Acoustics & Music Theory
why it sounds like that
Concept

Optimisation — where it appears

Choosing the best member of a set of candidates under a stated objective and stated constraints. It is used here where a musical decision has too many options to enumerate by hand, and it takes care to avoid objectives that need an exchange rate between two units.

Named by 28 essays across 4 fields — each of them below, with the objects they name alongside it.

Every assignment, at equal levels and at its own best balance. The 6 ways of putting 3 players on a chord, each drawn twice: hollow at equal levels, which is what an assignment ranking sees, and filled at the levels that balance the parts and then minimise roughness. Every scoring here is at the same total loudness, 23.3 sones, so two points are comparable. Solving the discrete problem first picks violin · clarinet · oboe; solving both at once picks violin · oboe · clarinet, and the two-stage answer costs 9.0 per cent more roughness. The orderings do not keep their places between the two columns, which is the whole of the argument: a ranking taken at equal levels is not a ranking.

Who plays what and how loud is one question

Two lines of argument, one about spectrum and one about loudness, each stopped at the same wall and each said so. One of them can choose who plays which note and has every player at the same level; the other can choose how loud each part is and has nobody assigned to anything. Put together they are a single problem with two kinds of variable, and solving it in stages picks a different answer from solving it at once — nine per cent rougher, at the same loudness, on an ordinary triad.

form · Orchestration What a competition decides when the two cues do not agree. Each spectrum with its two cue readings and the grouping the competition chooses. Harmonicity asks whether a partial is near enough a whole multiple to belong; common fate asks whether it decays at the same rate as the rest. Where they disagree there is no rule in this collection, so the published apparatus is used instead: every way of splitting the partials into one stream or two is scored for the partials each cue says it has wrongly grouped and wrongly separated, and the cheapest wins. an ideal string — harmonicity 100 per cent, common fate 20, and the competition says one stream; a piano string — harmonicity 90 per cent, common fate 20, and the competition says a cut after partial 2; a bell — harmonicity 88 per cent, common fate 13, and the competition says a cut after partial 1; a bar — harmonicity 33 per cent, common fate 67, and the competition says a cut after partial 4; a kettledrum — harmonicity 60 per cent, common fate 100, and the competition says one stream. The exchange rate between the two cues is the number nobody here can supply, so what is reported beside each is how many decades of it leave the answer unchanged.

The exchange rate nobody has

There are now two cues that disagree about how a spectrum divides, and every figure so far reports them separately because there is no principled way here to weigh one against the other. The published apparatus is a competition between grouping hypotheses with a cost per cue — and the useful thing it produces is not the winner but how much of the exchange rate the winner survives.

perception · Auditory scene A written dynamic is an instruction to the listener's impression. Every earlier scoring holds one chord still. A passage is a succession, and the running impression of loudness carries a chord into the one after it, so what a marking asks for and what playing the marking produces are different things. Here is a five-chord passage with a written shape. Playing each chord at its own written loudness gives the running impression 2.4, 3.0, 4.2, 5.6, 4.0 sones against the 2.4, 3.0, 4.2, 5.6, 2.0 that were asked for — right until the last chord, where it misses by 2.0. Solving for levels that make the impression arrive at the marking does not fix it: the last chord's target is I, two parts, and it is unreachable — the correction runs to silence and the impression still sits 1.1 sones above. A subito piano after a full chord is not a level a player can produce. It is a rate of change, and the smoother's two-second release is what refuses it.

A subito piano is a rate, not a level

All three earlier essays score one chord held still. An orchestration is a succession, and the running impression carries a chord into the one after it — so a written dynamic is an instruction to the listener's impression rather than to the instantaneous sound, and there are markings that cannot be produced at all. The correction runs to silence and the impression still sits above the target.

form · Orchestration The longest silence against the spread of the gaps. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.94, so the longest silence is not a second thing to know about a timeline; it is the same thing measured more crudely. Against the other axis on the frontier — how much of the cycle has to be heard before a listener knows where in it they are — it correlates at -0.02, which is nothing. The named timelines are marked, and all of them sit at 4, the shortest longest-gap any pattern of this size can have.

The longest silence is not a third axis

The frontier between evenness and locatability was drawn twice and both times against two quantities, with the third job a timeline does left uncomputed. Computed, neither candidate for it is a third quantity: the longest silence ranks the whole census with the spread of the gaps at 0.94, and syncopation is not a property of a cycle at all — every one of the 273 patterns changes its count when the bar line moves. What the request was actually asking for is a cap rather than an axis, and under the tightest one the census admits thirteen patterns and every named timeline is among them.

rhythm · Euclidean rhythm The schedule that hears every entrance best holds the high parts back. Six parts waiting to enter a five-chord passage over four sounding players, each entering once and staying: the schedule under which the least audible entrance is as audible as it can be made. brass on E3 enters at I, open with -1.4 decibels of mean margin over the mask; oboe on E4 enters at vi, close below with -2.6 decibels of mean margin over the mask; clarinet on G4 enters at I, open with -1.4 decibels of mean margin over the mask; voice on C5 enters at IV, close above with 2.1 decibels of mean margin over the mask; violin on G5 enters at V, bracketing with 6.9 decibels of mean margin over the mask; flue pipe on C6 enters at I, hollow with 4.0 decibels of mean margin over the mask. The least audible entrance is at -2.6 decibels and the margins sum to 7.6; of all 15625 schedules 0 have a better least audible entrance and 576 a larger sum.

Room is used up by whoever enters first

The chord with the most room for a part entering alone is a fact about that chord. It stops being a fact the moment two parts want it, because each part that comes in raises the mask over everybody after it. Given six parts waiting to enter a five-chord passage, choosing each part's moment the way one part's moment is chosen puts three of them into the same chord and lands in the bottom fifth of all 15,625 schedules. Placing them one at a time does no better. The schedule under which the least audible entrance is heard best is unique, and it brings the low and middle parts in while the texture is thin and holds the three highest back for the last three chords — because a high part keeps its room over a full texture and a middle part does not.

form · Orchestration Counted in beats, the abandoned chord comes back sooner and by steps. The piece in standard tuning with its G major, open chord given more and more of the piece, searched twice: once minimising the mean departure from just in cents, once minimising the mean beat rate. The vertical axis is how far that chord sits from just at each optimum. Counted in cents it stays at 13.29 cents until it holds exactly 44 beats — the same as the rest of the piece — and then drops to zero in one step. Counted in beats it drops at 30.4 beats to 4.96, then at 47.4 beats to 3.91, then at 73.1 beats to 0.00 cents. At the piece's own 8 beats the two counts agree exactly: the corner does not move, and what the beats change is where the steps are.

Counting beats moves the price of a chord, not the tuning

The search that found a guitar piece's best tuning counted every cent of error alike, and the obvious objection is that a cent of a third beats faster than a cent of an octave. Counted in beats instead, every piece gets exactly the same tuning back. What changes is how much of the piece the abandoned chord has to hold before it is rescued — thirty beats instead of forty-four, arriving in three steps instead of one.

tuning · Open strings Two shapes asking fifteen pairs of strings for different things. Every pair of the six strings, and the difference in cents between the two strings' offsets that would make the interval each shape puts on that pair exactly just. The piece's chords fall into two families that never disagree with themselves: E major, open and A major barred at 5 (44 beats), and G major, open (8 beats). They disagree on 13 of the 15 pairs: E2–A2 is asked for 1.96 and -13.69; E2–D3 is asked for 0.00 and 1.96; E2–G3 is asked for -13.69 and 0.00; E2–B3 is asked for 1.96 and -13.69; A2–D3 is asked for -1.96 and 15.64; A2–G3 is asked for -15.64 and 13.69; A2–E4 is asked for -1.96 and 13.69; D3–G3 is asked for -13.69 and -1.96; D3–B3 is asked for 1.96 and -15.64; D3–E4 is asked for 0.00 and -1.96; G3–B3 is asked for 15.64 and -13.69; G3–E4 is asked for 13.69 and 0.00; B3–E4 is asked for -1.96 and 13.69. The ring on each row is where the piece's cheapest tuning actually puts the pair. It sits on the first family's demand every time, which is what abandoning the other chord means: no weighting of the error can put a ring on two different places.

The chord a tuning gives up is a fingering

A guitar piece's best tuning makes two of its chords exactly just and abandons the third by thirteen cents, and no way of counting error — cents, beats, squared cents, or only the beats slow enough to hear — brings the third chord back. Fingered as a barre chord in the shape the other two already use, it comes back completely, and the whole piece is exactly just. The conflict was never between chords. It was between hands.

tuning · Open strings With 4 of six required at the end, the best schedule hands parts over. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 4 of the six sounding at the last chord. I, open: clarinet on G4 enters at 0.6 dB; vi, close below: oboe on E4 enters at 0.3 dB, and clarinet leaves; IV, close above: brass on E3 enters at -0.7 dB, voice on C5 enters at 2.2 dB, and oboe leaves; V, bracketing: violin on G5 enters at 7.2 dB; I, hollow: flue pipe on C6 enters at 4.0 dB. The least audible entrance is -0.72 dB against -2.65 for the best schedule in which nobody leaves; the walk has 2 exits and 6 entrances.

An exit is worth nothing until the tutti is given up

Six parts entering a five-chord passage have a best schedule when each enters once and stays, and letting parts leave and come back was supposed to improve it. Searched over every set of sounding parts at every chord, it improves it by exactly nothing, with or without the chord before still masking — as long as all six must be playing at the end. Let one part be missing from the final chord and the weakest entrance gains 1.4 decibels; let two be missing and it gains 1.9, by a relay in which the parts with least room come in, are heard for one chord, and give way.

form · Orchestration Tuned for the E shape, a cent of intonation is worth about five frets of hand movement. Every way through a ground on E major — I, IV, I, V, IV, I, for 16, 8, 8, 4, 4 and 8 beats — that is the cheapest route at some exchange rate between the piece's mean departure from just and the frets the hand moves, choosing each chord's fingering from its five moveable shapes; once with the guitar tuned to make the E shape exactly just and once on equal-tempered frets. Tuned for the E shape: 12.09 cents for 0 frets; 4.30 cents for 2 frets (worth it above 0.3 frets a cent); 2.09 cents for 10 frets (worth it above 3.6 frets a cent); 0.00 cents for 20 frets (worth it above 4.8 frets a cent). On the frets: 6.85 cents for 0 frets; 6.15 cents for 2 frets (worth it above 2.8 frets a cent); 5.95 cents for 4 frets (worth it above 10.2 frets a cent); 5.78 cents for 10 frets (worth it above 34.5 frets a cent); 5.60 cents for 20 frets (worth it above 57.5 frets a cent). Tuned for the shape, the route that keeps every chord in it is the cheapest once a cent is worth 4.8 frets; on the frets it needs 57.5.

Tuned for one shape, a cent is worth five frets

A guitar can be tuned so that one chord shape is exactly just wherever it is moved on the neck, and every other shape is then between twelve and fifteen cents out — worse than on the frets. So a player of such a tuning chooses, chord by chord, between intonation and the distance the hand travels. Searched exactly over every fingering of a I–IV–V ground, the route that keeps every chord in the one shape is the cheapest as soon as a cent of the piece's mean error is worth 4.8 frets of movement. On equal-tempered frets the same route needs a cent to be worth fifty-seven.

tuning · Open strings In every key with an open chord, the shape to tune for is that chord's. For a I–IV–I–V–IV–I ground in each of seven major keys, how far the hand moves on the route that keeps every chord exactly just, with the guitar tuned for each of the five moveable shapes, in frets. C: C 20, A 24, G 26, E 28, D 28; D: C 24, A 24, G 20, E 28, D 16; E: C 24, A 20, G 28, E 20, D 24; G: C 24, A 28, G 16, E 24, D 26; A: C 28, A 16, G 24, E 24, D 20; F: C 28, A 28, G 28, E 24, D 24; B♭: C 28, A 24, G 24, E 30, D 28. The keys whose tonic is an open chord — C, D, E, G and A — are served best by the shape of that chord in 5 of 5 cases; F and B♭ have no open tonic, and every shape leaves the hand further from the nut.

Tune a guitar for the shape of its key's open chord

A guitar can be tuned for any of five moveable chord shapes, and a player who then keeps every chord in that shape plays exactly in tune while the hand travels. Which shape should the tuning serve? Searched over seven keys and all five shapes, in every key whose tonic is an open chord the answer is that chord's own shape — C for C, D for D, G for G, A for A, and a tie between the E and A shapes in E. The reason is not the key and not the progression. It is that an open chord needs no barre: its hand sits one or two frets above the nut, and that saving is the whole of the difference.

tuning · Open strings On a guitar tuned for any major shape, no minor chord is within eight and a half cents of just. Each moveable minor shape — the open E minor, A minor and D minor chords — with its mean departure from just when the guitar is tuned to make one of the five major shapes exactly just, and on equal-tempered frets. E minor shape: 10.30 tuned for C, 10.95 tuned for A, 10.04 tuned for G, 9.78 tuned for E, 10.92 tuned for D; 5.74 on the frets. A minor shape: 10.17 tuned for C, 11.73 tuned for A, 14.47 tuned for G, 8.60 tuned for E, 12.20 tuned for D; 6.65 on the frets. D minor shape: 16.94 tuned for C, 9.78 tuned for A, 10.75 tuned for G, 11.41 tuned for E, 14.66 tuned for D; 8.15 on the frets. The least-bad is the A minor shape on a guitar tuned for the E shape, at 8.60 cents, and every one is worse than the frets.

A tuned guitar has no minor chord in tune

A guitar tuned to make one major chord shape exactly just can play every major chord in tune by moving that shape. It cannot play a minor chord in tune anywhere: every minor shape disagrees with every major shape about some pair of strings by eleven cents or more, and on any of the five tunings the least-bad minor chord — the A minor shape on a guitar tuned for the E shape — is 8.60 cents out, worse than on the frets. Put one minor chord into a I–vi–IV–V ground and the most-in-tune tuning in every key becomes the E shape's, because it is the one that hosts that chord least badly.

tuning · Open strings With its silent string placed for the minor chord, the C shape's tuning is the most in tune in every key. The I–vi–IV–V ground twice round in five major keys: the mean departure from just of the most-in-tune route under each major shape's tuning, with the strings that shape does not sound left where the one-shape tuning put them and placed for the minor chord. C: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. D: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. E: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. G: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. A: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. Under the C shape's tuning every major chord is exact and the minor chord is an E minor shape 6.26 cents out, for a quarter of the piece.

A silent string tunes the minor chord

A continuous search over a guitar's six strings, run on a piece with one minor chord in it, never settles between two chord shapes: at any share of the piece, the best tuning is exact for one chord and makes the other pay. But the tunings the earlier essays called exact for a shape were exact only on the strings that shape sounds. The C shape never sounds its low E string, and tuned to the C chord's own E it hosts an E minor chord at 6.26 cents — not the 8.60 that was said to be the best any tuned guitar could do. The pop ground's most in-tune tuning is then the C shape's, in every key, at 1.57 cents.

tuning · Open strings Made to stay 2 chords, the parts change over in groups rather than one by one. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord, and every part that enters staying at least 2 chords before it may leave. I, open: clarinet on G4 enters at 0.6 dB; vi, close below: nobody enters; IV, close above: brass on E3 enters at -0.7 dB, voice on C5 enters at 2.2 dB, and clarinet leaves; V, bracketing: violin on G5 enters at 7.2 dB; I, hollow: oboe on E4 enters at -1.1 dB, flue pipe on C6 enters at 4.7 dB, and brass and voice leave. The least audible entrance is -1.07 dB against -2.65 for the best schedule in which nobody leaves; the walk has 3 exits and 6 entrances.

A part that must stay changes over in shifts

The best way to keep every entrance audible, when the passage need not end with everybody in, was a relay: a part enters, is heard for one chord, and gives way. Make every part stay at least two chords once it is in and the relay does not collapse into overlaps, as predicted. It changes over in shifts: two parts enter together, one chord passes in which nobody enters, and both leave on the chord their successors arrive. Half of what exits were worth survives a two-chord stay and nearly two fifths a three-chord one — and with the chord before still masking, the stay costs less, not more, because it was the one-chord handover that memory had already made leaky.

form · Orchestration Tuned for the whole route, the I–vi–ii–V ground wants no shape at all until the hand is allowed to move. The best tuning for the I–vi–ii–V ground twice round, found by routing and retuning in turn from seventeen starts, in five keys and at eight exchange rates between a cent of mean error and a fret of hand movement: what chord shape it makes exact, its cents from just and the frets its route moves. C major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 30 frets); 50, C (3.13 cents, 30 frets). D major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (6.52 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 33 frets); 50, C (3.13 cents, 33 frets). E major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, E (4.60 cents, 15 frets); 12, A minor (4.11 cents, 20 frets); 20, A minor (3.78 cents, 26 frets); 50, C (3.13 cents, 49 frets). G major: 1, nothing (7.71 cents, 0 frets); 2, nothing (7.71 cents, 0 frets); 3, nothing (7.71 cents, 0 frets); 5, E minor (4.71 cents, 9 frets); 8, E minor (4.27 cents, 12 frets); 12, D (3.45 cents, 21 frets); 20, C (3.13 cents, 27 frets); 50, C (3.13 cents, 27 frets). A major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (5.90 cents, 3 frets); 8, C (4.11 cents, 15 frets); 12, C (4.11 cents, 15 frets); 20, C (4.11 cents, 15 frets); 50, C (3.13 cents, 35 frets). At the bottom of the range the best tuning is exact for no chord and the hand does not move; at the top it is one major shape with its silent string placed.

A guitar tuned for a still hand is tuned for nothing

Tune a guitar for a whole route through a ground rather than for one chord shape — routing, retuning and routing again until neither changes — and the answer depends on how much the player is willing to move. Valuing intonation highly, the best tuning is one major shape with its silent string placed, as before. In the middle it can be exact for a minor shape instead, because the route that keeps the hand near the nut plays its minor chords open. And for a player who will not move at all, the best tuning is exact for no chord whatever: every string within two cents of its fret, beating equal temperament by a few tenths of a cent. It is the compromise the earlier search went looking for, and it lives where nobody looked.

tuning · Open strings Scored as one arrival, a pair enters as a loud part carrying one that cannot be heard. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord. I, open: clarinet on G4 enters at -10.9 dB, violin on G5 enters at 19.6 dB; vi, close below: oboe on E4 enters at -1.5 dB, voice on C5 enters at 14.5 dB, and clarinet and violin leave; IV, close above: nobody enters, and oboe and voice leave; V, bracketing: brass on E3 enters at -0.0 dB, flue pipe on C6 enters at 25.0 dB; I, hollow: nobody enters. Scored with the parts entering on a chord taken together, at most 2 at a time, as one arrival against what was already sounding: the weakest arrival is 10.93 dB, and the most buried single entrant, measured as itself, is -10.94 dB.

A pair entering as one hides the part that cannot be heard

Scored as two entrances, a pair of parts coming in together is charged for masking itself; scored as one arrival — a new colour against what was already sounding — it is not, and the best schedules change completely. Allowed two at a time, the weakest arrival rises from 0.6 dB above the mask to 10.9, and every pair the search chooses is a loud part carrying a quiet one: a violin 19.6 dB clear beside a clarinet 10.9 under. Allowed any number, the six parts enter at once. The staggered build-up of every earlier schedule exists only because each part was meant to be heard as itself.

form · Orchestration The set's tuning is the D shape's on the strings it sounds, and the set's own on the two it does not. The six strings' offsets from equal temperament, low E pinned, of the single tuning found for the I–vi–IV–V ground in five keys at two, five and fifty frets a cent, beside the D shape's placed tuning, dashed. 2: low E +0.0, A +2.0, D +2.0, G +3.9, B +2.0, high E −11.7, exact for D; 5: low E +0.0, A +2.0, D +2.0, G +4.0, B +2.0, high E −11.7, exact for D; 50: low E +0.0, A +13.7, D +0.0, G +15.6, B +13.7, high E +0.0, exact for C. The D shape's placed tuning: low E +0.0, A +2.0, D +8.5, G +10.5, B +8.5, high E −5.2. On the D, G, B and high E strings, measured from the D string, the five-frets tuning and the placed D tuning agree; they differ only on the two strings the D shape leaves silent.

A guitar tuned for a set is tuned for the D shape

Tune a guitar once for an evening of songs in five keys rather than once per song, and the tuning the set wants is not the compromise the single-song search predicted. At the valuation a typical player puts on a cent, it is exact for the D shape — the one fingering every key's route uses for some chord — with that shape's two silent strings placed for the set rather than for any song. Not retuning between songs costs a fifth of a cent. Only a set that mixes two grounds, played by a player who will not move, wants a tuning exact for nothing.

tuning · Open strings Every decibel taken off the passenger comes off the arrival. The best walk with parts entering together scored as one arrival, at most two at a time, against a floor under each member: no entrant, heard as itself, more than the floor below its mask. Solid lines are the weakest arrival, dashed the most buried member, for two parts required at the end and for all six. 2 at the end: floor 0: arrival 0.59, most buried 0.59; floor 1: arrival 0.59, most buried -0.86; floor 2: arrival 0.59, most buried -1.57; floor 3: arrival 0.59, most buried -2.87; floor 4: arrival 0.59, most buried -3.47; floor 5: arrival 0.59, most buried -3.47; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.16; floor 10: arrival 10.93, most buried -9.90; floor 12: arrival 10.93, most buried -10.94; every part heard as itself, the earlier walks' score, 0.59. 6 at the end: floor 0: no walk; floor 1: no walk; floor 2: no walk; floor 3: arrival -2.65, most buried -2.65; floor 4: arrival 0.50, most buried -3.27; floor 5: arrival 0.59, most buried -4.24; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.13; floor 10: arrival 4.50, most buried -6.13; floor 12: arrival 4.50, most buried -6.13; every part heard as itself, the earlier walks' score, -2.65.

A floor under each member is paid for in steps

Score parts entering together as one arrival and the best pairs are a loud part carrying one that cannot be heard. Put a floor under each member — no part more than so many decibels under its own mask — and the prediction was a schedule between the two, neither staggered nor buried. There is none. The arrival's margin climbs in three steps as the floor is loosened, each step bought with a passenger buried about as deep as the step is high, and below a floor of five decibels nothing beats bringing the parts in one at a time, because the clarinet can clear its mask only alone.

form · Orchestration However many keys the set spans, its tuning is exact for a shape — the D shape from four keys on. The I–vi–IV–V ground in a set of songs whose keys are added one at a time round the circle of fifths from C — C, G, D, A, E, B, F♯, D♭, A♭, E♭, B♭, F — at five frets a cent: the set's mean cost with each song given its own tuning, with one tuning for the set, and on equal-tempered frets, with the shape the set's tuning is exact for above each point. 1 key: own 5.17, set 5.17 (C), frets 7.60; 2 keys: own 5.27, set 5.27 (C), frets 7.67; 3 keys: own 5.27, set 5.43 (C), frets 7.54; 4 keys: own 5.01, set 5.33 (D), frets 7.39; 5 keys: own 5.14, set 5.36 (D), frets 7.43; 6 keys: own 5.23, set 5.41 (D), frets 7.46; 7 keys: own 5.29, set 5.47 (D), frets 7.48; 8 keys: own 5.34, set 5.49 (D), frets 7.49; 9 keys: own 5.38, set 5.51 (D), frets 7.51; 10 keys: own 5.41, set 5.54 (D), frets 7.52; 11 keys: own 5.43, set 5.55 (D), frets 7.53; 12 keys: own 5.45, set 5.57 (D), frets 7.51. The set never stops sharing a shape: up to three keys it is the C shape's, and from four to all twelve the D shape's.

One guitar tuning serves every key, and it is the D shape's

A guitar tuned once for a set of pop songs in five keys is tuned exact for the D shape. The open question was the converse: which songs a given tuning serves, and how many keys a set can span before its best tuning stops being any shape's. Grown one key at a time round the circle of fifths, the set never stops sharing a shape: up to three keys it is the C shape's tuning, and from four keys to all twelve it is the D shape's, with the same offsets to a fifth of a cent. The five-key tuning serves the seven keys outside the set better than the two open keys inside it, and not retuning between songs costs less the more keys the evening spans.

tuning · Open strings A set in twelve keys keeps its shape through any number of rock songs and loses it to two minor ones. The I–vi–IV–V ground in all twelve keys, with songs outside it added one key at a time — minor-key songs on A, E, B, F♯ and on round the circle, or rock songs on G, D, A, E and on — at five frets a cent: the set's mean cost under its one best tuning, with the shape that tuning is exact for above each point. minor-key songs, i–iv–V–i: 0 added 5.57 (D), 1 added 5.77 (D), 2 added 5.97 (C), 3 added 6.01 (C), 4 added 6.04 (C), 6 added 6.08 (C), 8 added 6.12 (C), 12 added 6.03 (E minor). rock songs, I–♭VII–IV–I: 0 added 5.57 (D), 1 added 5.49 (D), 2 added 5.42 (D), 3 added 5.36 (D), 4 added 5.22 (D), 6 added 5.14 (D), 8 added 5.09 (D), 12 added 4.96 (D).

Two minor songs retune a guitar set for their dominant

A guitar tuned once for an evening of pop songs in every key is tuned exact for the D shape, because every key's route through the ground passes through it. Add songs that are not that ground and the tuning holds or breaks by a rule nobody would guess. Rock songs on I, ♭VII and IV never move it off the D shape, however many are added: they retune only the two strings the D shape leaves silent. One minor-key song leaves it alone too. Two move it to the C shape's tuning, and the chord they move it for is not one of their minor chords but their one major chord — the dominant, fingered in the C shape. Twelve move it again, to the E-minor shape's.

tuning · Open strings What each minor-key song gains from the C shape's tuning. For a minor-key song, i–iv–V–i, in each of the twelve keys, the saving in cents plus frets over five from playing it under the C shape's tuning rather than the twelve-key ground's own, the D shape's: C minor 1.69, C♯ minor 1.69, D minor 1.69, E♭ minor 1.69, E minor 1.69, F minor 1.69, F♯ minor 1.69, G minor 0.33, A♭ minor 1.69, A minor 1.69, B♭ minor 1.69, B minor 1.61. Switching the tuning costs the twelve ground songs 3.90 between them, so the switch pays when the added songs save more than that: the lines mark a half and a third of it, what each of two or three songs would have to save. Ten keys save 1.69 each — less than half, more than a third. G minor saves 0.33, because under the ground's tuning its dominant is the open D chord, already exact.

A guitar set keeps its tuning until the third minor song

Two minor-key songs were found to move a pop set's guitar tuning from the D shape to the C shape. Priced as a vote — what the move costs the twelve ground songs, against what it saves each minor song — two songs do not pay for it and three do. The earlier search had never tried keeping the tuning the set already had. And the key a minor song is in matters only at the nut: ten keys vote alike, and G minor, whose dominant is the open D chord, hardly votes at all.

tuning · Open strings The minor songs' tonic takes the tuning only at the ninth minor song. The twelve-key ground with minor-key songs added on A, E, B, F♯ and on round the circle: the set's mean cost, in cents plus frets over five, under three fixed tunings — the ground's own (the D shape's), the C shape's (the minor songs' dominant exact) and the E-minor shape's (their tonic exact). 0: 5.567, 5.892, 6.499; 1: 5.767, 5.937, 6.436; 2: 5.938, 5.975, 6.327; 3: 6.081, 6.008, 6.285; 4: 6.211, 6.037, 6.247; 5: 6.326, 6.062, 6.214; 6: 6.428, 6.085, 6.185; 7: 6.520, 6.105, 6.159; 8: 6.602, 6.123, 6.135; 9: 6.638, 6.102, 6.076; 10: 6.707, 6.118, 6.058; 11: 6.711, 6.134, 6.042; 12: 6.772, 6.148, 6.028. The ground's tuning is cheapest to two songs, the C shape's from three to eight, and the E-minor shape's from nine — when minor songs are nine of twenty-one.

A guitar set tunes for its minor chords only when nearly half its songs are minor

Add a third candidate to the vote — the tuning that makes the minor songs' own tonic chord exact — and a pop set's guitar passes through three tunings as minor songs are added: the ground's to the second, the dominant's from the third, and the tonic's only from the tenth, when ten songs in twenty-two are minor. Nothing in between is ever chosen: the two minor-song tunings differ on two strings, and every compromise between them costs the ground more than either end. Two keys break the pattern at the nut, and one earlier figure turns out to be wrong by a song.

tuning · Open strings One retuning is worth what the pop songs pay for the minor tuning, and no more. The twelve-key ground with minor-key songs on A added, and what the evening saves, in cents plus frets over five summed over its songs, if the guitarist may retune once between two songs, against tuning once for the whole set. With the minor songs together after the ground: 1 songs 2.48, 2 songs 4.95, 3 songs 6.25, 5 songs 7.81, 9 songs 10.94, 10 songs 11.18, 14 songs 11.18, 20 songs 11.18. From ten songs on it is 11.18 exactly — what the ground's twelve songs pay for the E-minor shape's tuning, which the one-tuning guitar has by then moved to — and it never grows again. With the same songs spread one at a time through the ground, one retuning saves 1.33 at 3, 1.07 at 5, 2.30 at 10, 1.30 at 20. The dashed line is what any number of retunings could save, every song in its own cheapest tuning: 12.04 from ten songs on.

One retuning is worth what the pop songs pay for the minor tuning

A guitarist allowed to retune once in an evening plays the pop songs in their own tuning and the minor songs in theirs. What that saves rises with every minor song until the tenth, and then stops: from there on it is exactly what the twelve pop songs would have paid to be played in the minor songs' tuning, and adding minor songs cannot raise it. The C shape's tuning, which a one-tuning guitar spends most of the range in, is never used. And the whole saving belongs to the running order. Put the same ten minor songs in two runs instead of one and a single retuning is worth three-fifths as much; spread them through the evening and it is worth a fifth.

tuning · Open strings Only three minor keys want a tuning of their own, and each is a key with open strings in its chords. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five: under the E-minor shape's tuning, the best of the one-tuning vote's three for every minor key, and under the tuning a search finds for that song alone. C minor 5.69 and 5.69; G minor 5.69 and 5.69; D minor 5.69 and 5.35; A minor 5.69 and 5.09; E minor 4.91 and 4.91; B minor 5.69 and 4.53; F♯ minor 5.69 and 5.69; C♯ minor 5.69 and 5.69; A♭ minor 5.69 and 5.69; E♭ minor 5.69 and 5.69; B♭ minor 5.69 and 5.69; F minor 4.89 and 4.89. Nine keys keep the E-minor shape's tuning. D, A and B minor each find a tuning of their own and save 0.34, 0.60, 1.16; A minor's makes its tonic and subdominant one shape, the open A-minor chord and the same shape at the fifth fret, both exact.

The minor half's own tuning belongs to one key

A guitarist who retunes once between the pop songs and the minor songs has been choosing among three tunings found for whole evenings. Searched for the minor songs alone, the best tuning is none of them — but only in A, B and D minor, where it gives the tonic and subdominant one shape at the nut and both are in tune. In A it saves six-tenths a song, and the refund a retuning buys stops having a ceiling. In the other nine keys, and in any minor half that mixes keys, the E-minor shape's tuning was already the answer.

tuning · Open strings With a capo up to fret 7, 8 of the twelve minor keys are played as B minor in one tuning. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five, under the tuning searched for the song on A (the B string raised 13.7 cents): at the nut, and at the capo position up to fret 7 that suits it best, with the E-minor shape's tuning at its best capo for comparison. C minor 5.72 → 4.53 (capo 1); G minor 5.72 → 5.72; D minor 5.72 → 4.53 (capo 3); A minor 5.09 → 5.09; E minor 5.72 → 4.53 (capo 5); B minor 4.53 → 4.53; F♯ minor 5.72 → 4.53 (capo 7); C♯ minor 5.72 → 4.53 (capo 2); A♭ minor 5.72 → 5.72; E♭ minor 5.72 → 4.53 (capo 4); B♭ minor 5.72 → 5.09 (capo 1); F minor 5.72 → 4.53 (capo 6). 8 keys reach 4.53, B minor's price, by putting the capo where B minor's open-string chords become theirs. E minor costs 4.53 at capo 5, against 4.91 in the E-minor shape's tuning.

A capo is a transposition the tuning cannot see

Tempered frets make a chord shape's error the same wherever it sits, so a song with a capo at the fifth fret is, to the tuning, the song five semitones lower on a shorter neck. That lets every minor key borrow B minor's open-string fingerings, and eight of the twelve take the raised-B tuning's best price with a capo no higher than the seventh fret. A minor half in mixed keys, which gave a retuning nothing, now nearly doubles its refund — but the capo pays the single-tuned evening as well, and a minor half all in A gets less from its retuning than before.

tuning · Open strings With the capo left at fret 5, E minor becomes B minor and the other keys pay for it. Twelve minor-key songs, i–iv–V–i, one in each key round the circle, in the raised-B tuning, in cents plus frets over five: at the nut, with the capo left at fret 5 for every song, and at each song's own best capo up to fret 7. C minor 5.72, 8.62 at fret 5, 4.53 at its own; G minor 5.72, 5.72 at fret 5, 5.72 at its own; D minor 5.72, 5.09 at fret 5, 4.53 at its own; A minor 5.09, 6.13 at fret 5, 5.09 at its own; E minor 5.72, 4.53 at fret 5, 4.53 at its own; B minor 4.53, 8.62 at fret 5, 4.53 at its own; F♯ minor 5.72, 5.72 at fret 5, 4.53 at its own; C♯ minor 5.72, 8.62 at fret 5, 4.53 at its own; A♭ minor 5.72, 5.72 at fret 5, 5.72 at its own; E♭ minor 5.72, 5.72 at fret 5, 4.53 at its own; B♭ minor 5.72, 7.82 at fret 5, 5.09 at its own; F minor 5.72, 5.72 at fret 5, 4.53 at its own. The twelve cost 66.82 at the nut, 78.04 with the capo at fret 5 and 57.87 with it moved for every song. At fret 5 the song on E takes B minor's fingerings and the song on D A minor's; every other key is moved to a key the tuning does not favour, on a neck 5 frets shorter.

A capo left in place serves one key

Moved for every song, a capo lets nearly every minor key borrow B minor's open-string fingerings. Left at one fret for a whole run of minor songs, as most guitarists leave it, it does something much weaker: it renames the keys. Over a run in every key that is a reshuffle on a shorter neck and can never beat the nut; over the four keys guitarists use most it gains nothing either, because two of them are already the tuning's best. A fixed capo pays only a run in very few keys, spaced the way the tuning's cheapest keys are.

tuning · Open strings Ordered so that neighbouring songs are a fifth apart, a set moves the capo most and recovers least. Twelve minor-key songs, three each in E, A, D and B minor, i–iv–V–i, in the best of the four tunings, the capo allowed to change fret 3 times between songs: the share of what moving the capo for every song saves over playing at the nut. grouped by key: 100%; a different key each song, at random: 51% on average over 40 orders, from 35 to 70; neighbours a fifth apart: 35%. The fifths order is B E B D A E B E D A D A.

A set ordered by key relation moves the capo most

Grouped by the fret each song wants, a set of twelve minor songs in E, A, D and B gets the whole of what a moving capo saves with three capo changes. A real set is not grouped by fret; it is ordered for the music, and the commonest rule for that order is key relation — neighbouring songs a fifth apart. That order is the worst the capo can face: no two neighbours want the same fret, and three changes recover thirty-five per cent of the saving against fifty-one for a random order. The capo groups keys a tone apart, and a set built on fifths is the order that keeps them furthest from each other.

tuning · Open strings Searched over every order, a set can step by fifths at each capo change for free, and pays about seven points for each fifth beyond. Twelve minor songs, three each in E, A, D and B minor, no key twice running: among all 41,304 orders, for each number of neighbouring songs a fifth apart, the largest share of a moving capo's saving recovered with 3 capo changes, with the mean over all orders at that count. 0: best 84%, mean 84% (2 orders); 1: best 84%, mean 81% (66 orders); 2: best 84%, mean 70% (590 orders); 3: best 84%, mean 62% (2630 orders); 4: best 78%, mean 55% (6738 orders); 5: best 70%, mean 50% (10626 orders); 6: best 63%, mean 46% (10626 orders); 7: best 61%, mean 42% (6738 orders); 8: best 54%, mean 39% (2630 orders); 9: best 46%, mean 37% (590 orders); 10: best 40%, mean 36% (66 orders); 11: best 37%, mean 37% (2 orders). The earlier essay's fifths order (9 fifths) recovers 35%, and a random order 51% on average.

A set steps by fifths where the capo moves

A set of twelve minor songs in E, A, D and B can be ordered for the capo or for key relation, and the two orders pull apart. Searched over every one of its 41,304 orders, the trade between them is exact and simple: a set can put a fifth between neighbouring songs at each capo change for nothing, because a fifth always crosses frets and a capo change is where crossing is paid for. Beyond that each extra fifth costs about seven points of the capo's saving. The tone steps the earlier essay suspected of carrying the compromise are where the capo stays still.

tuning · Open strings Letting songs move a tone adds between five and twenty points at any count of fifths, and never makes the trade disappear. Twelve minor songs, three each in E, A, D and B minor, with three capo changes and no key twice running, searched exactly. For each number of neighbouring songs a fifth apart, the largest share of a moving capo's saving, with every song in its own key and with up to 12 moved up or down a tone. 0: 84% and 92%; 1: 84% and 92%; 2: 84% and 92%; 3: 84% and 92%; 4: 78% and 83%; 5: 70% and 75%; 6: 63% and 75%; 7: 61% and 75%; 8: 54% and 66%; 9: 46% and 58%; 10: 40% and 57%; 11: 37% and 57%.

A set free to change key becomes a set in two

A set list orders its songs for the music and a capo wants them ordered by fret, and the trade between the two looked as if it might disappear if a singer would move a song a tone. It does not. Searched exactly, moving songs adds five to twenty points of the capo's saving at any count of fifths and stops improving at six moved; at its limit the best set rocks between B minor and E minor, because a count of fifths rewards a pendulum. A set that really walks the chain of fifths has three to spend, not eleven.

tuning · Open strings

Named alongside it

The objects these essays reach for when they reach for this one.

Just intonationEqual temperamentIntonationOpen stringVoicingOrchestrationTextureMaskingRegisterTranspositionDoublingEnumeration

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