Delta-rational chord: Difference between revisions
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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 ( | 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 | 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''<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'' dyads are rationally related in frequency space, which we call '''fully delta-rational''' (FDR). | [[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'' dyads are rationally related in frequency space, which we call '''fully delta-rational''' (FDR). | ||
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== Mathematical definitions == | == Mathematical definitions == | ||
# A chord C = α<sub>1</sub>:...:α<sub>''n''</sub> is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads α<sub>''k''<sub>1</sub></sub>:α<sub>''k''<sub>2</sub></sub> and α<sub>''k''<sub>3</sub></sub>:α<sub>''k''<sub>4</sub></sub>, such that the real intervals (α<sub>''k''<sub>1</sub></sub>, α<sub>''k''<sub>2</sub></sub>) and (α<sub>''k''<sub>3</sub></sub>, α<sub>''k''<sub>4</sub></sub>) are disjoint and (α<sub>''k''<sub>2</sub></sub> | # A chord C = α<sub>1</sub>:...:α<sub>''n''</sub> is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads α<sub>''k''<sub>1</sub></sub>:α<sub>''k''<sub>2</sub></sub> and α<sub>''k''<sub>3</sub></sub>:α<sub>''k''<sub>4</sub></sub>, such that the real intervals (α<sub>''k''<sub>1</sub></sub>, α<sub>''k''<sub>2</sub></sub>) and (α<sub>''k''<sub>3</sub></sub>, α<sub>''k''<sub>4</sub></sub>) are disjoint and (α<sub>''k''<sub>2</sub></sub> − α<sub>''k''<sub>1</sub></sub>)/(α<sub>''k''<sub>4</sub></sub> − α<sub>''k''<sub>3</sub></sub>) 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) | # 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) | ||
# A chord that has a delta signature with all entries +1 is called ''isodifferential'' or ''linear''. | # A chord that has a delta signature with all entries +1 is called ''isodifferential'' or ''linear''. | ||
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== Least-squares error == | == Least-squares error == | ||
=== Fully DR === | === Fully DR === | ||
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''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+ | 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''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> | ||
}}, i.e. a chord | }}, i.e. a chord | ||
<math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math> | <math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math> | ||
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|+ | 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|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> | ||
}} is taken to be the DR chord that is being approximated. | }} is taken to be the DR chord that is being approximated. | ||
Rewriting a bit, suppose the chord that is considered the approximation is 1:''f''<sub>1</sub>:''f''<sub>2</sub>:...:''f''<sub>''n''</sub>. Let <math>D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap|+ | Rewriting a bit, suppose the chord that is considered the approximation is 1:''f''<sub>1</sub>:''f''<sub>2</sub>:...:''f''<sub>''n''</sub>. Let <math>D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> | ||
}} written cumulatively. Then the resulting linear least-squares optimization problem is | }} written cumulatively. Then the resulting linear least-squares optimization problem is | ||
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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). | 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'' | 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'' | 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.--> | ||
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. | 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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| | | | ||
| 1 | | 1 | ||
| | | −1 | ||
| | | −1 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
| {{monzo|1 -2 1}} | | {{monzo|1 -2 1}} | ||
| 833.09 ( | | 833.09 (φ) | ||
| 36 | | 36 | ||
|- | |- | ||
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| | | | ||
| 2 | | 2 | ||
| | | −1 | ||
| | | −2 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| | | | ||
| 3 | | 3 | ||
| | | −2 | ||
| | | | ||
| | | −2 | ||
| 6:7:9 | | 6:7:9 | ||
| +1+2 | | +1+2 | ||
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| | | | ||
| | | | ||
| | | −1 | ||
| | | −2 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| | | | ||
| | | | ||
| | | −2 | ||
| | | −2 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| 2 | | 2 | ||
| | | | ||
| | | −4 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| 1 | | 1 | ||
| | | −4 | ||
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| | | −4 | ||
| | | −4 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| 1 | | 1 | ||
| | | −2 | ||
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| | | | ||
| | | −2 | ||
| | | −4 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| | | −4 | ||
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| | | −16 | ||
| 4:5:7 | | 4:5:7 | ||
| +1+2 | | +1+2 | ||
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| | | | ||
| | | | ||
| | | −1 | ||
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| | | −1 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| 1 | | 1 | ||
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| | | −1 | ||
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| | | −4 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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| | | | ||
| | | | ||
| | | −8 | ||
| 4:5:6 | | 4:5:6 | ||
| +1+1 | | +1+1 | ||
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== Higher-order differences of frequency == | == Higher-order differences of frequency == | ||
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|+(√2 | 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|+(√2 − 1) +√2 +(√2 + 1)}} chord. | ||
Precession being the second-order difference ( | Precession being the second-order difference (δ<sup>2</sup>) of frequency, we similarly have the theoretical notions of δ<sup>3</sup>-rationality, δ<sup>4</sup>-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. | ||
== Isodifferential chord == | == Isodifferential chord == | ||
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===== Class II ===== | ===== Class II ===== | ||
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 [[ | 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 [[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: {{nowrap|4:6:8:10 {{=}} 2:3:4:5}}. Thus, there is one class II series (the series of all odd harmonics): | ||
{| class="wikitable" | {| class="wikitable" | ||
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Some complex isoharmonic chords can be expressed with an offset from a simpler isoharmonic chord, so it is useful to notate them in a compact and readable way. For example, 41:51:61 is very similar to 4:5:6, so it can be notated as (4:5:6)[+0.1]. Similarly, (20:22:24:27:30:33:36)[+0.339] can be expanded to 20339:22339:24339:27339:30339:33339:36339. | Some complex isoharmonic chords can be expressed with an offset from a simpler isoharmonic chord, so it is useful to notate them in a compact and readable way. For example, 41:51:61 is very similar to 4:5:6, so it can be notated as (4:5:6)[+0.1]. Similarly, (20:22:24:27:30:33:36)[+0.339] can be expanded to 20339:22339:24339:27339:30339:33339:36339. | ||
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)[+ | 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 {{nowrap|(1 + φ):(2 + φ):(3 + φ)}}, which is approximately equal to 1.618:2.618:3.618. | ||
== Categorization of DR chords == | == Categorization of DR chords == | ||
Here is a table which uses the "delta ratio set" | 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. | ||
* '''How to tell a DR chord from a non-DR chord:''' a DR chord has at least one rational number in its delta ratio set. | * '''How to tell a DR chord from a non-DR chord:''' a DR chord has at least one rational number in its delta ratio set. | ||
* '''Within DR chords, how to tell an FDR chord from a non-fully DR chord:''' a FDR chord has ''only'' rational numbers in its delta ratio set. | * '''Within DR chords, how to tell an FDR chord from a non-fully DR chord:''' a FDR chord has ''only'' rational numbers in its delta ratio set. | ||
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|- | |- | ||
! colspan="2" rowspan="2" | Not JI, but isodifferential | ! colspan="2" rowspan="2" | Not JI, but isodifferential | ||
| style="white-space: nowrap;" | | | style="white-space: nowrap;" | φ:(φ + 1):(φ + 2):(φ + 3) | ||
| rowspan="7" | No, not all or none | | rowspan="7" | No, not all or none | ||
| +1+1+1 | | +1+1+1 | ||
|- | |- | ||
| style="white-space: nowrap;" | 1: | | style="white-space: nowrap;" | 1:φ:(2φ − 1):(3φ − 2) | ||
| style="white-space: nowrap;" | +( | | style="white-space: nowrap;" | +(φ − 1)+(φ − 1)+(φ − 1) | ||
|- | |- | ||
! colspan="2" rowspan="1" | Not JI or isodifferential | ! colspan="2" rowspan="1" | Not JI or isodifferential | ||
| style="white-space: nowrap;" | | | style="white-space: nowrap;" | φ:(φ + 1):(φ + 3) | ||
| +1+2 | | +1+2 | ||
| +1+2 | | +1+2 | ||
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|- | |- | ||
! colspan="3" rowspan="3" | (Incompletely) DR | ! colspan="3" rowspan="3" | (Incompletely) DR | ||
| style="white-space: nowrap;" | 4:5: | | style="white-space: nowrap;" | 4:5:τ:7:9 | ||
| style="white-space: nowrap;" | +1+( | | style="white-space: nowrap;" | +1+(τ − 5)+(7 − τ)+2 | ||
| style="white-space: nowrap;" | +1+( | | style="white-space: nowrap;" | +1+(τ − 5)+(7 − τ)+2 | ||
| rowspan="5" | (irrelevant for categorization) | | rowspan="5" | (irrelevant for categorization) | ||
| style="white-space: nowrap;" | {{(}}(7 | | style="white-space: nowrap;" | {{(}}(7 − τ)/(τ − 5), 7 − τ, τ − 5, 2/(τ − 5), 2, 2/(7 − τ){{)}} | ||
| rowspan="3" | no, but at least one | | rowspan="3" | no, but at least one | ||
|- style="white-space: nowrap;" | |- style="white-space: nowrap;" | ||
| 5:τ:8:(3 + | | 5:τ:8:(3 + τ) | ||
| +( | | +(τ − 5)+(8 − τ)+(τ − 5) | ||
| +1+(8 | | +1+(8 − τ)/(τ − 5)+1 | ||
| {{(}}1, (8 | | {{(}}1, (8 − τ)/(τ − 5) {{)}} | ||
|- style="white-space: nowrap;" | |- style="white-space: nowrap;" | ||
| 1:(1 + a):(1 + a + b):(1 + a + 2b):(1 + 3a + 2b),<br />with a/b irrational | | 1:(1 + a):(1 + a + b):(1 + a + 2b):(1 + 3a + 2b),<br />with a/b irrational | ||
| Line 1,216: | Line 1,216: | ||
|- | |- | ||
! colspan="4" rowspan="2" | Not DR | ! colspan="4" rowspan="2" | Not DR | ||
| 4:5: | | 4:5:τ:7 | ||
| style="white-space: nowrap;" | +1+( | | style="white-space: nowrap;" | +1+(τ − 5)+(7 − τ) | ||
| style="white-space: nowrap;" | +1+( | | style="white-space: nowrap;" | +1+(τ − 5)+(7 − τ) | ||
| style="white-space: nowrap;" | {{(}}(7 | | style="white-space: nowrap;" | {{(}}(7 − τ)/(τ − 5), 7 − τ, τ − 5{{)}} | ||
| rowspan="2" | No, none | | rowspan="2" | No, none | ||
|- style="white-space: nowrap;" | |- style="white-space: nowrap;" | ||
| 5:τ:7 | | 5:τ:7 | ||
| +( | | +(τ − 5)+(7 − τ) | ||
| +1+(7 | | +1+(7 − τ)/(τ − 5) | ||
| {{(}}(7 | | {{(}}(7 − τ)/(τ − 5){{)}} | ||
|} | |} | ||