Delta-rational chord: Difference between revisions

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{{Interwiki

|de=Delta-rationaler Akkord

|en=Delta-rational chord

|es=

|ja=

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

DR chords are a generalization of JI chords, in which all frequency differences of intervals 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 interval 8–10\13 is 0.994 times the frequency difference of the interval 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 interval 0–3\13.)

DR chords are a generalization of JI chords, in which all frequency differences of intervals 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{{c}}, 277{{c}}, 738{{c}}, 923{{c}}|med}}) is close to being delta-rational, because the frequency difference of the interval 8–10\13 is 0.994 times the frequency difference of the interval 0–3\13. (In the exactly DR chord {{dash|0\13, 3\13, 8\13, 924.159{{c}}|med}}, the 3rd and 4th notes have exactly the same frequency difference as the interval 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'' intervals are rationally related in frequency space, which we call '''fully delta-rational''' (FDR).

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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.

For example, with 5L 2s, 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.

Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meantone mapping, these are {{nowrap|''g''4/4 − 1}}, {{nowrap|''g'' − ''g''4/4}}. This is because in meantone, 1/1, 3/2, 5/4 are 1, ''g'', {{nowrap|''g''4/4}} 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!

To achieve this, we take the difference between the first two deltas and set it to zero, so {{nowrap|(''g''4/4 − 1) − (''g'' − ''g''4/4) {{=}} 0}}. Put in integer terms, it's {{nowrap|''g''4 − 2''g'' − 2 {{=}} 0}}. Solving for ''g'', the only root that makes sense is {{nowrap|''g'' ≈ 1.49453}}, which corresponds to 695.63{{c}}. 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¢}}.

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{{c}}}}.

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.

If instead we chose a Schismic mapping, the deltas would be {{nowrap|''g''8/8 − 1}} and {{nowrap|2/''g'' − ''g''8/8}}, which gives a generator of 498.308{{c}} 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

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 ('''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

<math>\begin{align}

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=== Fully DR triads ===

=== Partially DR tetrads ===

== DR and RTT ==

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.

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! rowspan="2" | Edos

|-

! g10

! ''g''10

! g9

! ''g''9

! g8

! ''g''8

! g7

! ''g''7

! g6

! ''g''6

! g5

! ''g''5

! g4

! ''g''4

! g3

! ''g''3

! g2

! ''g''2

! g1

! ''g''1

! g0

! ''g''0

|-

|

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! colspan="2" rowspan="2" | Not JI, but isodifferential

| style="white-space: nowrap;" | φ:(φ + 1):(φ + 2):(φ + 3)

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

| rowspan="8" | No, not all or none

| +1+1+1

|-

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== External links ==

* [~~https~~://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (Includes least-squares linear error calculation)]

* [http://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (Includes least-squares linear error calculation)]

[[Category:Chord]]

@@ Line 1: / Line 1: @@
+{{Interwiki
+|de=Delta-rationaler Akkord
+|en=Delta-rational chord
+|es=
+|ja=
+}}
 A '''delta-rational''' ('''DR''') chord is a [[chord]] that has integer ratios between frequency ''differences'' of some pair of intervals, called '''deltas''', with the intervals in question assumed to be between successive notes (Δ, capital delta, is often used to denote "difference").
-DR chords are a generalization of JI chords, in which all frequency differences of intervals 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 interval 8–10\13 is 0.994 times the frequency difference of the interval 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 interval 0–3\13.)
+DR chords are a generalization of JI chords, in which all frequency differences of intervals 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{{c}}, 277{{c}}, 738{{c}}, 923{{c}}|med}}) is close to being delta-rational, because the frequency difference of the interval 8–10\13 is 0.994 times the frequency difference of the interval 0–3\13. (In the exactly DR chord {{dash|0\13, 3\13, 8\13, 924.159{{c}}|med}}, the 3rd and 4th notes have exactly the same frequency difference as the interval 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''<sub>1</sub>}} : ... : {{nowrap|α + ''k''<sub>''n''</sub>}} for any positive (possibly irrational) number α and integers ''k''<sub>1</sub>, ..., ''k''<sub>''n''</sub>) are special cases of delta-rational chords, but in these chords ''all'' intervals are rationally related in frequency space, which we call '''fully delta-rational''' (FDR).
@@ Line 37: / Line 43: @@
 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.
+For example, with 5L&nbsp;2s, 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.
+Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meantone mapping, these are {{nowrap|''g''<sup>4</sup>/4 − 1}}, {{nowrap|''g'' − ''g''<sup>4</sup>/4}}. This is because in meantone, 1/1, 3/2, 5/4 are 1, ''g'', {{nowrap|''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!
+To achieve this, we take the difference between the first two deltas and set it to zero, so {{nowrap|(''g''<sup>4</sup>/4 − 1) − (''g'' − ''g''<sup>4</sup>/4) {{=}} 0}}. Put in integer terms, it's {{nowrap|''g''<sup>4</sup> − 2''g'' − 2 {{=}} 0}}. Solving for ''g'', the only root that makes sense is {{nowrap|''g'' ≈ 1.49453}}, which corresponds to 695.63{{c}}. 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¢}}.
+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{{c}}}}.
-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.
+If instead we chose a Schismic mapping, the deltas would be {{nowrap|''g''<sup>8</sup>/8 − 1}} and {{nowrap|2/''g'' − ''g''<sup>8</sup>/8}}, which gives a generator of 498.308{{c}} 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
+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'''&nbsp;''b'''''s'''{{angbr|''E''}} with generator range <math>I \subseteq (1, \sqrt{E})</math> (in the linear frequency domain), and consider a pair ('''u''',&nbsp;'''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
 <math>\begin{align}
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 === Fully DR triads ===
 {{main|Delta-rational triads in small edos}}
 === Partially DR tetrads ===
 {{main|Partially delta-rational tetrads in small edos}}
 == 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.-->
 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.
@@ Line 92: / Line 98: @@
 ! rowspan="2" | Edos
 |-
-! g<sup>10</sup>
+! ''g''<sup>10</sup>
-! g<sup>9</sup>
+! ''g''<sup>9</sup>
-! g<sup>8</sup>
+! ''g''<sup>8</sup>
-! g<sup>7</sup>
+! ''g''<sup>7</sup>
-! g<sup>6</sup>
+! ''g''<sup>6</sup>
-! g<sup>5</sup>
+! ''g''<sup>5</sup>
-! g<sup>4</sup>
+! ''g''<sup>4</sup>
-! g<sup>3</sup>
+! ''g''<sup>3</sup>
-! g<sup>2</sup>
+! ''g''<sup>2</sup>
-! g<sup>1</sup>
+! ''g''<sup>1</sup>
-! g<sup>0</sup>
+! ''g''<sup>0</sup>
 |-
 |
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 ! colspan="2" rowspan="2" | Not JI, but isodifferential
 | style="white-space: nowrap;" | φ:(φ + 1):(φ + 2):(φ + 3)
-| rowspan="7" | No, not all or none
+| rowspan="8" | No, not all or none
 | +1+1+1
 |-
@@ Line 1,190: / Line 1,196: @@
 == External links ==
-* [https://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (Includes least-squares linear error calculation)]
+* [http://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (Includes least-squares linear error calculation)]
 [[Category:Chord]]