19 October 2024

Father William


The Hatter's Diary
16 Oct 2024

The singalong took place, and I'm so grateful.  The thing could be evolving into a stage production, which means I should, perhaps, write more parts.  I plan to add a part to Walrus (I hear Tweedledum, a countertenor/soprano, in some descants) and a couple lines to the chorus of Looking-Glass (for at present, there aren't "hundreds of voices"), but that's barely a writing assignment - more like dusting a cake with a sprinkling of powdered sugar.  'Tis the business for mornings of leisure and tea, tea and leisure.  And chocolates.

But more clearly than ever before, I realized that Lobster and William don't work.  Lobster needs to go faster - it's manic, and I've clearly failed as a performer, but not necessarily as a composer.  Time and ambition will clarify this suspicion.

But William, dear William!  William is a total flop!!
Nobody minds the parallel fifths.  Nobody is absolutely horrified by a SEQUENCE of TRITONES.  Nobody thinks it sounds like Byrd, and what's more, nobody even cares about Byrd.  Nobody cleared their throat to say, "Why, William!  What ever are you doing here in Wonderland, with us?"  Nobody even thinks the plagal cadence is blasphemous.  Everyone just wants it to be over with.
Surely I'm not mistaken in the notion that this concept - that Father William is William Byrd - is the correct approach, and everyone ought to be delighted by a piece by Byrd from Wonderland.  And unfortunately, that means I have not done my job.
I need to do Byrd better.  And the first step in remedying that problem is to drop the piano.  William must break in and out of Byrdian vocal polyphony.  (The idea of rescoring some of these pieces has arrived with impeccable timing.)

I could scrap everything and start all over, but most of me doesn't want to totally erase the original composition, because I believe there's some magic trapped inside, even if nobody else can see it.
There's no real reason to rewrite the son's part (which does need the piano accompaniment) even though I'm not particularly attached to it, and some of the father's part might be salvageable.

Before I get started on that, I'll have to know some things that I don't know.
The first order of business is to review my own composition notes from a few years back.
To imitate Dowland/Byrd, I wrote William's accompaniment in a three-voice texture that
    Mostly avoided doubling the melody
    Mostly stayed on root position and first inversion triads
    Mostly avoided parallel 5ths and 8ves, and
    Mostly treated 2nds, 4ths, & 7ths from the bass as unaccented passing tones or suspensions.
William's phrases are punctuated with 4-3 suspensions on G major and D major cadences.
Each of William's melodic phrases appears in rhythmic augmentation in the bass.
Also, there's always some middle voice imitation.
Each of William's verses has four phrases.
    The first phrase remains relatively untouched throughout the verses to set the stage.
    The second verse ruins its 2nd phrase:
    The words, "I kept all my limbs very supple," are illustrated with a diverging melodic line.
    The third verse ruins its 2nd and 3rd phrases:
    The word "argued" is illustrated with a series of 3 ascending tritones in the 2nd phrase.
    The word "strong" displays strength with a high 5-beat note in the 3rd phrase.
    The fourth verse ruins its 2nd, 3rd, and 4th phrases, all of which have blatant parallel 5ths.
As Dr. David Neumeyer put it, "[parallel] fifths will irritate the pedantically minded."


That was worked out towards the beginning of 2021, when I was a being a big fat stupid crybaby.  "Dr. Neumeyer, help me!  Whatever shall I do?"  A lot of good that did me - now I'm sitting here, at the end of 2024 with a flop on my hands.  It's time to roll up my sleeves.  A really big girl would commence a massive interval analysis project on a hundred pieces by Byrd, but I'm not sure I'm that independent just yet.

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The Hatter's Diary
17-18 Oct 2024

In case it helps, I watched five lectures on 16th century 2-voice counterpoint by Kevin Ure to learn a little bit about species counterpoint, which I don't remember studying in school.  It's not the same as what I've set out to do, but it's a mindset.  A starting place.  I'm addressing the situation.

Chapter 1: Introductory Notes
    Cantus Firmus - "fixed voice," (CF).
    Teacher usually provided the CF, and it may not be changed.
    Counterpoint - the "solution" to the CF by the student; can be above or below.
    In counterpoint, horizontal voice leading is the point, not harmony.
    ---
    Fux wrote Gradus ad Parnassum (1725, Latin), a tutorial on 16th-century species counterpoint.
        First species - whole notes against whole notes.
        Second species - whole notes against half notes.
        Third species - whole notes against quarter notes.
        Fourth species - whole notes against tied half notes/syncopated whole notes.
        Fifth species combines the first four species.
    None of them actually have to be based on whole notes.
    As we move through the five species, more dissonances are permitted.
    Dissonance is a tool to create tension and move the piece forward.
    Although it must be treated carefully, it should be used as much as possible.
    Dissonances may not be exposed on strong beats.
    ---
    Steps - 2nds, skips - 3rds, leaps - 4ths or greater.
    Perfect consonances are unisons, 8ves, and 5ths.
    Imperfect consonances are 3rds and 6ths.
    Dissonances are 2nds, 4ths, 7ths, and tritones.
    The 4th is only considered a consonance if it fits in the harmonic series.
        (Fundamental, 8ve, 5th, 4th, M3rd, m3rd, so C G C is okay but C F C is not.)
    Diminished intervals, 4ths, and m7ths should be resolved inwards.
    Augmented intervals and M7ths should be resolved outwards.
    Always count intervals from the lowest sounding voice, even if voices have crossed.
    ---
    Modal counterpoint - all church modes except Locrian (tritone).
    Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian
    Tonal counterpoint - major or minor scales (not used in 16th or 18th centuries).
    ---
    Types of motion: similar, contrary, parallel, oblique.
    Both similar and parallel motion are called direct motion.
    Contrary is best, similar is next, and parallel and oblique are less desirable.
    Use a mix of different types of motion.
    ---
    Four Rules of Motion:
    Perfect consonance to perfect consonance - contrary or oblique motion.
    Imperfect consonance to perfect consonance - contrary or oblique motion.
    Perfect consonance to imperfect consonance - any motion.
    Imperfect consonance to imperfect consonance - any motion.
    In other words, never move in direct motion to a perfect consonance.
        Parallel motion into perfect consonances results in parallel 5ths or 8ves.
        Similar motion into perfect consonances results in hidden/direct 5ths or 8ves.
    In 4-part harmony, direct motion to an octave is "fixed" by stepwise motion in the soprano.
    ---
    Two-voiced counterpoint should stay within a 12th; 10th is better; 8ve is even better.
    Entire range (four parts) must be within F2-A5.

Chapter 2: Melodic Principles
    Establish tonality immediately; 1st scale degree begins and ends the piece.
    1st, 4th, and 5th scale degrees should be emphasized early.
    Aim for a V-I cadence.
    Avoid unresolved leading tones or tendency tones.
    ---
    Melodies have a natural contour of rise and fall.
    They should have one high point, which is consonant to the tonic (final).
    The high point is best placed between the halfway point and three quarters through the piece.
    Ideally, melodies should also have a low point, but this is less important.
    The low point should also be consonant to the final.
    ---
    Conjunct - stepwise
    Disjunct - skips or leaps.
    Counterpoint melodies should consist mostly of conjunct motion.
    Avoid conjunct motion that is followed by a leap in the same direction.
    ---
    Skips can be resolved by a step in either direction.
    Avoid consecutive skips unless outlining a triad.
    Dissonant leaps were not used in 16th or 18th-century counterpoint.
        Avoid melodic tritones, descending m6ths, M6ths, 7ths, and 9ths. 
    Leaps should be resolved by a step in the opposite direction.
    Leaps can also be resolved by smaller skips or leaps in the opposite direction.
    ---
    Melodic range should not exceed a 10th.
    Melodic leaps should not exceed an 8ve, and even 8ves are used rarely.
    Avoid consecutive skips and leaps that expand the melody over an 8ve.
    Avoid consecutive 4ths and 5ths.
    Avoid three or four intervals of the same type.
    ---
    With minor scales, use melodic (raised ascending, natural descending).
    Avoid chromatic relationships outside of the mode.
        Bb to B natural would not be acceptable in Ionian.
        Bb to B natural could be used in Lydian or Dorian (musica ficta).
        At least one measure should be placed between such tones.
    ---
    In Fux's CF, the last 3 bars are often stepwise and the last 2 bars are 2-1.    
    When writing counterpoint, begin with the opening, cadence, and high point.
    With all 2-voiced openings,
        If the CF is in the bass, begin with a unison, 5th or 8ve.
        If the CF is in the soprano, begin with a unison or 8ve.
    Fill in remaining voices with a lot of imperfect consonances.

Chapter 3: First Species
    Whole notes against whole notes.
    Dissonances are not acceptable.
    Avoid two unisons, 5ths, or 8ves on consecutive accented beats.
    5ths can move to 8ves and vice-versa, but only in contrary motion.
    ---
    Battuta - a 10-8 progression from downbeat to downbeat.
    Battutas are forbidden in the middle of a counterpoint because they sound like cadences.
    They're okay from a strong beat to a weak beat, and only forbidden in the first species.
    10-8-6 contrary motion is acceptable (voice exchange, 8ve is passing tones).
    Rests, Unisons, Repeated Notes
    Do not begin with a rest.
    Unisons to the CF may only be used in the first and last measures.
    Repeated notes are permitted.
    Cadences
    The cadence above and below the CF uses the modal degrees 1-7-1.
    This 7 is always sharped except in Phrygian mode.
    The third to last note cannot ascend by a 3rd to the leading tone.
    If the CF is in the bass, the cadence is M6-8.
    If the CF is in the soprano, the cadence is m10-8 or m3-unison.

Chapter 4: Second Species
    Whole notes against half notes.
    All downbeats must be consonant.
    Dissonance can be used on weak beats, but only when moving by step.
    Passing tones must begin and end with strong beat consonance.
    Consonant neighbor tones must also begin and end with strong beat consonance.
    All dissonances are passing tones, and they should be used often.
    ---
    Intervening thirds don't erase parallel fifths or octaves.
    However, leaps correct parallel 8ves - they "make the ear forget."
    Leaps also correct hidden 5ths or 8ves.
    ---
    If voices are too close together, 8ves or ascending m6th leaps can be used.
    If the counterpoint crosses voices with the CF, it must stay crossed.
    Rests, Unisons, Repeated Notes
    The first half of the measure may use a rest.
    Unisons are permitted on weak beats, beginning, and ending tones.
    Repeated notes are not permitted.
    Cadences
    The final 2 or 3 notes may revert to first species whole notes.
    If the CF is in the bass, penultimate measure is P5-M6 to an 8ve in the final measure.
    If the CF is in the soprano, penultimate measure is P5-m3 to an 8ve in the final measure.
    Avoid consecutive skips or leaps that outline a 7th.

Chapter 5: Third Species
    Whole notes against quarter notes.
    There should be about 2-4 skips and 1-3 leaps in 10-12 measures of CF.
    Melodic high point should be on a strong beat (1 or 3).
    It's still best to make the high point consonant with final, but less important than before.
    If the counterpoint crosses voices with the CF, it crosses back quickly.
    ---
    Avoid ascending skips from a strong beat.
    Descending skips from a strong beat are permitted.
    The disconnect - skipping or leaping and continuing in the same direction; avoid this.
    ---
    Downbeat to downbeat consecutive 8ve and 5th rules:
        There should be three intervening tones between consecutive 8ves and 5ths.
        Avoid emphasizing 8ves and 5ths with leaps, sequences, or devices.
        When one voice skips or leaps, the other voice should step.
    With 4 quarter notes between consecutive unisons, 5ths, or 8ves, you will always be safe.
    With 2-3, perfect consonances must not be emphasized with skips or leaps.
    ---
    Cambiata - "changing tone"
    If the CF is in the bass, 8-7-5-6 or 3-4-6-5 (first is preferred).
    If the CF is in the soprano, 3-4-6-5.
    The cambiata begins with a consonant interval and steps to a dissonance.
    The resolution to the dissonance is delayed with an interjected consonant interval.
    The 2nd beat note is often repeated on the next downbeat, consonant to the CF.
    Avoid outlining a tritone (B-A-F-G, F-G-B-A).
    Avoid creating a harmonic tritone, which sounds like a leading tone/cadence.
    ---
    Dissonant double neighbor tones begin and end with same consonance.
    They create dissonances on beats 2 and 3 (3-2-4-3, 8-9-7-8).
    Consonant neighbor tones can be used freely.
    ---
    Cambiatas and dissonant neighbor tones are "devices," and therefore exceptions.
    But contrary to Fux's teachings, there should generally be no dissonance on strong beats.
    Stranded dissonances are dissonances that are not passing or neighbor tones; avoid them.
    Rests, Unisons, Repeated Notes
    A quarter rest may be used on the first beat.
    Unisons are permitted on the downbeats of the first and last measure.
    Otherwise, they should be used sparingly and only on weak beats.
    Repeated notes are not permitted.
    Cadences
    Cadences - if the CF is in the bass, 3-4-5-6-8 or 8-7-5-6-8 cadences are most conventional.
    If the CF is in the soprano, use 3-5-4-3-1.
    The leading tone must be on the fourth beat, against scale degree 2 in the CF.

Chapter 6: Fourth Species
    Whole notes against tied half notes/syncopated whole notes.      
    Because half notes are tied over downbeats, dissonances are now permitted on strong beats.
    Suspensions have three parts:
        consonant preparation, suspension, consonant resolution.
    The suspension (the middle part) can be dissonant or consonant.
    Dissonant suspensions are preferable to consonant suspensions.
    Dissonant suspensions always resolve down by step; consonant suspensions don't have to.
    Never use more than 3 dissonant suspensions in a row.
    If the CF is in the bass, use 7-6 or 4-3 suspensions.
    If the CF is in the soprano, use 2-3 suspensions.
    If the CF is in the bass, 9-8, 2-1 are also possible, but must be handled more carefully. 
    Chains of 9-8 and 2-1 suspensions cause chains of parallel 5ths or 8ves.
    Therefore, they must be used one at a time with contrary motion (10-9-8, 3-2-1).
    Retardations (resolving upwards) were never used in the 16th century.
    Rests, Unisons, Repeated Notes
    A half note rest must be used at the beginning.
    Unisons are permitted at the beginning and the downbeat of the last measure.
    Repeated notes are not permitted.
    Cadences
    If the CF is in the bass, use a 7-6 sus to an 8ve.
    If the CF is in the soprano, use a 2-3 (not 9-10) sus to a unison.
    Second species cadences can be used as well.

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The Hatter's Diary
19-20 Oct 2024

After taking those notes, I looked up a bunch of similar stuff and started taking even more notes on species counterpoint.  But I stopped dead in my tracks and erased all of those when I suddenly ran into this book:
The Technique of Byrd's Vocal Polyphony
by Herbert Kennedy Andrews, 1966.

The discovery instantly created two problems.  The first is that when not enough people are reading The Technique of Byrd's Vocal Polyphony from 1966, it does not get digitized, or at least does not remain so, and the last thing I want is another book cluttering up a house that is shared by a gaggle of hoarders.  The second problem is that everybody who knows me, knows that I swore off reading and writing in the beginning of 9th grade.  But I would wager there have been a few exceptions to the reading claim, and I'm violating the writing principle this very moment, so the discomforts of owning another thing and the reading of a book are to be overruled.  It will arrive "between November 5th and November 21st," which gives me plenty of mornings for leisure and tea, tea and leisure.  And chocolates.

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The Hatter's Diary
21-23 Oct 2024

I was going to stop taking notes on all the different species rules.  But of course I downloaded the English translation of Gradus because I was hoping to take notes on the fifth species' clausula vera.  "What's that?" you may be wondering.  Well, I don't actually know, because the English translation floating around from about 1750 is partial.  So until my Andrews comes, I'll be taking a look at A Study of Counterpoint from Johann Joseph Fux's Gradus ad Parnassum, translated and edited by Alfred Mann, 1943.

Fux (1660-1698) was an Austrian composer and theorist.  Gradus is made up of play-like dialogue, which you'd never have guessed from the bad (partial) translation.  Aloysius is the master (Palestrina), and Josephus is the pupil (the Hatter channeling the world Byrd visited before he came to Wonderland to write Father William).  This is as close as I come to reading graphic novels.

As I'm making my way through the first species, it's becoming clear that Kevin Ure did a pretty good job, because these conversations between Aloysius and Josephus are hardly giving me pause.  It's not a particularly fast read, but that doesn't mean it's not a page turner.  And for what it's worth, Aloysius and Josephus are a great deal more charismatic than Kevin.  I would recommend this approach - bore yourself with a college professor and then sit back and relax as Fux's play unfolds before you.  (Mann also translated all the movable C clefs to treble and tenor clefs, bless him.)

Page numbers are based on the 1971 copyright edition.

First Species
p. 32 Fux explains why direct 5ths and 8ves are offensive [by imagination].
    Stepwise motion to the perfect consonances would reveal parallel 5ths or 8ves.
p. 39 Avoid progressing to or from a unison with a skip or a leap in either voice.
    (Similarly, avoid skipping/leaping inwards to an 8ve, which also presents a range problem.)
p. 39 Using a sharp before an upwards resolution (musica ficta) is encouraged.

Second Species
p. 46 When cadencing in Phrygian, a B against the F in the CF is a tritone; use C instead.

Third Species
p. 50 Indeed, Fux approves dissonant passing tones on beat 3.
p. 52 Similarly to p. 32, Fux explains why the cambiata should be pardoned [by imagination].
    Stepwise eighths on beat 2 results in 7-6-5-6, which is so much nicer than 7-5-6.
p. 54 musica ficta examples in Lydian, resulting in transposed Ionian.

Fourth Species ("The Ligature")
p. 56 Nothing is new, but Fux explains why suspensions are permitted [by imagination].
    Syncopated whole notes can be shifted a half note to the right to check their consonances.
p. 58 If the CF is in the soprano, 4-5 and 9-10 suspensions are possible in addition to 2-3s.
p. 58 7-8 suspensions are off limits simply because the masters did not use them.
p. 60 It is possible to cut off a tie for good reasons, like avoiding melodic repetition.

Fifth Species
p. 62 Ornamentation - can add stepwise eighths on weak beats.
p. 62 7-6 suspensions can be decorated with two eighths on beat 2 (7-6-5-6).
p. 62 7-6 suspensions can be decorated with leaps to and from a consonance on beat 2 (7-3-6).
p. 62 Fifth Species is called florid counterpoint.
p. 65 Ties over the downbeat "brings about the greatest beauty in counterpoint."
p. 65 Sneakily, three voiced examples started popping up with the CF in the middle.
p. 67 This is not a rule, but a piece of advice.
    A measure of quarter quarter half isn't that great, so consider one of these alternatives:
    Half quarter quarter, all quarters, or tying the half over the barline to a quarter or half.

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The Hatter's Diary
28 Oct - 04 Nov 2024

I've been on sabbatical to celebrate two birthdays and Halloween, make my daughter a witch's dream-catcher, carve a jack-o-lantern, paint a Haewood conjecture mug, give some haircuts, and design a holiday card.  The art that has simply demanded my attention.  The projects creep up onto my shoulder and whisper into my ear, "If not for you, I will never exist."

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The Hatter's Diary
05 Nov 2024

Three Voice First Species
p. 71 Three voiced counterpoint is the "most perfect of all" because triads have three parts.
    Fux's triads are 5/3/1 (no inversions of triads are considered triads).
p. 71 Both upper voices must be consonant with the lowest.
p. 72 To avoid parallel perfects, 6/3/1, 8/3/1, or 8/6/1 might be used instead of 5/3/1.
p. 73 Voice leading principles are much more important than the completion of triads.
p. 74 If the CF is in the bass and moving by step, it's better to harmonize with a 6th than a 5th.
p. 76 Rules (such as no tritones) should be followed between upper voices if possible.
    However, in three voices, rules must sometimes be broken for good reasons.
p. 77 Ascending 6ths on downbeats sound harsh, but they work on upbeats.
p. 78 Sometimes adhering to a rule forces more awkwardness further along.
p. 79 8ves are preferred to unisons.
p. 80 In a scale with a m3rd from the final, cadence without any 3rd.
    Picardy 3rds are considered disturbing and m3rds are considered inconclusive.
    If you must choose a 3rd to avoid parallel 5ths and tritones, choose the M3rd.

Three Voice Second Species
p. 86 Two voices of whole notes in consonance, one voice of half notes.
p. 86 In three voices, parallel 5ths can be "fixed" with a third to promote harmonic triads.
p. 87 Examples use suspensions at the cadences.
p. 87 Downbeats should have triads.
p. 88 It is often necessary to end with M3rds.
p. 90 In this species, write only when considering 1-2 measures ahead.

Three Voice Third Species
p. 91 Two voices of whole notes in consonance, one voice of quarter notes.
p. 91 Downbeats should have triads, but if they cannot, triads should be on beats 2 or 3.
p. 93 One voice of whole notes, one of half notes, one of quarters.

Three Voice Fourth Species ("The Ligature")
p. 94 Wait, what?
    What a rejection of democracy.
    What a rejection of women.
    What a rejection of progress.
    What a shithole of a country.
    What an absolute and utter disgrace.

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The Hatter's Diary
06 Nov 2024

It's a good thing my mild interest in Fux outweighs my attachment to the US, or I'd be broken.

Three Voice Fourth Species ("The Ligature")
p. 94 






The Hatter's Diary
Nov 2024

The Technique of Byrd's Vocal Polyphony

11 October 2024

A Card for Merethis

Before I get to the card for Merethis, let's take a look at the shawl I bought The Volkert.
It's a 100% cotton 60" x 80" jacquard woven tapestry from personalthrows.com.
It is so pretty!!

And it has a rainbow fringe!!

And my favorite Tenniel drawing.  I'm in love.

Meanwhile, this is what Merethis looked like at the party.
But my Cheshire had lamented that I had thrown away my last handmade hat card!
Then last night, Milli told me she missed my handmade hat card, too.
So that was enough to get me going.

So I printed it again.
Then I stained it with a layer of brown food coloring with water.
Then I stained it with a layer of black food coloring with water.
Then I washed it out with water and baked it to dry.
Then I colored in the printing with some inky black markers.

Then I trimmed it down and burned the edges with a lighter.
I glued it to a backing card to make it stronger and glazed it three times.
Finally, I painted the edges in black acrylic paint.

And it turned out like this.

Merethis & Me

25 September 2024

How To Tie a Scarf Into a Bow

Scare Glow wants to take Pooh Bear out to the fall festival.

And he wants to borrow my Mad Hatter's scarf to tie as a bow tie.

First he has to make the sides a little bit uneven - maybe two squares uneven.

Then he has to put the longer side on top.

Then he has to tie it just like the beginning of tying shoelaces.
Now the long part is only one square longer than the other side.
This is when he adjusts the scarf against his purple collar until it's just so.

The knot has naturally made a cross.
We will put the longer tail that's going upwards over his head.

Once the piece on his head comes down, it will form the center knot.
So next, he has to make a bunny ear with the tail that was hanging down.

Then he brings the longer tail over the bunny ear.

And tucks it behind the bunny ear.

And pulls it through the loop it just made to form a bow.

Every good bow needs some floofing.

All ready to go.

Pooh says he likes honey better than flowers.

That might have hurt Scare Glow's feelings, but he was too busy singing
"I'm too sexy for my bow."

17 September 2024

The Celebrity Irrationals

Let it be known that I know nothing of the irrational celebrities.
Last time, I mentioned the square root of 2 (√2), pi (Ï€), the golden ratio phi (φ), and Euler's number (e).  I hadn't meant to leave us all hot and bothered.

   √2   
The square root of some number x is a number, that when multiplied by itself, is x.
So the positive square root of 64 is 8.
The positive square root of 25 is 5.
The positive square root of 2 is bigger than 1 but less than 2.
It is about 1.4142, and it's an irrational number.
In fact, the square roots of all non-square natural numbers are irrational.

That was all I had to say about it, but Sarah absolutely insists upon geometrical graphics.
(Sarah and I have been the best of friends since 1988.)
Below is a square with a side of unit 1.
It's bisected along the diagonal with the blue line.
The Pythagorean theorem states that for all right triangles,
    the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
So in the right triangle below, the sum of the squares of the two shorter sides, 12 + 12, is 2.
And that means that the hypotenuse squared is 2, which means that the hypotenuse = √2.
Put another way, the ratio of the length of the blue line over one of the red lines is √2.



   Ï€   
Pi is everyone's favorite.
It is about 3.14159, and it's an irrational number.
All circles are the same, except for the fact that some are bigger than others.
So when something is true of one circle, it is true of all circles.
Pi is really helpful when discussing circles, because:
    The circumference of a circle with radius r is 2rÏ€, and
    the area of a circle with radius r is Ï€r2.

Wouldn't you know it, but Sarah insisted on another geometrical graphic.
When I said that the circumference of a circle with radius r is 2rÏ€, this is what that looks like.

The circumference is red, and two radii make up the blue line.
This means that the ratio of the length of the red line over the length of the blue line is π.

And when I said that the area of a circle with radius r is Ï€r2, this is what that looks like.
The blue square has an area of r2.
This means that the ratio of the area of the circle over the area of the blue square is π.

2Ï€ is another irrational number called tau (Ï„).
It is about 6.28328, and some people like it better than π.
Generally speaking, however, π won the popularity contest.
In a rebellious fury to this harsh reality, you could make two pies on June 28th.
But until then, here's my pi song: Old McDonald Had a Pie

   Ï†   
Two numbers x and y are said to be in the golden ratio φ if x > y > 0, and
    (x/y) = (x+y)/x.
Wait, don't go on until you understand that.
There's a number x, and it's bigger than y.
The ratio between these numbers, x/y, is the same ratio as their sum and x.
Think of a ratio and how this is almost always not true.
4/3 is NOT equal to 7/4.
It's only true for φ.

Let's solve for phi, which is x/y.
    Ï† = x/x + y/x
    Ï† = 1 + y/x
    Ï† = 1 + 1/φ
    Ï†2 = φ + 1
    0 = -φ2 + φ + 1

I don't know if you remember the quadratic equation.
One time I put it to the tune of "Pop Goes the Weasel" and it was helpful.
    x equals negative b
    plus or minus the square root
    of b squared minus four ac
    all over two a
Did you sing it?

If the format for the quadratic equation is 0 = ax2 + bx + c,
and we have 0 = -φ2 + φ + 1,
then x = φ, a = -1, b = 1, and c = 1.
So according to "Pop Goes the Weasel,"
    Ï† = [-1 ± √(1 - -4)]/[-2]
    Ï† = (-1 ± √5)/(-2)
    φ = about -.618 or 1.618
But because we defined φ as a positive ratio, we can throw out -.618.
(Meanwhile, notice how the √5 guaranteed we were dealing with an irrational.)
So let's double check our findings.
    about 1.618 = x/y, so we could call x about 1.618 and we could call y 1.
    1.618 should = about (1.618 + 1)/(1.618), and that checks out.

This is a rectangle with the height of 1 and the width of 1.618.

It looks pretty nice, doesn't it?
We can cut a golden ratio rectangle into a square and another golden ratio rectangle.

And we can keep going.


If we fill each of those squares with quarter circle arcs, we create the golden spiral.
It is pretty, and does not actually approximate nautilus seashell as legend claims.

Coincidentally, George and I, the most harmonious couple ever, got married on June 18th.

But wait!  Sarah wanted to tell us about one of her high school exes, Fibonacci.
The Fibonacci sequence begins thusly: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
    Let's call the nth term in the sequence Fn.
    Fn = F(n-1) + F(n-2)
If we take a look at the ratio Fn/ (Fn-1), something very interesting happens.
As n approaches infinity, the ratio approaches φ.
So 34/21 is much closer to φ than 2/1, and 832040/514229 is closer still.

   e   
Let's say I have $100.
And I get 4% interest one time a year.
At the end of that year, my $100 has turned into $104.
Let's call this amount, the money I get back, "m."
The equation to get m was
    m = $100(1 + .04).

Now let's say I have $100.
And I get 4% interest, but the interest is calculated or "compounded" twice a year.
So after 6 months, I'll get .02 of my $100, and after 12 months, I'll get .02 of what's in the bank.
At the end of the year, m = $104.04.
The equation to do that was
    m = $100(1 + .02) + [$100(1 + .02)](.02)
Simplified,
    m = $100[(1 + .02) + (1 + .02)(.02)].
Simplifying further,
    m = $100[(1 + .02)2].

Now let's say I have $100.
And I get 4% interest, but the interest is compounded four times a year.
Every quarter, I will get .01 added to what's in the bank.
At the end of the year, m = $104.06.
The equation to do that was:
    m = $100(1 + .01) at the end of the first quarter
    + [$100(1 + .01)](.01) at the end of the second quarter
    + [$100(1 + .01) + $100(1 + .01)(.01)](.01) at the end of the third quarter
    + [$100(1 + .01) + $100(1 + .01)(.01)(.01)](.01) at the end of the fourth quarter
And that's
    $101 + $1.01 + $1.0201 + 1.030301 = $104.060401
Simplified,
    m = $100[(1 + .01)4].

So it seems that the more often we compound our interest, the bigger m gets.
(However, it also seems that the amount it's growing is decreasing.)
What happens if we never stop compounding interest?
That's called "continuous compounding."

If we call our 4% "r" for rate,
and the number of times we compound interest "n" for number,
we get this.
    m = $100(1 + r/n)n.

When the interest rate is 100% instead of 4%,
we get this.
    m = $100(1 + 1/n)n.
And "continuously compounding" means we calculate that as n gets bigger and bigger.
In other words, we find the limit "as n approaches infinity."
The good news is that there's already number that equals "[(1 + 1/n)n] as n approaches infinity."
And that number is called Euler's number, or e.
It's an irrational number that begins 2.71828.

Furthermore,
    m = $100ert,
    where r is your rate per unit of time
    and t means how much time, measured in the same units, that you left your $100 in the bank.

So if I put my $100 at 4% and compound it continuously for 1 year,
    m = $100e(.04)(1) = a little less than 104.082.

And if I stopped continuously compounding to take a sip of coffee, I would make a little less.

   e   
There are lots of ways to discover e.
I asked my dad to tell me what e was, and he did that in two ways in under 5 minutes.
This is the first of them.

! means "factorial."
"3 factorial" or "3!" means 3*2*1.
10! means 10*9*8*7*6*6*5*4*3*2*1.

As n approaches infinity,
    e = (1 + 1/1! + 1/2! + 1/3! + 1/4!... + 1/n!)
Futhermore, as n approaches infinity,
    e= (1 + x/1! + x2/2! + ... xn/n!)

   e   
Here's the second, but you might want to make yourself comfortable.

We're not very good at understanding big numbers or small numbers.
We're really only kind of good at understanding the smaller positive numbers.
Even better if they're smaller positive integers!
    How many eggs shall I cook?
    How many sticks did you find?
    How many people are coming to the party?
But when we start talking about how many atoms or grains of sand, the numbers not only mean very little to us, but they're also difficult to compare and difficult to compute.
To get around that, we sometimes translate very big and very small numbers into smaller positive numbers times 10 to something, and that's called scientific notation.

A quick internet search will tell us that there are between 1078 and 1082 atoms in the universe.
And there are between 10111 and 10123 positions (including illegal moves) in the game of chess.
Did you know that there are more chess positions than atoms in the universe?
And by a LOT?
But if we're only counting legal moves in chess, there are 1040 positions.
That's sort of the square root of the number of atoms in the universe.

When we're thinking of big numbers in this way, we're thinking in powers.
And when we're thinking in powers, logarithms are helpful.
Here's something to stare at for a second.
    If log10(100) = x,
    then x = 2.
In other words, "log10(100)" means "10 to what power = 100?"

Here's a little chart expanding on this idea.
    log10(1000) = 3
    log10(100) = 2
    log10(10) = 1
    log10(1) = 0
    log10(.1) = -1
    log10(.01) = -2
    log10(.001) = -3

Here are some other examples.
    log2(2) = 1
    log2(4) = 2
    log2(8) = 3
    log2(16) = 4

    log5(5) = 1
    log5(25) = 2
    log25(5) = 1/2
    log125(5) = 1/3

    log7(7) = 1
    log7(1) = 0
    log7(1/7) = -1
    log7(1/49) = -2

    log3(3) = 1
    log3(9) = 2
    log3(1/9) = -2
    log3(1/√3) = -1/2

While we’re at it, here are some properties of logarithms.
    logx(1) = 0
    logx(x) = 1
    logx(ab) = logx(a) + logx(b)
    logx(a/b) = logx(a) - logx(b)
    logx(ab) = (b)logx(a)
    x^(logx(a)) = a
    logx(a) = logy(a)/logy(x)

The two most common bases to use for logarithms are 10 and e.
"loge" is called the "natural log" and is abbreviated "ln."
So ln(e) = 1.
loge or ln is so common, in fact, that "log" means "ln" to a mathematician.
Confusingly, "log" means “log10” to a scientist or engineer.

Notice that our last property of logarithms shows us how to change bases.
Because
    logx(a) = logy(a)/logy(x),
Then we can translate between log10 and ln.
    log10(x) = ln(x)/ln(10).

It is nearly beyond the scope of this post to understand why this matters.
But let's try to remember derivatives from high school.
We learned that when we're graphing curves, we can measure how quickly we're moving up or down.
That's done by drawing a tangent line to the curve at a given point, and then measuring its slope.

We learned that the derivative of xis 2x.
Do you remember that?  More generally speaking, if
    f(x) = xn, then the derivative (notated f'(x)) = nx(x-1).
And if you did lots of this stuff, you might remember that if
    f(x) = 3x4 + x3 + 4x2 x1 + 5, then
    f'(x) = 12x3 + 3x2 + 8x1 + 1.
That wasn't important, but I did sneak in 31415 just to be cute.

The point is that f’(x) gives us the slope of the line tangent to f(x) at x.
(This is worth a reread if it doesn't readily make sense.)

Sometimes, instead of graphing f(x) = xn, people graph equations like these:
    f(x) = 1x
    f(x) = 2x
    f(x) = 3x
And as it so happens, if
    f(x) = ax, then
    f'(x) = axln(a).

With the first equation, f(x) = 1x will always yield 1.
Courtesy of desmos graphing calculator, that looks like the picture below.
Its derivative, f'(x), is 1xln(1), and since ln(1) = 0, the slope is always 0.
f(x) is graphed in red and f'(x) is graphed in blue.

f(x) = 2x looks like this (in red).
f'(x) = 2xln(2) looks like this (in blue).
ln(2) is about .693, which is less than 1, so the slope of the tangent is less than the y value.

f(x) = 3x looks like this (in red).
f'(x) = 3xln(3) looks like this (in blue).
ln(3) is about 1.099, so the slope of the tangent is a LITTLE more than the y value.

But what about f(x) = ex?  That looks like this.
f'(x) = exln(e).
ln(e) = 1, so the slope of the tangent is always exactly the y value.
In other words, the derivative of f(x) = ex is ex.
What?  That was like Goldilocks and the 3 bears where f'(x) = 2xln(2) was too small,
f'(x) = 3xln(3) was too big, and f'(x) = exln(e) was just right.

   e   
Here's a fourth way to think about e.
f(x) = 1/x looks like this.

When x = 0, there is no y value; in fact, 1/x is undefined.
When x is very little, there is a y value, and it is very large.
When x = 1, y = 1.
What if we were to measure the space under the curve starting at x = 1?
That would look like this.

And now for the punchline.
If we stop that measurement at x = e, the area under the curve is exactly 1 square unit.

This is beautiful because 1/x is messy and e is messy, and you can combine them to get 1.
It's basically a magic trick, don't you think?
Please do not assume this is everything there is to know about e.
Mom read an entire book on e this year, and my reports don't even cover half the wiki article.
This was like a first date.

   i   
In a fit of whimsy, I'll leave you with a bonus celebrity irrational, i.
i is called an imaginary number because it's not real, and it is √-1.
Even though it's imaginary, we can compute things like this.
    i-2 = -1
    i-1 = -i
    i= 1
    ii
    i= -1
    i= -i
    i= 1
    (2i)= -4
    (3i)= -9

After 4 chapters on e, that was like a pillow mint.
Pillow mints are made with butter, powdered sugar, whipping cream, and peppermint extract.