Delta-rational chord: Difference between revisions

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

=== Layman guide ===

We start by choosing the [[MOS scale]] and equave, and the DR chord.

For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.

Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meantone mapping, these are g4/4 -1, g-g4/4. This is because in meantone, 1/1, 3/2, 5/4 are 1, g, g4/4 respectively, so the deltas are identified by subtracting the each term with the one before it.

In this case, we want the difference between our deltas to become 1, so the delta signature will be +1+1.

To achieve this, we take the difference between the first two deltas and set it to zero, so (g4/4 -1) - (g-g4/4) = 0. Put in integer terms, it's g4 - 2g - 2 = 0. Solving for g, the only root that makes sense is g≈1.49453, which in cents is 695.630c. And thus, with this generator, we will have a DR ~4:5:6 meantone chord!

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¢}}.

If instead we chose a Schismic mapping, the deltas would be g8/8 -1, 2/g - g8/8, which gives a generator of 498.308c for 4:5:6.

=== Mathematical definition ===

Let ''a'' and ''b'' be positive integers and suppose {{nowrap|gcd(''a'', ''b'') {{=}} 1}}. Let {{nowrap|''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 ({{nowrap|'''u''', '''v'''}}) of notes from the root of a given triad in the MOS, {{nowrap|'''0''' (unison) < '''u''' < '''v'''}}. Let '''p''', '''g''' be a basis formally representing the MOS scale's period and generator. Write

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== DR and RTT ==

== 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 {{nowrap|''g'' − {{sfrac|''g''4|4}}}}, and the major third in the same triad has a delta of {{nowrap|{{sfrac|''g''4|4}} − 1}}. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial {{nowrap|''g''4 − 2''g'' − 2}} (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: {{nowrap|''g''4 +~~ 2~~''g'' − 8 {{=}} 0}} The latter equation has solution {{nowrap|''g'' {{=}} 1.4960 {{=}} 697.3¢}}. ~~

As stated above, one can tune a rank-2 regular temperament or a MOS scale in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI mapping.

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.

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 == Finding a tuning of a MOS scale with an exact DR chord ==
+=== Layman guide ===
+We start by choosing the [[MOS scale]] and equave, and the DR chord.
+For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.
+Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meantone mapping, these are g<sup>4</sup>/4 -1, g-g<sup>4</sup>/4. This is because in meantone, 1/1, 3/2, 5/4 are 1, g, g<sup>4</sup>/4 respectively, so the deltas are identified by subtracting the each term with the one before it.
+In this case, we want the difference between our deltas to become 1, so the delta signature will be +1+1.
+To achieve this, we take the difference between the first two deltas and set it to zero, so  (g<sup>4</sup>/4 -1) - (g-g<sup>4</sup>/4) = 0. Put in integer terms, it's g<sup>4</sup> - 2g - 2 = 0. Solving for g, the only root that makes sense is g≈1.49453, which in cents is 695.630c. And thus, with this generator, we will have a DR ~4:5:6 meantone chord!
+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''<sup>4</sup> + 2''g'' − 8 {{=}} 0}} The latter equation has solution {{nowrap|''g'' {{=}} 1.4960 {{=}} 697.3¢}}.
+If instead we chose a Schismic mapping, the deltas would be g<sup>8</sup>/8 -1, 2/g - g<sup>8</sup>/8, which gives a generator of 498.308c for 4:5:6.
+=== Mathematical definition ===
 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
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 {{main|Partially delta-rational tetrads in small edos}}
-== DR and RTT ==
+== DR and RTT ==<!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->
-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 {{nowrap|''g'' − {{sfrac|''g''<sup>4</sup>|4}}}}, and the major third in the same triad has a delta of {{nowrap|{{sfrac|''g''<sup>4</sup>|4}} − 1}}. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial {{nowrap|''g''<sup>4</sup> − 2''g'' − 2}} (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: {{nowrap|''g''<sup>4</sup> + 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.-->
+As stated above, one can tune a rank-2 regular temperament or a MOS scale in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI mapping.
 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.