User:FloraC/Fokker analysis of rank-3 scales: Difference between revisions
Modal chroma > residual chroma, redefine residual stability accordingly |
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Fokker analysis refers to the analysis of scales as Fokker blocks. In this essay, Fokker analysis of rank-3 scales is illustrated and the related issues are discussed in later chapters. | Fokker analysis refers to the analysis of scales as Fokker blocks. In this essay, Fokker analysis of rank-3 scales is illustrated and the related issues are discussed in later chapters. | ||
Rank-1 scales i.e. equal-step scales and rank-2 scales i.e. mos scales have been well explored. The scene of rank-3 scales has been a mess. Fragmented definitions and theories are all over the place. | Rank-1 scales i.e. equal-step scales and rank-2 scales i.e. mos scales have been well explored. The scene of rank-3 scales has been a mess. Fragmented definitions and theories are all over the place. So it will benefit from Fokker analysis very prominently. | ||
== Notation == | == Notation == | ||
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Two chromas are to be named: the residual chroma | Two chromas are to be named: the residual chroma | ||
<math>\displaystyle | <math>\displaystyle C_\text{r} = L - S = l - s</math> | ||
and the domal chroma | and the domal chroma | ||
<math>\displaystyle | <math>\displaystyle C_\text{d} = L - l = S - s</math> | ||
These form the chroma basis {''C''<sub>r</sub>, ''C''<sub>d</sub>} of the rank-3 Fokker block. The residual chroma is assumed to be always larger than the domal chroma – since the chroma basis is essentially orderless, no Fokker block is missed or duplicate by the assumption. | These form the chroma basis {''C''<sub>r</sub>, ''C''<sub>d</sub>} of the rank-3 Fokker block. The residual chroma is assumed to be always larger than the domal chroma – since the chroma basis is essentially orderless, no Fokker block is missed or duplicate by the assumption. | ||
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<math>\displaystyle | <math>\displaystyle | ||
C_\text{r} + C_\text{d} = L - s \\ | |||
C_\text{r} - C_\text{d} = l - S | |||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
C_1 = | C_1 = C_\text{r} + C_\text{d} \\ | ||
C_2 = | C_2 = C_\text{r} \\ | ||
C_3 = | C_3 = C_\text{r} - C_\text{d} \\ | ||
C_4 = | C_4 = C_\text{d} | ||
</math> | </math> | ||
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The 2-step clearly shows us that the four chromas involved are 2187/2048, 135/128, 25/24 and 81/80, so we find | The 2-step clearly shows us that the four chromas involved are 2187/2048, 135/128, 25/24 and 81/80, so we find | ||
<math>\displaystyle | <math>\displaystyle C_\text{r} = 135/128, C_\text{d} = 81/80</math> | ||
Therefore, we find | Therefore, we find | ||
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First we may assume | First we may assume | ||
<math>\displaystyle | <math>\displaystyle C_\text{r} = 135/128, C_\text{d} = 81/80</math> | ||
It is the same as Aura's diatonic scale. Therefore, we find | It is the same as Aura's diatonic scale. Therefore, we find | ||
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; Fokker block 2 | ; Fokker block 2 | ||
<math>\displaystyle | <math>\displaystyle C_\text{r} = 25/24, C_\text{d} = 81/80</math> | ||
Therefore, we find | Therefore, we find | ||
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; Fokker block 3 | ; Fokker block 3 | ||
<math>\displaystyle | <math>\displaystyle C_\text{r} = 135/128, C_\text{d} = 25/24</math> | ||
Therefore, we find | Therefore, we find | ||
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The different part is that still four distinct chromas are involved. They are 7168/6561, 784/729, 343/324, 64/63. The set defines a unique chroma basis. It is | The different part is that still four distinct chromas are involved. They are 7168/6561, 784/729, 343/324, 64/63. The set defines a unique chroma basis. It is | ||
<math>\displaystyle | <math>\displaystyle C_\text{r} = 784/729, C_\text{d} = 64/63</math> | ||
Therefore, we find | Therefore, we find | ||