Delta-rational chord

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A delta-rational (DR) chord is a chord that has integer ratios between frequency differences of some pair of dyads, called deltas, with the dyads in question assumed to be between successive notes (Δ, capital delta, is often used to denote "difference").

DR chords generalize JI chords, in which all frequency differences of dyads are exactly integer ratios. But unlike JI chords, a DR chord need not have integer ratios between frequencies of notes. For example, the 13edo chord 0-3-8-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational, because the frequency difference of the dyad 8-10\13 is 0.994 times the frequency difference of the dyad 0-3\13. (In the exactly DR chord 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes have exactly the same frequency difference as the dyad 0-3\13.)

JI chords and chords that are subsets of isodifferential chords (these correspond to all chords of the form α : α + k1 : ... : α + kn for any positive (possibly irrational) number α and integers k1, ..., kn) are special cases of delta-rational chords, but in these chords all dyads are rationally related in frequency space, which we call either fully delta-rational (FDR) or linear.

Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads (when measured as absolute frequency differences) tend to be perceived as more concordant than other chords. This acoustic effect is thought to be caused by synchronized interference beating among the fundamentals and among lower harmonics of the fundamentals; the effect may be more or less pronounced depending on register, timbre, the complexity of the linear relationship, etc. For example, the delta-rational acoustic effect is expected to be weaker in chords with wider voicings, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious). The justification for only considering dyads between adjacent notes is that the resulting notes within the dyads could psychoacoustically interfere with the beating of the dyads.

Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicing of JI chords stay JI. However, unlike the case with most JI chords, a 2/1-equivalent MOS scale can tune a DR chord exactly, provided that the chord's delta signature specifies two integer entries.

Denoting a delta-rational chord

Delta signature

A delta-rational chord is determined by two things:

  • the dyad formed by its lowermost two notes;
  • its delta signature which has integer ratios, i.e. a list of (scaled) frequency increases between successive notes, their ratios showing the simple rational relationships, with a + before each increase. Note that it is whether the deltas are rationally related to each other that defines DR, not whether the deltas are related to the frequency of the root. If we divide every term by the first term to make the first term 1, the result is called a normalized delta signature.
  • Two delta signatures are equivalent if one can be obtained from the other by scaling by a positive real number. For example, +2+e+3 is equivalent to +2φ+eφ+3φ, and both signatures imply a delta-rational chord.

For example, a chord with a +1+2+1 delta signature is a:(a+1):(a+3):(a+4) for some possibly irrational a.

Deltas that are free, i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have almost equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1.

If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, c or use one fewer letter by writing one set with positive integers without variables: an +a +b +a +b chord can also be written +1 +c +1 +c where c = b/a.

Fully delta-rational chords always have a delta signature with no irrational ratios between terms.

Mathematical definitions

  1. A chord C = α1:...:αn is delta-rational (DR) or partially delta-rational (PDR) when the chord has two distinct dyads αk1k2 and αk3k4, such that the real intervals (αk1, αk2) and (αk3, αk4) are disjoint and (αk2 − αk1)/(αk4 − αk3) is rational. Equivalently, a chord is delta-rational if it has a delta signature with some integers showing up.
  2. When all dyads are linearly related, equivalently when the chord has a delta signature with all entries integers, we call the chord fully delta-rational (FDR) or linear.
  3. A chord that has a delta signature with all entries +1 is called isodifferential.

Due to the aforementioned equivalence of delta signatures under scaling, delta signatures of n terms are really elements of a projective space [math]\mathbb{R}\mathbf{P}^{n-1};[/math] they are specifically in the subset that is the image of the all-positive orthant of [math]\mathbb{R}^n.[/math]

In practice these terms can loosely refer to approximations of mathematically exact PDR and FDR chords, for example in edo tunings.

Finding a tuning of a MOS scale with an exact DR chord

Let a, b be positive integers and suppose gcd(a, b) = 1. Let E > 1 be the frequency ratio of the equave. Consider a MOS aLbsE with generator range [math]I \subseteq (1, \sqrt{E})[/math] (in the linear frequency domain), and consider a pair (u, v) of notes from the root of a given triad in the MOS, 0 (unison) < u < v. Let p, g be a basis formally representing the MOS scale's period and generator. Write

[math]\begin{align} \mathbf{u} &= u_p \mathbf{p} + u_g \mathbf{g} \\ \mathbf{v} &= v_p \mathbf{p} + v_g \mathbf{g} \end{align} [/math]

as elements of [math]\mathbb{Z}^2\langle \mathbf{p}, \mathbf{g}\rangle[/math].

Define the rational function [math]r_{\mathbf{u}, \mathbf{v}} : I \to (0,\infty)[/math] by

[math]\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{E^{v_p}x^{v_g}- E^{u_p}x^{u_g}}{E^{u_p}x^{u_g} - 1} }.[/math]

Then, provided that the positive rational number [math]m/n[/math] lies in the image [math]r_{\mathbf{u}, \mathbf{v}}(I),[/math] we can solve for the frequency ratio [math]g \in I[/math] that satisfies [math]r_{\mathbf{u}, \mathbf{v}}(g) = m/n,[/math] making the specified chord (0, u, v) a +n+m DR chord.

The existence of an exact tuning for a delta signature specification is only guaranteed to hold when we only care about a ratio between two terms in the delta signature being exact. If we want to optimize an arbitrary specified delta signature (with some deltas possibly held free), we can use a least-squares-error solution instead to minimize the error.

Measuring the error of an approximate fully DR chord

Least-squares error measures

Say we want the error of a chord 1:r1:r2:...:rn (in increasing order) in the linear domain as an approximation to a fully delta-rational chord with signature +δ12 ... +δn, i.e. a chord

[math] \alpha : \alpha + \delta_1 : \cdots : \alpha + \sum_{l=1}^n \delta_l [/math]

where we vary α and ask, "By at least how much do the deltas have to be off for any α?"

Solving the linear least-squares optimization problem

[math] \displaystyle{ \min_\alpha \sqrt{\sum_{i=1}^n \Bigg( \alpha r_i - \alpha - \sum_{l=1}^i \delta_l \Bigg)^2 } }[/math]

yields

[math] \alpha = \displaystyle{\frac{\sum_{i=1}^n \delta_i (r_i - 1)}{\sum_{i=1}^n (r_i - 1)^2},} [/math]

which can be plugged back into the error formula to obtain the error. (We multiply the 1:r1:r2:...:rn chord by α in order to compare it to the target DR chord on the same isodifferential series.)

This error measure is called least-squares delta error. Least-squares delta error does not depend on whether the chord whose error is being measured is 1:r1:r2:...:rn or the same chord linearly shifted to have root α. Unfortunately, this error measure does not form a metric on the set of delta signatures with a fixed number of terms.

DR and RTT

One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. The value to be solved for is the generator's frequency ratio (not its cent value).

If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the triad's root and letting [math]g[/math] be the frequency ratio for the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of [math]g-g^4/4[/math], and the major third in the same triad has a delta of [math]g^4/4-1[/math]. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial [math]g^4-2g-2[/math] (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.

Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: [math]g^4+2g-8=0.[/math] The latter equation has solution g = 1.4960 = 697.3c.

Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the relevant DR polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients, etc. In the case of negative coefficients, only the absolute value is considered.

Coefficients of terms Chord Delta signature Temperament Generator (cents) Edos
g^10 g^9 g^8 g^7 g^6 g^5 g^4 g^3 g^2 g^1 g^0
1 -1 -1 4:5:6 +1+1 [1 -2 1 833.09 (phi) 36
2 -1 -2 4:5:6 +1+1 Father 428.42 14
3 -2 -2 6:7:9 +1+2 Beep 258.65 33, 42, 51
1 -1 -2 4:5:6 +1+1 Mavila 523.66 23, 39
1 -2 -2 4:5:6 +1+1 Meantone 695.63 19, 31, 50
1 2 -4 4:5:6 +1+1 Porcupine 160.89 15
1 -4 12 5:6:9 +1+3 Mavila 674.90 16, 25
1 -4 -4 4:5:6 +1+1 Avila 660.23 20
1 -2 2 4:5:6 +1+1 Hanson 317.96 19
1 -2 -4 4:5:6 +1+1 Uncle 467.46 18
3 -4 -16 4:5:7 +1+2 Mabila 527.66 25
1 -1 -1 4:5:6 +1+1 Tetracot 176.54 34
1 -1 -4 4:5:6 +1+1 Sensi 442.74 19, 65
1 2 -8 4:5:6 +1+1 Orson 271.51 22, 31, 53

Limitations

Interpreting simple JI chords as signatures/templates for delta-rational chords is reasonable, as the psychoacoustic effect of DR is more robust to detuning than that of JI for many people, but it is no panacea. If the temperament is too inaccurate, other inversions and voicings of a given JI chord will not have the DR signatures preserved acceptably, and compromising will make all of them inaccurate, though this of course depends on what error tolerance you prefer. This is because DR simplicity is only preserved by inversion and revoicing if the chord in question is low-complexity JI.

For example, take 0.0570 (the LSD error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with LSD error 0.0681; the 7:8:12 has an even higher error of 0.123. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, all three inversions are barely acceptable to unacceptable as simple-DR approximations: 4:6:7 has LSD error 0.0574 from +2+1, 6:7:8 has LSD error 0.0652 from +1+1, and 7:8:12 has LSD error 0.0570 from +1+4.

Finding approximate DR chords in edos

Todo: add more heuristics

Some heuristics for finding delta-rational chords:

  • If a chord with a given step numbers in an edo is delta-rational it or similar numbers will usually also be delta-rational in nearby edos.

Incomplete list of approximate DR chords in small edos

(Chords whose range is < 1200¢ and only contains ratios of 1's and 2's between related dyads. Decide on error bound and search programmatically)

9edo

  • 0-4-7 (+1 +1)

10edo

  • 0-4-7 (+1 +1)

11edo

  • 0-2-4-7 (+1 +? +2)
  • 0-3-7 (+2 +3)
  • 0-3-6-8 (+1 +? +1)
  • 0-3-6-10 (+1 +? +2)
  • 0-3-7-9 (+1 +? +1)
  • 0-3-8-10 (+1 +? +1)
  • 0-4-7 (+1 +1)
  • 0-4-10 (+1 +2)
  • 0-4-6-9 (+1 +1)
  • 0-5-7 (+2 +1)
  • 0-5-9 (+1 +1)
  • 0-5-8-10 (+3 +? +2 (?))
  • 0-6-9 (+3 +2)
  • 0-6-10 (+1 +1)
  • 0-6-8-10 (+2 +? +1)

12edo

  • 0-4-10 (+1 +2)

13edo

  • 0-2-7 (+1 +3)
  • 0-2-11 (+1 +6)
  • 0-2-4-7 (+1 +? +2)
  • 0-2-4-11 (+1 +? +5)
  • 0-2-5-8 (+1 +? +2)
  • 0-2-6-10 (+1 +? +3)
  • 0-2-7-9 (+2 +? +3)
  • 0-2-7-11 (+1 +? +3)
  • 0-2-8-10 (+2 +? +3)
  • 0-3-8 (+1 +2)
  • 0-3-10 (+1 +3)
  • 0-3-7-9 (+1 +? +1)
  • 0-3-7-11 (+1 +? +2)
  • 0-3-8-10 (+1 +? +1)
  • 0-3-9-11 (+1 +? +1)
  • 0-4-10 (+1 +2)
  • 0-4-6-8 (+3 +? +2)
  • 0-4-6-9 (+1 +? +1)
  • 0-4-7-9 (+3 +? +2)
  • 0-4-7-10 (+1 +? +1)
  • 0-5-7 (+2 +1)
  • 0-5-9 (+1 +1)
  • 0-5-7-9 (+2 +1 +1)
  • 0-6-9 (+3 +2)
  • 0-6-9-11 (+2 +? +1)

14edo

  • 0-3-9-11 (+1 +? +1)
  • 0-5-9 (+1 +1)

18edo

  • 0-3-8 (+1 +2)
  • 0-3-11-13 (+1 +? +1)
  • 0-3-12-14 (+1 +? +1)
  • 0-3-13-15 (+1 +? +1)
  • 0-4-7-10-13 (+1 +? +1 +1)
  • 0-5-7-11 (+1 +? +1)
  • 0-5-9 (+1 +1)
  • 0-5-13-16 (+1 +? +1)
  • 0-5-14-17 (+1 +? +1)
  • 0-6-11 (+1 +1)
  • 0-6-11-15 (+1 +1 +1)
  • 0-7-10 (+2 +1)
  • 0-7-17 (+1 +2)
  • 0-7-9-14 (+1 +? +1)
  • 0-7-10-15 (+2 +1 +2)

21edo

  • 0-3-6-11 (+1 +1 +2)
  • 0-6-11-19 (+1 +1 +2)

Higher-order differences of frequency

Generalizing, one could consider chords where differences between its frequency deltas (as Tom Price has called them, precessions) are rationally related, while the deltas themselves may not be. This corresponds to chords where differences between various interference beatings go in and out of sync in a periodic manner. One precession-rational chord is 5:5.4142...:6.8284...:9.2426..., a +(sqrt(2) − 1) +sqrt(2) +(sqrt(2) + 1) chord.

Precession being the second-order difference (Δ2) of frequency, we similarly have the theoretical notions of Δ3-rationality, Δ4-rationality, and so on. The practical consequences of higher-order differences are as of yet speculative, though a few people have reported finding precession psychoacoustically meaningful.

Isodifferential chord

In an isodifferential chord (known variously by linear chord, equal-hertz chord, equal-beating chord, and proportional-beating chord), the frequencies of the pitches are in an arithmetic sequence, or in other words, there is an equal difference in cycles per second between successive pitches.

Isoharmonic chord

An isoharmonic chord is a specific type of isodifferential chord, where the ratios between the notes are rational numbers, and therefore the chord is in just intonation. Such a chord can be built by successive jumps up the harmonic series by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape—with diminishing step size as one ascends.

An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but the word "chord" is used here for consistency.

Classification

Class i

The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:

harmonic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
cents diff 1200 702 498 386 316 267 231 204 182 165 151 139 128 119 112

Some "scales" built this way: otones12-24, otones20-40...

Class ii

The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the Bohlen-Pierce tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):

harmonic 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
cents diff 1902 884 583 435 347 289 248 217 193 173 157 144 133 124 115
Class iii

Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:

harmonic 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46
cents diff 2400 969 617 454 359 298 254 221 196 176 160 146 135 125 117
harmonic 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47
cents diff 1586 814 551 418 336 281 242 212 189 170 155 142 132 122 114
Class iv
harmonic 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61
cents diff 2786 1018 637 464 366 302 257 224 198 178 161 147 136 126 117
harmonic 3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63
cents diff 1467 782 537 409 331 278 239 210 187 169 154 141 131 122 114
Class v
harmonic 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76
cents diff 3102 1049 649 471 370 306 259 225 199 179 162 148 136 126 118
harmonic 2 7 12 17 22 27 32 37 42 47 52 57 62 67 72 77
cents diff 2169 933 603 446 355 294 251 219 195 175 159 146 134 125 116
harmonic 3 8 13 18 23 28 33 38 43 48 53 58 63 68 73 78
cents diff 1698 841 563 424 341 284 244 214 190 172 156 143 132 123 115
harmonic 4 9 14 19 24 29 34 39 44 49 54 59 64 69 74 79
cents diff 1404 765 529 404 328 275 238 209 186 168 153 141 130 121 113

Notation

Some complex isoharmonic chords can be expressed with an offset from a simpler isoharmonic chord, so it is useful to notate them in a compact and readable way. For example, 41:51:61 is very similar to 4:5:6, so it can be notated as (4:5:6)[+0.1]. Similarly, (20:22:24:27:30:33:36)[+0.339] can be expanded to 20339:22339:24339:27339:30339:33339:36339.

Irrational isodifferential chords can be expressed with the same notation by using irrational numbers within the square brackets. For example, the chord (1:2:3)[+φ] can be expanded to (1+φ):(2+φ):(3+φ), which is approximately equal to 1.618:2.618:3.618.

Categorization of DR chords

Here is a table which uses the "delta ratio set" — the set of unique undirected ratios between the deltas of a chord's delta signature — to categorize chords.

  • How to tell a DR chord from a non-DR chord: a DR chord has at least one rational number in its delta ratio set.
  • Within DR chords, how to tell an FDR chord from a non-fully DR chord: a FDR chord has only rational numbers in its delta ratio set.
  • Within FDR chords, how to tell an isodifferential chord from a non-isodifferential chord: an isodiffential chord has only 1 in its delta ratio set.

All JI chords are FDR chords, because JI chords are rational, and therefore their delta ratio sets will include only rational numbers.

If an FDR chord is both JI and isodifferential, then it is an isoharmonic chord.

chord type illustrative examples
actual chord deltas delta ratio set
frequency ratio are items all integers? delta signature reduced delta signature (class) are items all the same? unique undirected ratios between the deltas are items all rational?
DR FDR JI, not isodifferential 4:5:7:8 yes, all +1+2+1 +1+2+1 no, not all { 1, 2 } yes
3:5:9:11 +2+4+2
3:4:7:9 +1+3+2 +1+3+2 { 3/2, 2, 3 }
isoharmonic

(JI and isodifferential)

class i 4:5:6 +1+1 +1+1 yes, all { 1 }
4:5:6:7 +1+1+1 +1+1+1
3:4:5:6
class ii 3:5:7:9 +2+2+2
5:7:9:11
class iii 1:4:7:10 +3+3+3
2:5:8:11
... ... ...
not JI, but isodifferential ɸ:(ɸ+1):(ɸ+2):(ɸ+3) no, not all or none +1+1+1
1:ɸ:(2ɸ-1):(3ɸ-2) +(ɸ-1)+(ɸ-1)+(ɸ-1)
not JI and not isodifferential ɸ:(ɸ+1):(ɸ+3) +1+2 +1+2 no, not all { 2 }
(incompletely) DR 4:5:τ:7:9 +1+(τ-5)+(7-τ)+2 +1+(τ-5)+(7-τ)+2 (irrelevant for categorization) { (7-τ)/(τ-5), 7-τ, τ-5, 2/(τ-5), 2, 2/(7-τ) } no, but at least one
5:τ:8:(3+τ) +(τ-5)+(8-τ)+(τ-5) +1+(8-τ)/(τ-5)+1 { 1, (8-τ)/(τ-5) }
1:(1+a):(1+a+b):(1+a+2b):(1+3a+2b), with a/b irrational +a+b+b+2a +a+b+b+2a { a/b, 1, 2, 2a/b }
not DR 4:5:τ:7 +1+(τ-5)+(7-τ) +1+(τ-5)+(7-τ) { (7-τ)/(τ-5), 7-τ, τ-5 } no, none
5:τ:7 +(τ-5)+(7-τ) +1+(7-τ)/(τ-5) { (7-τ)/(τ-5) }