Definitions
- Class 1 numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163
- Lucas numbers: …2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199...
- 12-tone octave ratios: $\frac{16}{15}, \frac{9}{8}, \frac{6}{5}, \frac{5}{4}, \frac{4}{3}, \frac{45}{32}, \frac{3}{2}, \frac{8}{5}, \frac{5}{3}, \frac{9}{5}, \frac{15}{8}, 2$
Relating Class 1 Numbers And Lucas Numbers
The following are three links between class 1 numbers and Lucas numbers. The lucky numbers of Euler and Fibonacci numbers have similar mathematical coincidences to Links #1-3 (here, pages 43-44).
Link #1: derived from $Fibonacci(n-1) + Fibonacci(n+1) = Lucas(n)$
A mathematical coincidence relating class 1 numbers and Lucas numbers that has stuck with me is:
$$ Class1(k-1) + Class1(k+1) - 7 + \frac{-1 + (-1)^{k}}{2} \approx Lucas(n) $$
- $Class1(1) + Class1(3) - 7 + 0 \hspace{12pt} = \hspace{12pt} 1 + 3 - 7 + 0 \hspace{12pt} = \hspace{12pt} -3 \hspace{12pt} = \hspace{12pt} Lucas(-3) + 1$
- $Class1(2) + Class1(4) - 7 - 1 \hspace{12pt} = \hspace{12pt} 2 + 7 - 7 - 1 \hspace{12pt} = \hspace{12pt} 1 \hspace{12pt} = \hspace{12pt}$ Lucas(1)
- $Class1(3) + Class1(5) - 7 + 0 \hspace{12pt} = \hspace{12pt} 3 + 11 - 7 + 0 \hspace{12pt} = \hspace{12pt} 7 \hspace{12pt} = \hspace{12pt}$ Lucas(4)
- $Class1(4) + Class1(6) - 7 - 1 \hspace{12pt} = \hspace{12pt} 7 + 19 - 7 - 1 \hspace{12pt} = \hspace{12pt} 18 \hspace{12pt} = \hspace{12pt}$ Lucas(6)
- $Class1(5) + Class1(7) - 7 + 0 \hspace{12pt} = \hspace{12pt} 11 + 43 - 7 + 0 \hspace{12pt} = \hspace{12pt} 47 \hspace{12pt} = \hspace{12pt}$ Lucas(8)
- $Class1(6) + Class1(8) - 7 - 1 \hspace{12pt} = \hspace{12pt} 19 + 67 - 7 - 1 \hspace{12pt} = \hspace{12pt} 78 \hspace{12pt} = \hspace{12pt} Lucas(9) + 2 \hspace{12pt} = \hspace{12pt} 3^4 - 3$
- $Class1(7) + Class1(9) - 7 + 0 \hspace{12pt} = \hspace{12pt} 43 + 163 - 7 + 0 \hspace{12pt} = \hspace{12pt} 199 \hspace{12pt} = \hspace{12pt}$ Lucas(11)
Where the bolded indices for the exact Lucas numbers (i.e. 1, 4, 6, 8, 11) will be revisited in the context of music theory.
Link #2: derived from the Ulam spiral
Since a single piece of circumstantial evidence is typically ignored, here is a relationship based on prime-generating polynomials from the Ulam spiral:
$$ 4n^2 + p $$
- $4n^2 + 3, n = 1...2$
- $4n^2 + 19, n = 1...1$
- $4n^2 + 1, n = 1...3$
- $4n^2 + 43, n = 1...4$
- $4n^2 + 67, n = 1...7$
- $Lucas(5) = 11$ is absent
- $4n^2 + 163, n = 1...18 + 1$
Link #3: derived from the Riemann zeta function zeroes
Another connection based on "$1.27 \cdot (1.557)^n$" (from here) is:
$$ y = \lfloor \frac{7}{4} \cdot (1 + \frac{\pi}{2e})^{n-1} \rceil $$
- 1, 2, 3, 4, 7, 11, 17, 27, 43, 67, 106, 167…
Where the first six values are the sorted Lucas numbers (with less accuracy afterwards), and 1, 2, 3, 7, 11, 17, 43, 67, 167 are essentially the class 1 numbers (with errors for 19 and 163).
Relating Lucas Numbers And The 12-Tone Octave
Motivation: major notes
A connection between number theory and music theory relates to a bolded subset of the 5-smooth (or regular) numbers:
- 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75…
Where the bolded terms correspond to the major notes in the octave when divided by 24:
| Semitone | Transition | Interval | Harmony |
|---|---|---|---|
| 0 | 1 = $\frac{\pmb{24}}{24}$ | unison | major |
| 1 | $\frac{16}{15}$ | semitone | minor |
| 2 | $\frac{9}{8}$ = $\frac{\pmb{27}}{24}$ | major second | major |
| 3 | $\frac{6}{5}$ | minor third | minor |
| 4 | $\frac{5}{4}$ = $\frac{\pmb{30}}{24}$ | major third | major |
| 5 | $\frac{4}{3}$ = $\frac{\pmb{32}}{24}$ | perfect fourth | major |
| 6 | $\frac{45}{32}$ | diatonic tritone | minor |
| 7 | $\frac{3}{2}$ = $\frac{\pmb{36}}{24}$ | perfect fifth | major |
| 8 | $\frac{8}{5}$ | minor sixth | minor |
| 9 | $\frac{5}{3}$ = $\frac{\pmb{40}}{24}$ | major sixth | major |
| 10 | $\frac{9}{5}$ | minor seventh | minor |
| 11 | $\frac{15}{8}$ = $\frac{\pmb{45}}{24}$ | major seventh | major |
| 12 | 2 = $\frac{\pmb{48}}{24}$ | octave | major |
Link #4: derived from minor notes
A natural follow-up question would be if a similar pattern exists for the minor notes:
| Semitone | Transition | Interval | Harmony |
|---|---|---|---|
| 1 | $\frac{16}{15}=\frac{(\frac{\pmb{128}}{5})}{24}$ | semitone | minor |
| 3 | $\frac{6}{5}=\frac{(\frac{\pmb{144}}{5})}{24}$ | minor third | minor |
| 6 | $\frac{45}{32}=\frac{(\frac{168.75}{5})}{24}$ | diatonic tritone | minor |
| 8 | $\frac{8}{5}=\frac{(\frac{192}{5})}{24}$ | minor sixth | minor |
| 10 | $\frac{9}{5}=\frac{(\frac{\pmb{216}}{5})}{24}$ | minor seventh | minor |
Where $128 = 2^7 = (\frac{12}{3!})^{(3! + 1)}$, $144 = 12^2 = (\frac{12}{1!})^{(1! + 1)}$, and $216 = 6^3 = (\frac{12}{2!})^{(2! + 1)}$. Interpolating, one might stumble upon:
| Semitone$, s_0$ | ($s_0$-2) mod 12$, s_{-2}$ | n | f(n) = $(\frac{12}{n!})^{(n!+1)}$ | Transition | Interval |
|---|---|---|---|---|---|
| 1 | 11 | 3 | $2^7$ = $\frac{\pmb{16}^2}{2}$ | $\frac{16}{15}=\frac{(\frac{2^7}{5})}{24}$ | semitone |
| 3 | 1 | 1 | $\pmb{12}^2$ | $\frac{6}{5}=\frac{(\frac{12^2}{5})}{24}$ | minor third |
| 6 | 4 | $\approx \frac{\pmb{3}}{\pmb{2}}$ | $\approx$ $\pmb{13}^2$ - $\frac{1}{4}$ | $\frac{45}{32}=\frac{(\frac{13^2 - \frac{1}{4}}{5})}{24}$ | diatonic tritone |
| 8 | 6 | $\approx \sqrt{\pmb{\pi}}$ | $\approx$ $\pmb{14}^2$ - $2^2$ | $\frac{8}{5}=\frac{(\frac{14^2-2^2}{5})}{24}$ | minor sixth |
| 10 | 8 | 2 | $6^3$ = $\pmb{15}^2$ - $3^2$ | $\frac{9}{5}=\frac{(\frac{6^3}{5})}{24}$ | minor seventh |
Where $s_{-2}$ is set up such that $n$ is monotonically increasing and the $s_{-2}$ values (i.e. 1, 4, 6, 8, 11) correspond to the indices of the exact Lucas numbers from Link #1.
Comments
I asked two similar questions previously (before the music theory discovery), here and a deleted one. The content is also based on my paper.
Questions
- Is there a connection between any of: class 1 numbers, Lucas numbers, and the 12-tone octave?
- Is there a way to relate music theory to square roots?
- What strategies could be used to formalize musical axioms within the above context?