Delta-rational chord: Difference between revisions

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

# 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) or ''linear''.

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

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 ''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 ('''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  

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.

Say we want the error of a chord 1:''r''₁:''r''₂:...:''r''_''n'' (in increasing order) in the linear domain as an approximation to a fully delta-rational chord with signature +δ₁ +δ₂ ... +δ_''n'', i.e. a chord

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:''r''₁:''r''₂:...:''r''_''n'' 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.

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When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.

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Another possible approach is to view delta signatures as vectors of positive reals by fixing the first element as 1 and measure the induced <math>L^p</math> distance.