JP Monteagudo

Lissajous Curves   Lissajous curves are created by plotting two oscillations on perpendicular axes  and ​​​. These oscillations, represented by sinusoidal functions, intersect to create different patterns. When the ratio of these frequencies is equal to — the oscillations are equally phased— the curve is a straight line.

Beautiful movement—the music and beauty of Lissajous(Liss–uh–joo) curves

Visualization &amp; Pearson’s correlation coefficient   The 30DayChartChallenge is a data visualization community that hosts daily challenges for April. Today’s challenge involves dinosaurs. I used the Datasaurus package to create an animated visualization to demonstrate the importance of graphing data, and the effects of outliers on statistical properties.

Animating the Datasaurus dataset as part of the 30 Day Chart Challenge

TL; DR:   Don’t be an absolutist– use relative paths. Use the here package instead of setwd() or getwd() to increase reproducibility and avoid wasting your and other people’s time.

Don’t be an absolutist. Use the here package for reproducible workflows

monthly changes in her circle orb
<i>
</i>
Tip  If you’re wondering where I’m getting my titles and headings, they come from Shakespeare’s sonnets.  The moon has always fascinated me. Eversince I was a child, I was entranced by it; her constant pursuit everywhere I went and her sneaky tactics during the day. This year, before the solar eclipse, I decided to do a little digging and replicate one algorithm used to determine the phases of the moon.

swear not by the moon, the fickle moon

not so quickly! Recently, I become intrigued by game theory algorithms, not because I absolutely love playing games, but because it offers a quantifiable, systematic way to understand human relationships. I find the certainty of science and systematization reassuring to some degree so when I learned about this game called “the prisoner’ dilemma”, I thought it would fun to experiment and figure out the answer to this classic scenario.

the iterated prisoners’ dilemma

Quantile Beta Regression   The quantile beta regression model discussed in the 2021 paper by Running Code   When you click the
<strong>
 Render
</strong>
button a document will be generated that includes both content and the output of embedded code.

A parametric quantile beta regression for modeling case fatality rates of COVID-19. Building models using the betareg and RBE3 packages

RQ: What exactly is the CLT and what’s the deal with n  30?   CLT - mathematical definition and simple terms definition  How we use CLT  How to know when empirical distribution converges to normal distribution? How did we come up with n $ $ 30?  Cohen (1991) Taught that at least 30 observations were needed to use critical-ratio approach used in t-tables when comparing groups.

“{Can I have the CLT, please– a straightforward explanation of the Central Limit Theorem and the magic (n )}”

JP's blog

::: {#lissajous-curves .section .level2}
## Lissajous Curves {.anchored anchor-id="lissajous-curves"}

Lissajous curves are created by plotting two oscillations on
perpendicular axes ![](https://latex.codecogs.com/png.latex?x) and
![](https://latex.codecogs.com/png.latex?y)​​​. These oscillations,
represented by sinusoidal functions, intersect to create different
patterns. When the ratio of these frequencies is equal to
![](https://latex.codecogs.com/png.latex?1)--- the oscillations are
equally phased--- the curve is a straight line. When the frequencies on
both axes differ, the oscillations are out of phase---one moves faster
than the other at a specific angle---and our Lissajous figure will show
a curve. Each frequency produces a specific musical note that will
intercept and either form a pleasant sound, harmonious consonance or
discordance. Consonant notes have a perfect ratio.
:::

:::::::::: {#music-perfect-ratios .section .level2 .page-columns .page-full}
## Music & Perfect Ratios {.anchored anchor-id="music-perfect-ratios"}

Think of a guitar string vibrating at a base frequency of
![](https://latex.codecogs.com/png.latex?329%20Hz)---![](https://latex.codecogs.com/png.latex?329)
cycles per second. By altering our position along the string, we change
the frequency and thus produce different notes. If we moved from the nut
(fret ![](https://latex.codecogs.com/png.latex?0)) to between frets
![](https://latex.codecogs.com/png.latex?11) and
![](https://latex.codecogs.com/png.latex?12) and plucked the string, the
frequency would now be twice as high---we call this an octave. If we
moved ![](https://latex.codecogs.com/png.latex?3/4) down from the nut,
our new note would oscillate at three times the frequency of
E4(![](https://latex.codecogs.com/png.latex?329Hz)). We can produce an
infinite amount of notes by moving along the string and altering the
speed of the oscillations. *You are creating music with movement!*

:::: {#theory-of-just-intonation .section .level3 .page-columns .page-full}
### Theory of Just Intonation {.anchored anchor-id="theory-of-just-intonation"}

However, keeping track of an infinite, unordered sequence of notes is
impossible; instead, we use systems that allow us to measure and obtain
the frequency of a small number of notes. One such system is called
"just intonation" and a commonly seen tuning method is 5-limit tuning.
This method allows us to find the exact, pure frequency of other notes
by multiplying our base frequency by products of the power of prime
numbers ![](https://latex.codecogs.com/png.latex?2),
![](https://latex.codecogs.com/png.latex?3), and
![](https://latex.codecogs.com/png.latex?5) (e.g.,
![](https://latex.codecogs.com/png.latex?2%5E%7B-1%7D%20%5Ccdot%203%5E%7B1%7D%20%5Ccdot%205%5E%7B1%7D)).^1^

::: {.no-row-height .column-margin .column-container}
^1^ In 5--limit tuning we only multiply integers to obtain other musical
notes.

^2^ octaves:
![](https://latex.codecogs.com/png.latex?f_%7Bx%7D%20=%202%5E%7Bk%7D);
perfect fifths: ![](https://latex.codecogs.com/png.latex?3%5E%7Bk%7D),
major thirds:
![](https://latex.codecogs.com/png.latex?5%5E%7Bk%7D,k%20%5Cin%20%5Cmathbb%7BZ%7D).

^3^ Sometimes the resulting note is outside our octave, and we simply
divide it by two and bring it back to our interval.
:::

We call the sinusoidal movements in intervals of powers of
![](https://latex.codecogs.com/png.latex?2) -- octaves, powers of
![](https://latex.codecogs.com/png.latex?3) represent perfect fifths,
and powers of ![](https://latex.codecogs.com/png.latex?5) are called
major thirds^2^. By multiplying our base frequency by different
combinations of these powers,^3^ we get sets of harmoniously consonant
and discordant notes.
::::

::::::: {#the-perfect-ratios .section .level3 .page-columns .page-full}
### The Perfect ratios {.anchored anchor-id="the-perfect-ratios"}

So how can we make sure every note is consonant? Thankfully, the ancient
Greeks discovered a way to create beautifully sounding and naturally
occurring notes using perfect^4^ ratios for octaves, major fifths, and
major thirds. These perfect ratios make music blend, flow, and feel
"right". These are the ratios that create all of the Lissajous figures
we'll explore today.

::: {.no-row-height .column-margin .column-container}
^4^ perfect refers to ratios that produce "just" or "pure" melodic
sounds.
:::

::::: {.cell .tbl-cap-location-bottom layout-align="center" tbl-colwidths="[50,50]"}
Code

::: {#cb1 .sourceCode .cell-code cap-location="bottom" style="background: #f1f3f5;"}
``` {.sourceCode .numberSource .r .number-lines .code-with-copy}
ratios <- c(1,"9/8", "5/4","4/3","3/2", "5/3","15/8",2,240,270,300,320,360,400,450,480)
cols <- c("C","D","E","F","G","A","B","C'")
rows <- c("Perfect Ratios", "Frequency(in Hz)")
perfect_ratios <- matrix(ratios, ncol = 8, byrow = TRUE)
colnames(perfect_ratios) <- cols
rownames(perfect_ratios) <- rows
perfect_ratios
##                  C     D     E     F     G     A     B      C'   
## Perfect Ratios   "1"   "9/8" "5/4" "4/3" "3/2" "5/3" "15/8" "2"  
## Frequency(in Hz) "240" "270" "300" "320" "360" "400" "450"  "480"
knitr::kable(perfect_ratios)
```
:::

::: {.cell-output-display}
|                  | C   | D   | E   | F   | G   | A   | B    | C'  |
|:-----------------|:----|:----|:----|:----|:----|:----|:-----|:----|
| Perfect Ratios   | 1   | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2   |
| Frequency(in Hz) | 240 | 270 | 300 | 320 | 360 | 400 | 450  | 480 |

Perfect ratios in Just Intonation using 5-limit tuning method
:::
:::::
:::::::
::::::::::

::::::::::: {#the-code-animation .section .level2 .page-columns .page-full}
## The Code & Animation {.anchored anchor-id="the-code-animation"}

We'll create the Lissajous curves using two sinusoidal waves on
different phases ^5^. The numerator of the perfect ratio will be the
coefficient ![](https://latex.codecogs.com/png.latex?a) of the
![](https://latex.codecogs.com/png.latex?sin) function ---
![](https://latex.codecogs.com/png.latex?sin(t)%20=%20sin(at%20+%20%5Cdelta))---
and the denominator ![](https://latex.codecogs.com/png.latex?b) will go
to the ![](https://latex.codecogs.com/png.latex?y) coordinate ---
![](https://latex.codecogs.com/png.latex?sin(t)%20=%20sin(bt)). We'll
define our interval from ![](https://latex.codecogs.com/png.latex?0) to
![](https://latex.codecogs.com/png.latex?2%5Cpi)​​.

::: {.no-row-height .column-margin .column-container}
^5^ the curves can also be created with
![](https://latex.codecogs.com/png.latex?sin(t)) and
![](https://latex.codecogs.com/png.latex?cos(t)) functions since
![](https://latex.codecogs.com/png.latex?cos(t)%20=%20sin(t%20+%20%5Cfrac%7B%5Cpi%7D%7B2%7D)),
but the phase difference will now be
![](https://latex.codecogs.com/png.latex?%5Cfrac%7Ba%7D%7Bb%7D*%5Cfrac%7B%5Cpi%7D%7B2%7D).
:::

::::::: {.callout .callout-style-simple .callout-note .callout-titled}
::::: {.callout-header .d-flex .align-content-center}
::: {.callout-icon-container}
:::

::: {.callout-title-container .flex-fill}
Helpful Tip
:::
:::::

::: {.callout-body-container .callout-body}
You can familiarize yourself with Lissajous figures and their sine
functions by using this [interactive virtual
oscilloscope](https://dood.al/oscilloscope/).
:::
:::::::

Using the `gganimate` package in `RStudio`, we'll build a data frame
with a set number of times steps between
![](https://latex.codecogs.com/png.latex?0) and
![](https://latex.codecogs.com/png.latex?2%5Cpi)​; and the figures with
coordinates for each perfect ratio. Then, we'll stitch each plot with
the time steps and coordinates to get a final
![](https://latex.codecogs.com/png.latex?8%20%C3%97%208) animated plot.

:::: {.cell layout-align="center"}
Code

::: {#cb2 .sourceCode .cell-code style="background: #f1f3f5;"}
``` {.sourceCode .numberSource .r .number-lines .code-with-copy}
## ------------------------------- ##
## Creating Lissajous figures      ##
## ------------------------------- ##
library(gganimate)# Loading libs   ##
library(tidyverse)                 ##
library(here)                      ##
library(av)                        ##
## ------------------------------- ##

params <- data.frame(
  id <- seq(1,500,1), # no. of steps
  t <- seq(0,2*pi, length.out = 500)  # equally spaced intervals 0-2*pi
)

figs <- function(t){
  x1 <- sin(t*1) # We're only using perfect ratios
  y1 <- sin(t*1)
  
  x2 <- sin(t*9)
  y2 <- sin(t*8)
  
  x3 <- sin(t*5)
  y3 <- sin(t*4)
  
  x4 <- sin(t*4)
  y4 <- sin(t*3)
  
  x5 <- sin(t*3)
  y5 <- sin(t*2)
  
  x6 <- sin(t*5)
  y6 <- sin(t*3)
  
  x7 <- sin(t*15)
  y7 <-  sin(t*8)
  
  x8 <- sin(t*2)
  y8 <- sin(t*1)
  
  time <- seq_along(t)
  
  data.frame( # Create df with x, y variables accross time
    time, 
    x1, x2, x3, x4, x5, x6, x7, x8,
    y1, y2, y3, y4, y5, y6, y7, y8
  )
}

actual_figs <- figs(params$t)

df <- bind_cols(params, actual_figs) |> # bind two dfs
  select(everything()) |> # make data long
  pivot_longer(x1:x8, names_to = "x_group", values_to = "x") |>
  pivot_longer(y1:y8, names_to = "y_group", values_to = "y") |>
  mutate(x_group = str_remove(x_group, "x"),
         y_group = str_remove(y_group, "y")) |>
  unite("group_id", x_group, y_group, remove = FALSE)

plot <- df|>
  ggplot(aes(x = x, y = y, color = group_id, group = group_id)) +
  geom_point(size = 3) +
  geom_path() +
  facet_grid(x_group ~ y_group) +
  coord_equal() +
  guides(color = "none") +
  theme_void() +
  transition_reveal(time) +
  ease_aes("linear")

animate(plot, duration = 30, fps = 24, height = 1080, width = 1080,
        renderer = av_renderer())

anim_save(filename = "lissajous_figs.mp4",
          path = here::here("blog","2024","04","lissajous","documents"),
          height = 1080, width = 1080)
#| Code adapted from Kieran Healy's blog "Pi Day Circles"
```
:::
::::
:::::::::::

::: {#the-final-product .section .level2}
## The Final Product {.anchored anchor-id="the-final-product"}

Here's the final, animated
![](https://latex.codecogs.com/png.latex?8%C3%978) plot displaying
Lissajous figures using only the perfect ratios.
:::

::::::: {#quarto-appendix}
:::::: {#quarto-citation .section .quarto-appendix-contents}
## Citation {#citation .anchored .quarto-appendix-heading}

<div>

::: {.quarto-appendix-secondary-label}
For attribution, please cite this work as:
:::

::: {#ref-monteagudo2024 .csl-entry .quarto-appendix-citeas}
Monteagudo, JP. 2024. "Beautiful Movement---the Music and Beauty of
Lissajous(Liss--Uh--Joo) Curves." April 15, 2024.
https://doi.org/<https://doi.org/10.59350/y9kmp-zqh66>.
:::

</div>
::::::
:::::::
