Delta-rational chord: Difference between revisions

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").  

* 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''.

The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''₁:''r''₂:...:''r''_''n'' (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ₁ +δ₂ ... +δ_''n''

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. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->

| -1

| -1

| -1

| -2

| -2

| -2

| -1

| -2

| -2

| -2

| -4

| -4

| -4

| -4

| -2

| -2

| -4

| -4

| -16

| -1

| -1

| -1

| -4

| -8

For example, take 0.00807 (the least-squares 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 least-squares error '''0.0118'''; the 7:8:12 has an even higher error of '''0.0178'''. 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, the 4:6:7, 6:7:8, and 7:8:12 have errors '''0.0108''', '''0.0106''', and 0.00737. If we use the average of the CTE and the perfect +2+1 generator, the errors become 0.00535, '''0.0112''', and '''0.0126'''.

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 {{nowrap|+(sqrt(2) &minus; 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.

===Isoharmonic chord ===

 

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:

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 [[BP|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):

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:

===== Class iv =====

 

===== Class v =====

 

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.

 

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

* <b>How to tell a DR chord from a non-DR chord:</b> a DR chord has at least one rational number in its delta ratio set.

* <b>Within DR chords, how to tell an FDR chord from a non-fully DR chord:</b> a FDR chord has <i>only</i> rational numbers in its delta ratio set.

* <b>Within FDR chords, how to tell an isodifferential chord from a non-isodifferential chord:</b> an isodiffential chord has only 1 in its delta ratio set.

! colspan="2" rowspan="3" | JI, <i>not</i> isodifferential

! rowspan="8" | Isoharmonic

(JI ''and'' isodifferential)

! rowspan="3" | Class i

! rowspan="2" | class ii

! rowspan="2" | class iii

| ɸ:(ɸ+1):(ɸ+2):(ɸ+3)

| 1:ɸ:(2ɸ-1):(3ɸ-2)

| +(ɸ-1)+(ɸ-1)+(ɸ-1)

| ɸ:(ɸ+1):(ɸ+3)

| { 2 }

| 4:5:τ:7:9

| +1+(τ-5)+(7-τ)+2

| +1+(τ-5)+(7-τ)+2

| {{(}}(7-τ)/(τ-5), 7-τ, τ-5, 2/(τ-5), 2, 2/(7-τ){{)}}

|-

| 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

| 4:5:τ:7

| +1+(τ-5)+(7-τ)

| +1+(τ-5)+(7-τ)

| {{(}}(7-τ)/(τ-5), 7-τ, τ-5{{)}}

|-

| +(τ-5)+(7-τ)

| +1+(7-τ)/(τ-5)

| {{(}}(7-τ)/(τ-5){{)}}