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"). Here ''dyad'' refers not to a [[Dyad|chord of two pitch classes]], but to an interval between two notes.  

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 {{dash|0, 3, 8, 10|med}}\13 ({{dash|0¢, 277¢, 738¢, 923¢|med}}) 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 {{dash|0\13, 3\13, 8\13, 924.159¢|med}}, 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 [[Delta-rational chord#Isodifferential chord|isodifferential chord]]s (these correspond to all chords of the form α : {{nowrap|α + ''k''₁}} : ... : {{nowrap|α + ''k''_''n''}} for any positive (possibly irrational) number α and integers ''k''₁, ..., ''k''_''n'') are special cases of delta-rational chords, but in these chords ''all'' dyads are rationally related in frequency space, which we call '''fully delta-rational''' (FDR).

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.

* The dyad formed by its lowermost two notes;

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 "{{nowrap|+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 {{nowrap|+1 +? +1}}.

# A chord C = α₁:...:α_''n'' is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads α_''k''₁:α_''k''₂ and α_''k''₃:α_''k''₄, such that the real intervals (α_''k''₁, α_''k''₂) and (α_''k''₃, α_''k''₄) are disjoint and (α_''k''₂ − α_''k''₁)/(α_''k''₄ − α_''k''₃) is rational. Equivalently, a chord is delta-rational if it has a delta signature with some integers showing up.

# 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)

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

== Least-squares error ==

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'' &gt; 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ₁ +δ₂ ... +δ_''n''

 

 

We can vary ''x'' and ask, "By at least how much (in the linear domain) does the approximating chord have to be off for any ''x'' > 0?" When a specific ''x'' > 0 achieves this minimum, the resulting chord with delta signature {{nowrap|+δ₁ +δ₂ ... +δ_''n''

 

 

 

 

 

which can be plugged back into the error formula to obtain the error. (We multiply the target DR chord by ''x'' in order to compare it to the approximation on the same isodifferential series.)

 

 

 

 

 

d1 = sympy.Symbol("\\delta_{1}", real=True)

d3 = sympy.Symbol("\\delta_{3}", real=True)

 

== DR and RTT ==

 

 

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: {{nowrap|''g''⁴ + 2''g'' − 8 {{=}} 0}} The latter equation has solution {{nowrap|''g'' {{=}} 1.4960 {{=}} 697.3¢}}. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->

! g¹⁰

! g⁹

! g⁸

! g⁷

! g⁶

! g⁵

! g⁴

! g³

! g²

! g¹

! g⁰

| rowspan="7" | No, not all or none

[[Category:Chords]]