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<div id="header">
<h1 class="title toc-ignore">Musical scales</h1>
</div>
<div id="introduction" class="section level1">
<h1>Introduction</h1>
<p>The purpose of this notebook is to display notes used in various
music scales.</p>
<div id="see-also" class="section level2">
<h2>See also</h2>
<p>Packages that work with chords:</p>
<ul>
<li><p>R package <a
href="https://github.com/r-music/chorrrds">chorrrds</a> provides “tools
to use in combination with the music chords” such as the
“<code>deg_maj</code> and <code>deg_min</code> data frames, that
contains all of the minor and major main scales and its respective
degrees”. I did not try it.</p></li>
<li><p>R package <a
href="https://mikl.dk/post/2019-07-27-pichor/">pichor</a> displays piano
chords and chord inversions. I did not try it.</p></li>
</ul>
</div>
</div>
<div id="setup" class="section level1">
<h1>Setup</h1>
<p>Create a data frame that contains the twelve notes of the chromatic
scale used later in the analysis.</p>
<pre class="r"><code>notes <- tribble(
~it, ~en,
"do", "C",
"do#", "C#" ,
"re", "D",
"mib", "Eb",
"mi", "E",
"fa", "F",
"fa#", "F#",
"sol", "G",
"sol#","G#",
"la", "A",
"sib", "Bb",
"si", "B",
)
notes$n = as.integer(rownames(notes))
# Duplicate the notes so that increments of scale number below don't fall out of the range.
notes2 <- rbind(notes, notes)
# A function to generate the line of a markdown table separated by pipes "|"
display <- function(x){cat("", x, "\n", sep="|")}
# The scale corresponding to the white notes of the piano
major_heptatonic <- c(0, 2, 4, 5, 7, 9, 11)
notes[major_heptatonic + 1,]</code></pre>
<pre><code>## # A tibble: 7 × 3
## it en n
## <chr> <chr> <int>
## 1 do C 1
## 2 re D 3
## 3 mi E 5
## 4 fa F 6
## 5 sol G 8
## 6 la A 10
## 7 si B 12</code></pre>
<pre class="r"><code># A function to display a table with the given scale shifted to all the note of
# the major heptatonic scale
display_scale <- function(input_scale){
display(input_scale)
display(rep("---", length(input_scale)))
for (i in major_heptatonic + 1){
display(notes2$it[input_scale + i])
}
for (i in major_heptatonic + 1){
display(notes2$en[input_scale + i])
}
}</code></pre>
</div>
<div id="heptatonic-scale" class="section level1">
<h1>Heptatonic scale</h1>
<p><a href="https://en.wikipedia.org/wiki/Heptatonic_scale">Heptatonic
scale</a></p>
<blockquote>
<p>“A heptatonic scale is a musical scale that has seven pitches (a.k.a.
tones) per octave.”</p>
</blockquote>
<div id="heptatonic-major-scale" class="section level2">
<h2>Heptatonic major scale</h2>
<pre class="r"><code>display_scale(major_heptatonic)</code></pre>
<table>
<thead>
<tr class="header">
<th>0</th>
<th>2</th>
<th>4</th>
<th>5</th>
<th>7</th>
<th>9</th>
<th>11</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>do</td>
<td>re</td>
<td>mi</td>
<td>fa</td>
<td>sol</td>
<td>la</td>
<td>si</td>
</tr>
<tr class="even">
<td>re</td>
<td>mi</td>
<td>fa#</td>
<td>sol</td>
<td>la</td>
<td>si</td>
<td>do#</td>
</tr>
<tr class="odd">
<td>mi</td>
<td>fa#</td>
<td>sol#</td>
<td>la</td>
<td>si</td>
<td>do#</td>
<td>mib</td>
</tr>
<tr class="even">
<td>fa</td>
<td>sol</td>
<td>la</td>
<td>sib</td>
<td>do</td>
<td>re</td>
<td>mi</td>
</tr>
<tr class="odd">
<td>sol</td>
<td>la</td>
<td>si</td>
<td>do</td>
<td>re</td>
<td>mi</td>
<td>fa#</td>
</tr>
<tr class="even">
<td>la</td>
<td>si</td>
<td>do#</td>
<td>re</td>
<td>mi</td>
<td>fa#</td>
<td>sol#</td>
</tr>
<tr class="odd">
<td>si</td>
<td>do#</td>
<td>mib</td>
<td>mi</td>
<td>fa#</td>
<td>sol#</td>
<td>sib</td>
</tr>
<tr class="even">
<td>C</td>
<td>D</td>
<td>E</td>
<td>F</td>
<td>G</td>
<td>A</td>
<td>B</td>
</tr>
<tr class="odd">
<td>D</td>
<td>E</td>
<td>F#</td>
<td>G</td>
<td>A</td>
<td>B</td>
<td>C#</td>
</tr>
<tr class="even">
<td>E</td>
<td>F#</td>
<td>G#</td>
<td>A</td>
<td>B</td>
<td>C#</td>
<td>Eb</td>
</tr>
<tr class="odd">
<td>F</td>
<td>G</td>
<td>A</td>
<td>Bb</td>
<td>C</td>
<td>D</td>
<td>E</td>
</tr>
<tr class="even">
<td>G</td>
<td>A</td>
<td>B</td>
<td>C</td>
<td>D</td>
<td>E</td>
<td>F#</td>
</tr>
<tr class="odd">
<td>A</td>
<td>B</td>
<td>C#</td>
<td>D</td>
<td>E</td>
<td>F#</td>
<td>G#</td>
</tr>
<tr class="even">
<td>B</td>
<td>C#</td>
<td>Eb</td>
<td>E</td>
<td>F#</td>
<td>G#</td>
<td>Bb</td>
</tr>
</tbody>
</table>
<p>The Bb or Sib scale</p>
<pre class="r"><code>notes2$it[major_heptatonic+11]</code></pre>
<pre><code>## [1] "sib" "do" "re" "mib" "fa" "sol" "la"</code></pre>
</div>
<div id="heptatonic-minor-scale" class="section level2">
<h2>Heptatonic minor scale</h2>
<pre class="r"><code>display_scale(c(0, 2, 3, 5, 7, 8, 10))</code></pre>
<table>
<thead>
<tr class="header">
<th>0</th>
<th>2</th>
<th>3</th>
<th>5</th>
<th>7</th>
<th>8</th>
<th>10</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>do</td>
<td>re</td>
<td>mib</td>
<td>fa</td>
<td>sol</td>
<td>sol#</td>
<td>sib</td>
</tr>
<tr class="even">
<td>re</td>
<td>mi</td>
<td>fa</td>
<td>sol</td>
<td>la</td>
<td>sib</td>
<td>do</td>
</tr>
<tr class="odd">
<td>mi</td>
<td>fa#</td>
<td>sol</td>
<td>la</td>
<td>si</td>
<td>do</td>
<td>re</td>
</tr>
<tr class="even">
<td>fa</td>
<td>sol</td>
<td>sol#</td>
<td>sib</td>
<td>do</td>
<td>do#</td>
<td>mib</td>
</tr>
<tr class="odd">
<td>sol</td>
<td>la</td>
<td>sib</td>
<td>do</td>
<td>re</td>
<td>mib</td>
<td>fa</td>
</tr>
<tr class="even">
<td>la</td>
<td>si</td>
<td>do</td>
<td>re</td>
<td>mi</td>
<td>fa</td>
<td>sol</td>
</tr>
<tr class="odd">
<td>si</td>
<td>do#</td>
<td>re</td>
<td>mi</td>
<td>fa#</td>
<td>sol</td>
<td>la</td>
</tr>
<tr class="even">
<td>C</td>
<td>D</td>
<td>Eb</td>
<td>F</td>
<td>G</td>
<td>G#</td>
<td>Bb</td>
</tr>
<tr class="odd">
<td>D</td>
<td>E</td>
<td>F</td>
<td>G</td>
<td>A</td>
<td>Bb</td>
<td>C</td>
</tr>
<tr class="even">
<td>E</td>
<td>F#</td>
<td>G</td>
<td>A</td>
<td>B</td>
<td>C</td>
<td>D</td>
</tr>
<tr class="odd">
<td>F</td>
<td>G</td>
<td>G#</td>
<td>Bb</td>
<td>C</td>
<td>C#</td>
<td>Eb</td>
</tr>
<tr class="even">
<td>G</td>
<td>A</td>
<td>Bb</td>
<td>C</td>
<td>D</td>
<td>Eb</td>
<td>F</td>
</tr>
<tr class="odd">
<td>A</td>
<td>B</td>
<td>C</td>
<td>D</td>
<td>E</td>
<td>F</td>
<td>G</td>
</tr>
<tr class="even">
<td>B</td>
<td>C#</td>
<td>D</td>
<td>E</td>
<td>F#</td>
<td>G</td>
<td>A</td>
</tr>
</tbody>
</table>
<p>Note how the notes of the C major scale are the same as the notes of
the A minor scale. The notes of the F major scale are the same as the
notes of the D minor scale. This is explained in <a
href="https://en.wikipedia.org/wiki/Minor_scale">minor scale</a> and the
correspondance between minor and major scales is visible on the <a
href="https://en.wikipedia.org/wiki/Circle_of_fifths">circle of
fifths</a>.</p>
</div>
<div id="chords-in-the-scale-of" class="section level2">
<h2>Chords in the scale of</h2>
<p>For each scale, show the major, minor and 7 chords that are in that
scale.</p>
<pre class="r"><code>major <- c(0, 4, 7)
minor <- c(0, 3, 7)
major7 <- c(0, 4, 7, 10)
notes2$it[major + 1]</code></pre>
<p>[1] “do” “mi” “sol”</p>
<pre class="r"><code>notes2$it[minor + 1]</code></pre>
<p>[1] “do” “mib” “sol”</p>
<pre class="r"><code>major_chords <- notes
major_chords$m4 = slice(notes2,(4+1):(4+12))
major_chords$m7 = slice(notes2,(7+1):(7+12))
minor_chords <- notes
minor_chords$m3 = slice(notes2,(3+1):(3+12))
minor_chords$m7 = slice(notes2,(7+1):(7+12))
major7_chords <- notes
major7_chords$m4 = slice(notes2,(4+1):(4+12))
major7_chords$m7 = slice(notes2,(7+1):(7+12))
major7_chords$m10 = slice(notes2,(10+1):(10+12))
for (i in 1:nrow(notes)){
cat(sprintf("\nThe %s scale: ", notes2$en[major_heptatonic + i][1]))
cat(notes2$it[major_heptatonic + i])
major_match = c()
minor_match = c()
major7_match = c()
for (j in 1:nrow(notes)){
if (length(setdiff(notes2$n[major + j], notes2$n[major_heptatonic + i])) == 0L){
major_match = c(major_match, j)
}
if (length(setdiff(notes2$n[minor + j], notes2$n[major_heptatonic + i])) == 0L){
minor_match = c(minor_match, j)
}
if (length(setdiff(notes2$n[major7 + j], notes2$n[major_heptatonic + i])) == 0L){
major7_match = c(major7_match, j)
}
}
cat(" contains the major chords ")
major_slice <- slice(major_chords, major_match)
cat(major_slice$it)
cat(", the minor chords ")
minor_slice <- slice(minor_chords,minor_match)
cat(paste0(minor_slice$it, "(m)"))
cat(" and the major7 chord ")
major7_slice = slice(major7_chords,major7_match)
cat(paste0(major7_slice$it, "7"))
chords <- unnest(major_slice, cols=c(m4,m7), names_repair="universal") %>%
bind_rows(unnest(minor_slice, cols=c(m3,m7), names_repair="universal")) %>%
bind_rows(unnest(major7_slice, cols=c(m4,m7,m10), names_repair="universal"))
names(chords) <- paste0(rep(c("it", "en","n"),4),rep(1:4,each=3))
options(knitr.kable.NA = '')
chords %>% knitr::kable() %>% print()
}</code></pre>
<p>The C scale: do re mi fa sol la si contains the major chords do fa
sol, the minor chords re(m) mi(m) la(m) and the major7 chord sol7</p>
<table>
<thead>
<tr class="header">
<th align="left">it1</th>
<th align="left">en1</th>
<th align="right">n1</th>
<th align="left">it2</th>
<th align="left">en2</th>
<th align="right">n2</th>
<th align="left">it3</th>
<th align="left">en3</th>
<th align="right">n3</th>
<th align="left">it4</th>
<th align="left">en4</th>
<th align="right">n4</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">do</td>
<td align="left">C</td>
<td align="right">1</td>
<td align="left">mi</td>
<td align="left">E</td>
<td align="right">5</td>
<td align="left">sol</td>
<td align="left">G</td>
<td align="right">8</td>
<td align="left"></td>
<td align="left"></td>
<td align="right"></td>
</tr>
<tr class="even">
<td align="left">fa</td>
<td align="left">F</td>
<td align="right">6</td>
<td align="left">la</td>
<td align="left">A</td>
<td align="right">10</td>
<td align="left">do</td>
<td align="left">C</td>
<td align="right">1</td>
<td align="left"></td>
<td align="left"></td>
<td align="right"></td>
</tr>
<tr class="odd">
<td align="left">sol</td>
<td align="left">G</td>
<td align="right">8</td>
<td align="left">si</td>
<td align="left">B</td>
<td align="right">12</td>
<td align="left">re</td>
<td align="left">D</td>
<td align="right">3</td>
<td align="left"></td>
<td align="left"></td>
<td align="right"></td>
</tr>
<tr class="even">
<td align="left">re</td>
<td align="left">D</td>
<td align="right">3</td>
<td align="left">fa</td>
<td align="left">F</td>
<td align="right">6</td>
<td align="left">la</td>
<td align="left">A</td>
<td align="right">10</td>
<td align="left"></td>
<td align="left"></td>
<td align="right"></td>
</tr>
<tr class="odd">
<td align="left">mi</td>
<td align="left">E</td>
<td align="right">5</td>
<td align="left">sol</td>
<td align="left">G</td>
<td align="right">8</td>
<td align="left">si</td>
<td align="left">B</td>
<td align="right">12</td>
<td align="left"></td>
<td align="left"></td>
<td align="right"></td>
</tr>
<tr class="even">
<td align="left">la</td>
<td align="left">A</td>
<td align="right">10</td>
<td align="left">do</td>
<td align="left">C</td>
<td align="right">1</td>
<td align="left">mi</td>
<td align="left">E</td>
<td align="right">5</td>
<td align="left"></td>
<td align="left"></td>
<td align="right"></td>
</tr>
<tr class="odd">
<td align="left">sol</td>
<td align="left">G</td>
<td align="right">8</td>
<td align="left">si</td>
<td align="left">B</td>
<td align="right">12</td>
<td align="left">re</td>
<td align="left">D</td>
<td align="right">3</td>
<td align="left">fa</td>
<td align="left">F</td>
<td align="right">6</td>