friction locks mostly to the first. The Bowl's modes are a rim's. Both swell slowly, and the wheel or the aftertouch opens the friction and makes it louder, an octave and 6 dB at full pressure.

When an object asks more than this, the MODES display shows what you asked and what you will get. There are several honest answers.

Let it be. A bell's top modes dying sooner than they would is not what the ear objects to, as long as mode 1 is long. Move the object down. One octave down the ceiling doubles. Give the time to the hall. A hall of 3 to 5 s behind a mode at its ceiling is what the Church Bell does, and the tail you hear is largely the hall's. Thin the modes up top. A mode at ×10 that lasts 90 ms at C4 lives far inside the ceiling; it is only the long ones that hit it.

KEY DECAY says how the rings follow the keys: T60 ∝ f^−KEY DECAY. At 1 a ring halves an octave up, and every key plays the same Res, which is how a struck resonator rings. At 0 the ring stays the same and the Res rises half a dB a key. MATERIAL sets KEY DECAY to its own a: wood falls faster up the keyboard than metal.

Velocity is two things at once, and that is the difference between a model and a sample set played louder: the level moves by the object's VELOCITY in dB, and the brightness moves by VEL SOFT in octaves. On a marimba a soft note is 30 dB down and two octaves darker.

The tone's brightness is the FILTER's and its attack and release are the AMP ENV's: on a breath those sections take over from what they do on a strike.

When you turn a knob the LINK sends only what changed: a few parameter messages for most knobs, the whole program when the layers change, and the Info a second later. When the K2700 is sitting on another program the display says so, and your edits stay in the app until you send them with LINK or by clicking that display.

A resonant low pass's peak is a mode: the higher the Res, the taller and narrower the peak and the longer the filter rings after a click. The first test on the K2700 struck a single 2POLE LOWPASS following the keys and measured its ring at C2, C4 and C6. At Res +48 dB it rang 7.3, 2.05 and 0.54 s, at +36 dB 2.07, 0.54 and 0.13 s: 12 dB less Res, a ring four times shorter, just as Q = 10^(Res/20) says. Every ring was a clean sine at the key's pitch, within a few cents.

Later sessions found one correction: under 1 kHz the K2700's resonator rings a little shorter than the law, 0.90 to 0.93 of it at 130 to 260 Hz and 0.81 at 65 Hz. The compiler counts it and gives the bass modes the Res they need.

The compiler turns each mode's ring into Res: Res = 20 · log10(T60 × f / 2.2), less what the layer's own envelope takes, in the K2700's half-dB steps. The Res follows the keys too. With KEY DECAY at 1 a ring halves an octave up and the Res stays the same on every key, as a struck resonator's would; at 0 the ring stays the same and the Res rises half a dB a key.

Move the mode and the ring it asks. The numbers are the ones the compiler works out: the Res it writes, the ring the K2700 plays with that Res, and the ceiling at that frequency.

+48 dB is as far as the K2700's Res goes, and it sets the longest ring a mode can have: 2.2 × 251 / f, 553 / Hz seconds, a little less in the bass. Seven seconds at C2, two at C4, half a second at C6. A marimba, a xylophone, a tom or a wood block fit easily. A vibraphone, a glockenspiel, a bell or a glass bowl ask for 6 to 11 s in the middle of the keyboard and get 2; in the bass they ring long. KEDITOR PHYSICAL draws the ceiling under every mode and says when a ring is cut to it.

Ways round it were tried on the K2700. Two 2POLEs in series rang 2.06 s at C4 against 1.88 for one, too little to matter; a slow, quiet re-excitation could not hold a mode up; and a narrow BANDPASS, at 0.02 of an octave, saturated into a square wave and went on sounding after the notes.

The strike layer (ALG 2) starts with a SQUARE oscillator at −120 semitones, 0.26 Hz, its phase reset on Note On: through the few milliseconds of a strike it is one steady level. Two GAINs opened by ENV2 for 5 ms cut it into a pulse. Between them a one-pole LOPASS is the mallet's hardness, opened two octaves by the velocity and following the keys half a semitone a key; the second GAIN carries the level's key tracking. The layer's own AMP stays at −96 dB.

Each mode is a layer of its own (ALG 107) whose input is layer 1, taken after its four functions and before its AMP. Its 2POLE LOWPASS sits at the mode's ratio of the note and follows the keys 100 cents a key, so the whole object is tuned by the key. Its AMP is the mode's level, with the object's range of velocity (30 dB on the Marimba), and its envelope lets it ring to the end. A mode that would pass 16 kHz stops at the key where it would, so high notes use fewer voices. The strike's envelope outlasts the modes' by 10 %, so no mode loses its input before it ends.

The first strike was a click of noise, and on the K2700 three identical strikes rang 2.5 to 15 dB apart: the share of a noise burst that falls on a mode's frequency is a matter of chance, by about 5.6 dB. A real mallet does not do that. With the pulse, 29 sets of identical strikes on the K2700 differed by a median of 0.06 dB. The pulse's length, 2.5 to 20 ms, did not change its level at 262 or 1046 Hz on the K2700, so the hardness is all in the LOPASS.

Every layer of a strike has an AMPENV in User mode: an attack of nothing to 100 %, then a single Dec1 down to 0 %, as long as 15 times the longest ring the program gives at C4, at most 60 seconds. A K2700 Dec1 falls straight in dB, 96 dB in its time, so its T60 is 0.625 of what the page says, and the ring you asked for is the resonator's and the envelope's together: 1 / T60 = 1 / ring + 1 / (0.625 × Dec1). The compiler gives the resonator a little more Res to pay for it.

Under the page's Decay KeyTrk, 2.000x makes the envelope twice as fast each octave up, exactly as a ring does. Without dampers every layer has Ignore Release on, so the note rings on after the key; with dampers it is off, and Rel1 is the damper's time. The exciter's own envelope is 10 % longer than the modes': a layer whose input stops takes its reader's voice with it.

A sustain is written through the left-hand curve. If the AMP ENV asks for −3 dB the page says 91 %, because 90 % plays −3.4 dB and 100 % plays 0.

Layer 1 (ALG 2) is NOISE, a LOPASS that the velocity and the pressure open, a GAIN gate that ENV2 holds open while the key is down, and a level GAIN that the pressure raises. Its AMP stays at −96 dB. The modes are the strike's layers, ALG 107, tuned by the key.

A resonator's share of a noise grows with its frequency and its Q, because its peak is Q tall and f / Q wide. Each mode's AMP takes 10 log10(π f Q / 2) off for it, the friction layer's GAIN key tracks by −(2 − KEY DECAY) × 3 dB an octave to take the rest back, and each mode's AMP gets back the 1.2 dB an octave of its frequency by which rubbed modes fell short of that law.

The AMPENV's release is the modes' ring, not the friction's. Let the key go and ENV2's release shuts the friction in the time of the AMP ENV's RELEASE, while the modes go on ringing.

Sharp bands run into a limit of the K2700's DSP. Its blocks are fixed point, and a block's output clips 6 dB over full scale: a band sharper than about Res +18, fed the full saw, has its 2POLE stuck near +20 dB at the peak, the band near +14, and distorts. So the first HIPASS of every band over Res +15 pads its input by 6, 12 or 18 dB, one step for every 6 dB its Res passes +15, and the layer's AMP makes them up. Each band then plays its true peak within 0.4 dB up to Res +36; the broad bands, the trumpet's among them, are untouched.

Each formant is a 2POLE LOWPASS at its frequency, its Res its Q: 20 log10(Hz / width). A chain of low passes passes the source whole under the first formant, and each formant's level then comes from where it is and how wide it is, which is how a tract behaves. A soprano's A has formants at 800, 1150, 2900, 3900 and 4950 Hz.

The source is a saw (ALG 7: a GAIN, a two-block SAW, NOISE+ for the breath through the tract), 18 dB under full scale on its Pad, so that a formant on a strong harmonic keeps under the blocks' ceiling; the last layer's AMP makes it up. A choir is three saws (ALG 5: TRIPLE SAW) and a second layer with the tone's filter and the breath; with that filter out of the tract, the tract is one layer shorter and the choir is still five voices.

On the K2700 what a layer hands on through the Alt Input comes 24 dB under what it plays when heard, and a voice has four such passages. Left at their defaults, the voices play 58 to 73 dB down. The answer is in the Pad: the second 2POLE of each layer that another layer reads goes at Pad 0, which gives the 24 dB back, and the first formant keeps its Pad 3, without which a soprano's G5 on A distorts. The levels came up to −12 to −27 dB, within 1.5 to 3.7 dB of plan.

A second vowel is the same chain with each formant moved. The wheel drives each frequency by the cents between the two vowels and each Res by the difference between their Qs. With AUTO on, FUN1 = max(the wheel, ASR2) drives them instead, and ASR2 is the diphthong: its delay is HOLD, its attack is SLIDE, a straight ramp in cents, and its release is 30 seconds, so the note's tail stays on the second vowel.

It is a layer of its own (ALG 102) reading the tone: a HIPASS at 1.5 kHz, a DIST at +24 dB, the HIPASS again and a LOPASS at 3 kHz. On the Clarinet the DIST comes first, squaring its fundamental, then two HIPASSes and the LOPASS. The DIST is always torn; what changes is whether you hear it. A drive that followed the blowing jumped in level as it started to clip.

The opening is three FUNs. FUN2 = (FUN1 + 2 × velocity) / 3 is how hard the note is being blown. FUN3 = diode(FUN2 − 0.6). FUN4 = FUN3 + FUN3. FUN4 drives the AMP by 80 dB, so the layer is 64 dB under its top at 0.6 of the blowing, opens a little at full velocity, and all of it with full aftertouch.

GROWL is LFO2 at 24 Hz on the Edge's AMP, by knob 8 (CC 3). An open bore has no reed to tear, so on the Flute GROWL swings the sines' and the air's levels instead, up to 12 dB: a flutter-tongue.

The hall is the K2700's factory Hall Reverb 1 on aux 1, which every K2700 has; its time is the K2700's knob 9, CC 9, 0.5 to 10 s, and the panel's TIME sends the value that gives it. The delay is PHYS Delay, the app's own chain on aux 2: one delay box, its repeat from a quarter of a beat to a beat and a half at the K2700's tempo on CC 84, its feedback 0 to 90 % on CC 19.

A K2700 loads a program's aux chains when it enters that program, not when its edit buffer changes. So when the LINK goes on, KEDITOR PHYSICAL reads which program the K2700 is on; if that program's effects are not its own, the display says whose you are hearing, and STORE with SELECT puts the K2700 on the stored program, after which everything plays with its own hall and delay.

Renaming PHYS Delay on the K2700 makes the app lose it, and it would offer to write another. Renaming a program does nothing: the name is yours.