First you're another Sloe-eyed vamp.
Then someone's mother, then you're camp.
Then you career from career to career.
I'm almost through my memoirs.
And I'm here.
This week, Dad asked me to find an algorithm for turning base 10 decimals into binary. He's so annoying.
To review what binary is, we first need to review base 10. We have a counting system called base 10, where we organize our things into piles of 10's using digits 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9. Once we get too many of those piles, we group 10 of them together and call them "hundreds." And we organize those into bigger piles of 10 hundreds, and those are called "thousands." This goes on for quite some time - enough to take care of our counting needs, anyway. We stop understanding numbers easily by one million, which is why we haven't already started a revolution, despite the fact that one person in our country has a trillion dollars.
103 - one thousand
106 - one million
109 - one billion
1012 - one trillion 1015 - one quadrillion 1018 - one quintillion 1021 - one sextillion
1024 - one septillion 1027 - one octillion 1030 - one nonillion 1033 - one decillion
But the choice of the number "10" and the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9 aren't based on some sort of universal truth, and in fact, they're a bit random. Binary or base 2 is what we call it when we start organizing our things into piles of 2 instead. We only need two digits when we do that, so we use 0 and 1. We can group those piles of 2 into bigger piles of two 2's, and the unit 22 is like our hundreds unit. The unit 23 is like our thousands.
Let's explore the number: 111,111.
In base 10, that number means 1 pile of 105 + 1 pile of 104 + 1 pile of 103 + 1 pile of 102 + 1 pile of 101 + 1 pile of 100. In base 2, it means 1 pile of 25 + 1 pile of 24 + 1 pile of 23 + 1 pile of 22 + 1 pile of 21 + 1 pile of 20. Notice that 100 = 20 = 1.
Now consider this number: .111
In base 10, that's one tenth and one one-hundredth and one one-thousandth.
Another way to look at this is one 10-1 + one 10-2 + one 10-3.
In base 2, that's one half and one fourth and one eighth, or one 2-1 + one 2-2 + one 2-3.
So back to Dad's question. Let's turn some long ugly number with a decimal into binary.
I'll choose 654.32̅1̅.
(If you were dying to convert an irrational number into binary and found yourself reading this post, I'm sorry to tell you that you'll have to round. Life is full of disappointments, and you may as well get used to it now.)
The first step is to work on the part before the decimal.
To do that, divide 2 into the integer and write the remainder in the ones (20) unit.
654/2 = 327 R 0
0.
That's the whole algorithm, so now we'll repeat it for the next unit to the left.
Divide 2 into the new number (327) and write the remainder into the 2's (21) unit.
327/2 = 163 R 1
10.
The next remainder goes into the 4's (22) unit.
163/2 = 81 R 1
110.
8's (23) unit.
81/2 = 40 R 1
1,110.
16's (24) unit.
40/2 = 20 R 0
01,110.
32's (25) unit.
20/2 = 10 R 0
001,110.
64's (26) unit.
10/2 = 5 R 0
0,001,110.
128's (27) unit.
5/2 = 2 R 1
10,001,110.
256's (28) unit.
2/2 = 1 R 0
010,001,110.
512's (29) unit.
2/1 = 0 R 1
1,010,001,110.
And that takes care of the part before the decimal.
Before we go on, we ought to check.
512 + 128 + 8 + 4 + 2 = 654
Now we turn our attention to the piece after the decimal.
It is: .32̅1̅
Any repeating decimal can be expressed as a fraction.
Before we begin let's make .32̅1̅ a fraction.
There's a trick to this!
First, name your repeating decimal.
x = .32̅1̅
Then multiply it by 10whatever so you're set up to subtract and erase the repeating part.
100x = 32.12̅1̅
and
100x - x = 31.8
so
99x = 31.8
and
x = 318/990 = 106/330 = 53/165
We're all ready to get started.
The after-the-decimal algorithm is quite similar to the first one.
We'll be working from the decimal moving towards the right this time.
Multiply the fraction by 2.
If the result is ≥ 1, put a 1 in the halves (2-1) unit; if not, put a 0.
Once the fraction is greater than 1, subtract 1 to find the new fraction.
53*2/165 = 106/165 < 1
1,010,001,110.0
1/4's (2-2) unit.
106*2/165 = 212/165 ≥ 1
1,010,001,110.01
1/8's (2-3) unit.
47*2/165 = 94/165 < 1
1,010,001,110.010
1/16's (2-4) unit.
94*2/165 = 188/165 ≥ 1
1,010,001,110.0101
1/32's (2-5) unit.
23*2/165 = 46/165 < 1
1,010,001,110.01010
1/64's (2-6) unit.
46*2/165 = 92/165 < 1
1,010,001,110.010100
1/128's (2-7) unit.
92*2/165 = 184/165 ≥ 1
1,010,001,110.0101001
1/256's (2-8) unit.
19*2/165 = 38/165 < 1
1,010,001,110.01010010
1/512's (2-9) unit.
38*2/165 = 76/165 < 1
1,010,001,110.010100100
1/1024's (2-10) unit.
76*2/165 = 152/165 < 1
1,010,001,110.0101001000
1/2048's (2-11) unit.
152*2/165 = 304/165 ≥ 1
1,010,001,110.01010010001
That looks like a pretty good place to stop, so we're truncating now.
I would have stopped at the 1/512's (2-9) unit, but I thought I'd wait for the next 1 to lock in the 2,048th.
Now that we've solved the riddle, it's time to reacquaint ourselves with the layout of Stuff & Nonsense VII: Mother Goose Suite, which was planned in 2024. The piece will be scored in 6 movements, written in the key of G major for "goose." I normally quote the themes of each set in the postludes of songs 4 and 7. In this case, each of the Stuff & Nonsense themes from songs 1-6 will be paired with a nursery rhyme.
Mother Goose Suite
Here We Go Round the Mulberry Bush
From Through the Looking-Glass and What Alice Found There, Chapter 4.
Alice and the Tweedles were recalled to have sung it.
This one will be used in each of the 6 movements.
Here we go round the mulberry bush,
The mulberry bush, the mulberry bush,
Here we go round the mulberry bush,
All on a frosty morning.
VII.1 Twinkle, Twinkle
From Alice's Adventures in Wonderland, Chapter 7, sung by The Hatter.
The Dormouse interrupts and sings the last 4 twinkles.
Quotes S&N I. The Mouse's Tale, sung by The Mouse.
Twinkle, twinkle, little bat!
How I wonder what you’re at!
Up above the world you fly
Like a tea-tray in the sky.
Twinkle, twinkle—
Twinkle, twinkle, twinkle, twinkle—
VII.2 The Queen of Hearts
From Alice's Adventures in Wonderland, Chapter 12, sung by The White Rabbit.
Quotes S&N IV. The White Rabbit's Evidence, also sung by The White Rabbit.
The Queen of Hearts, she made some tarts,
All on a summer day:
The Knave of Hearts, he stole those tarts,
And took them quite away!
VII.3 Tweedeldum and Tweedeldee
From Through the Looking-Glass and What Alice Found There, Chapter 4, sung by Alice.
Quotes S&N VI. The White Queen's Riddle, sung by The White Queen & The Red Queen.
Tweedledum and Tweedledee
Agreed to have a battle;
For Tweedledum said Tweedledee
Had spoiled his nice new rattle.
Just then flew down a monstrous crow,
As black as a tar-barrel;
Which frightened both the heroes so,
They quite forgot their quarrel.
VII.4 Humpty Dumpty
From Through the Looking-Glass and What Alice Found There, Chapter 6, sung by Alice.
Quotes S&N V. Humpty Dumpty's Poem, sung by Humpty Dumpty.
Humpty Dumpty sat on a wall:
Humpty Dumpty had a great fall.
All the King’s horses and all the King’s men
Couldn’t put Humpty Dumpty in his place again.
VII.5 The Lion and The Unicorn
From Through the Looking-Glass and What Alice Found There, Chapter 7, sung by Alice.
Quotes S&N III. The Lobster Quadrille, sung by The Mock Turtle.
The Lion and the Unicorn were fighting for the crown:
The Lion beat the Unicorn all round the town.
Some gave them white bread, some gave them brown:
Some gave them plum-cake and drummed them out of town.
VII.6 Hush-a-by Lady
From Through the Looking-Glass and What Alice Found There, Chapter 9.
Sung by The Red Queen, who then requested that Alice sing it as well.
Quotes S&N II. The Duchess' Lullaby, sung by The Duchess & The Cook.
I'm aware that I finished the last post less than a minute ago.
However, the first post was all about cold, hard facts, and this one will contain some initial ideas and reference material; where the previous post set up the boundaries of my playground, this one will unleash the monsters we'll be playing with.
One idea I had had previously was to build each movement in such a way that it quoted its predecessors, similar to the way many children's stories and songs are structured. Like "There's a Hole in the Bottom of the Sea."
An example of such a structure might be:
1. Mulberry
Twinkle
Mulberry
Twinkle & Mouse's Tale
2. Mulberry
Queen of Hearts
Twinkle & Mouse's Tale
Mulberry
Queen of Hearts & White Rabbit's Evidence
3. Mulberry
Tweedles
Queen of Hearts & White Rabbit's Evidence
Twinkle & Mouse's Tale
Mulberry
Tweedles & The White Queen's Riddle
4. Mulberry
Humpty
Tweedles & The White Queen's Riddle
Queen of Hearts & White Rabbit's Evidence
Twinkle & Mouse's Tale
Mulberry
Humpty & Humpty
5. Mulberry
Lion Unicorn
Humpty & Humpty
Tweedles & The White Queen's Riddle
Queen of Hearts & White Rabbit's Evidence
Twinkle & Mouse's Tale
Mulberry
Lion Unicorn & Lobster Quadrille
6. Mulberry
Hush-a-by Lady
Lion Unicorn & Lobster Quadrille
Humpty & Humpty
Tweedles & The White Queen's Riddle
Queen of Hearts & White Rabbit's Evidence
Twinkle & Mouse's Tale
Mulberry
Hush-a-by Lady & Duchess' Lullaby
Another idea I had was to make sure to quote Twinkle in minor, because it appears that way in the ending fugal section of Father William. How fortunate that this passage was originally scored in G minor, while this suite happens to be in G as well. I'm never reaching for the stars, and yet I'm constantly being inundated with stardust. Somebody or something must love me tremendously, and to quote Twinkle in major would not only be a clumsy misstep, but a rude rejection of such a beautiful gift. We must mind our manners.
In 2024, I had decided that Mulberry Bush, Twinkle, and Hush-a-by (as sung to Purcell's Lillibullero) were worthy melodic quotes, meaning that their scores had been published before 1871 (Through the Looking-Glass and What Alice Found There), and they appear to have been popular enough that Alice readers would have brought them to mind.
My initial conclusions were that the Queen of Hearts "did not pass the test of time," and the Tweedles "didn't lend themselves well to singing." Humpty only passed the margin by 1 year, and there's no indication that either Humpty or The Lion and The Unicorn ever made it big. Though these four contenders aren't as strong as the first three, it's likely I'll use them.
Library of Potential Nursery Rhymes
Sheet Music, 1879
The Singing Master, 1840
Nursery Rhymes with Old Tunes, 1846
British Library Music Collections English Songs Vol. 7, 1775
I'm aware, again, that I finished the last post less than a minute ago. Sometimes it takes three posts before we're organized and ready for hibernation.
I have scores for every nursery rhyme except the Tweedles. But would you know it? Fortune favors the hatters. English poet John Byrom (1692-1763) is credited with the epigram that named the Tweedles in the first place, and those two just so happen to be composers. From 1725:
Some say, compar'd to Bononcini
That Mynheer Handel's but a Ninny
Others aver, that he to Handel
Is scarcely fit to hold a Candle.
Strange all this Difference should be
'Twixt Tweedle-dum and Tweedle-dee!
The nursery rhyme would be printed 80 years later, in Original Ditties for the Nursery. Further investigation into Giovanni Bononcini (1670-1747) reveals that he lived in London from 1720 to 1732, and for that time, he was basically in a popularity contest with Handel. In fact, there was a political divide; the Tories favored Handel while the Whig party favored Bononcini. One of Bononcini's most famous works was his opera Xerse, written in 1694.
Originally, the opera Xersewas composed by Francesco Cavalli (1602-1676) in 1655. The libretto, written by Nicolò Minato, was loosely based on Book 7 of The Histories by Herodotus (484-425 BCE). Nearly 40 years after Cavalli's opera, Bononcini wrote a new opera by the same name, and Minato's libretto was adapted by Silvio Stampiglia. But then in 1738, after another 40 years, Handel (1685-1759) wrote Serse, with Stampiglia's work adapted by an unknown librettist. All three of these operas begin with the aria, "Ombra Mai Fù," in which Xerxes I, the King of Persia, sings tenderly to a tree for providing him shade.
To recap, the original Tweedles were none other than Bononcini and Handel, and both of them are remembered today in part for their renditions of an opera about Xerxes (518-465 BCE), the King of Persia. Coincidentally, the most famous aria in those operas happens to be a song about a tree.
Chapter 4 of Through the Looking-Glass, begins, "They [the Tweedles] were standing under a tree, each with an arm around the other's neck..." Alice then recalls that she and the Tweedles had been singing Here We Go Round the Mulberry Bush, as the music "seemed to come from the tree under which they were dancing..." The white rattle that begins the battle is spotted under a tree. And when the brothers battle, they hit trees. As the monstrous crow enters the scene, Alice hides among the trees. I do believe we have run into some scores for the Tweedles, and composed by the original Tweedles, no less.
Today I've been falling ever more in love with Bononcini, but I don't know why I bother, for I don't intend to keep a shred of the beauty (or even the dignity) woven so carefully into these masterful scores.
Meanwhile, I've started thinking about what kinds of mulberries we'll be bringing to this party.
VII.1 - Mode
Twinkle shall make an appearance in G minor, so we'll probably be needing minor mulberries.
Not to be forgotten is Byrd's glorious Picardy third.
VII.2 - Inversion
The White Rabbits' Evidence is all about the horizontal symmetry of the Dorian mode.
I therefore inverted the Mulberry intervals, which resulted in C minor.
Another way to think of this is reflecting the Mulberries over the horizontal G axis.
VII.3 - Retrograde
We are in the presence of the great The White Queen!
I therefore ordered our Mulberries in retrograde.
Another way to think of this is reflecting the Mulberries over the vertical double barline.
VII.4 - Scale
For Humpty Dumpty, I altered the scale to G whole tone.
The harmonies have morphed into augmented triads.
VII.5 - Meter
"Les Lignets" or "Les Lanciers" from The Lobster Quadrille plans to pay a visit.
Meanwhile, our nursery rhyme here isThe Lion and the Unicorn.
They're both in duple time!
VII.6 - Tempo
This one is a lullaby, with the gentle lilt of Purcell's dotted rhythm.
With the closing of KLCC at the end of December, Malinda and I finally had the time to return to our own programs. We've done it before:
16 songs from 1916
Performed 02 July 2016
Wine, Chocolate & Medleys
Performed 23 December 2017
A Dozen Duets
Performed with Celeste Coburn 29 June 2019
Then covid hit, and with it, the KLCC choir pretty much collapsed. On a good day, we had a choir of five, and after two deaths, there wasn't much to salvage. The congregation itself wasn't all that much bigger than the choir had been, and we continued to meet this way, like a couple of stray cats in an abandoned stadium, for five more years. During that time, Malinda and I funneled our efforts into designing the KLCC Christmas cantatas. We didn't have many resources, but we did what we could to make them special. The idea of Rainbow Follies had been in the works for ages, but finding the time to learn the music, write the script, and plan the party was another thing. Fast forward to 2026, a decade after our first production, and we hardly missed a Sunday. There were the occasional trips or celebrations that prevented us from meeting up, but little by little, we built the program. It's hard to explain exactly what we did on Sundays, but it was always some combination of:
Learning songs - working on the hard parts
Trying them in different ways with different ideas
Changing the order of the program and testing the flow
Writing or editing the script
Shopping for props, stationary, rainbow plates, etc.
Just hanging out and being friends
The hardest step is always to pick a date. One of our goals was to give the KLCC people a reason to get together again, but mostly, we just wanted everyone to have fun. Pianistically, my focus was to feel more secure under pressure, particularly with stride pattern and jumping. I think I achieved that, or at least progressed, but that hardly mattered because I still freaked out due to my kryptonite - page turns - and fumbled pretty badly twice. Anyway, nobody seemed to mind too much.
Things we'll keep:
The craft table!
4:00 pm - people hate driving at night, and the party started 15 minutes early
Programs with questions and clipboards - we loved all the different thoughts
A few days ago, I gave my kids a lecture in the car, proving that B♭ major has the most irregular of all the scale fingerings. I found my own sermon so riveting that I decided to write it down. So now, I will teach you my method to understanding/generating the major and minor piano scale fingerings. We'll begin with major and harmonic minors, and then proceed to the natural and melodic minors. We shall have charts and colors, and it will be a gay old time.
Before we begin, a bit of housekeeping. First, if you don't know how to form scales, try my chord and scales relationship post. I also have a more comprehensive Oom-Pah post, if you're insatiable.
Second, I am of the opinion that we must learn hands together patterns. In my teaching experience, students would rather practice hands separately for some time, blindly throw the hands together, and hope that some combination of muscle memory and their favorite religion will see to the rest. Because our brains love symmetry, our natural preferences will be obliterating our hands separate practice in a hurry. In other words, the way nearly everyone learns piano scale fingerings is a pretty good way not to learn them. Each of my children, for example, has been practicing not learning scales for over 6 years, and they have succeeded.
Last, I am of the belief that we do not need to memorize every finger for every note of every scale. We only need to learn the fingerings or concepts that leave us no choices. My Rubik's Cube posts are based on this line of thinking - if certain structures force certain things to happen, there is absolutely no reason to memorize or even keep track of them. I will be telling you everything you need to know, and if you can remember which scale belongs to which category, you should be able to figure out the rest.
These terms were originally inspired by the book, "N. Jane Tan's Scales, Chords, & Cadences." The top row of these charts show right hand fingerings, and the bottom row shows the left. They're all very easy, except for the BRIDGE .
Claws
Hooves
Duck Feet
THREES
3
3
ONES
1
1
CLAWS
1
2
2
1
BRIDGE
4
1
2
2
1
4
W's
2
3
4
4
3
2
V's
2
3
3
2
DUCK FEET
1
2
3
3
2
1
HOOVES
3
4
4
3
Note to self: I will never again figure out how to center a chart within a chart. The inserted chart must be formatted as "top centered" under "table properties" outside of blogger before copying and pasting.
My guess is that everybody reading this post will know the fingering of the C major scale, so we'll start there. Befitting to my reputation of being generous, I'll include two octaves. This is an ascending C major scale.
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
The first step is to put in our 3's .
3
3
3
3
E
A
E
A
3
3
3
3
Next, let's take a look at our CLAWS .
1
2
1
2
F
G
F
G
2
1
2
1
This one is the hardest - the BRIDGE . Bridges can "bridge" us from one octave to the next, and they need a bit more practice than the rest.
4
1
2
B
C
D
2
1
4
Every pattern except for the BRIDGE is excellent for block practice. Remember, practicing 2 octaves gives you only one iteration of the BRIDGE , so you're robbing yourself of opportunity. Jump to 3 or 4 octaves the very moment you summon the bravery. If you like practicing 3 octaves, consider a triplet rhythm so that your beats line up. (I actually like following the beat when it doesn't line up, too.)
Altogether, the standard BRIDGE pattern looks like this:
1
2
4
5
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
5
4
2
1
Working through the 12 major and 12 harmonic minor scales, this BRIDGE scale pattern accounts for 10 of the 24.
They are: C, c, G, g, D, d, A, a, E, and e.
1
2
4
5
5
4
2
1
There are also 3 additional 3 BRIDGE scales, which start on 3's and look like this. Notice that in this case, bridges do not bridge us across the octave.
They are: A♭, c♯, and g♯.
A note to close the chapter. On September 23rd of 2018, I suddenly realized that c♯harmonic scale and the g♯ harmonic scale feel exactly the same. I recently asked my 10-year-old daughter to find the two scales that feel exactly the same, and she first answered G major and a harmonic minor. Although the pair c♯ harmonic minor and g♯ harmonic minor feel more the same to me due to the fact that they sound more the same, technically, she's also correct.
That reminds me of something funny George said last week. " You're not like everybody else; there are only one and a half of you in the world."
Notice that the BRIDGE SCALES work for almost all of the white keys. We'd love for them to work for the other ones (B, b, F, f) as well, but alas, those lay some black keys out under our thumbs. We avoid that because moving our hands in towards the fallboard when not necessary is not efficient.
And so we come to our first rule of scale fingering:
Never place a first finger on a black key.
We call these the WAVE SCALES because we put our W's on the groups of 3 black keys and our V's on the groups of 2 black keys. That way, our thumbs can stay on the white keys between those groupings.
Let's take a look at B major, which has the same fingering as b minor.
1
5
B
C♯
D♯
E
F♯
G♯
A♯
B
C♯
D♯
E
F♯
G♯
A♯
B
4
1
F is a little less intuitive, but it relies on the exact same principles. I'll demonstrate in f minor because it has more black notes to ground us into the WAVE pattern, but it shares the same fingering as F major.
1
4
f
g
a♭
b♭
c
d♭
e♮
f
g
a♭
b♭
c
d♭
e♮
f
5
1
Notice that with WAVE SCALES , we set up by following groups of black notes with our long fingers, regardless of our starting notes. WAVE SCALES can start on any finger, unlike the 10 BRIDGE scales, which always start with 5's and 1's, or the 3 3 BRIDGE scales, which start with 3's . While WAVE SCALES are based on a pattern that could work with almost any scale, the BRIDGE and 3 BRIDGE scales have a cookie cutter solution over the shifting terrain below.
The WAVE SCALES are: B, b, F, f, D♭, G♭, b♭, e♭
Remember that you have to find your starting point!
The hoof fingering divides the four long fingers into halves like hooves. Or like this.
So if the WAVE pattern could work for almost any scale, why don't we just use them for everything? Well, a lot of the time, as with the BRIDGE and 3 BRIDGE scales, something else is more comfortable. The easiest possible cross under is from a black key to a white key, so any time we get that chance, we'll take it.
This takes us to the second rule of scale fingering: Always prioritize crosses under a black key to a white key.
Notice that the 2 HOOF scales have no groupings of 3, but only pairings of fingers.
Congratulations! You know all of your major and harmonic minor fingerings!
For 9 of the 12 keys, the harmonic minor fingerings are the same as the fingerings for the natural and melodic minor scales. But that's not true for these 3.
These scales also happen to be the ones in which the CLAWS become exceptionally lobster-like in the harmonic form, since they stretch from white key over adjacent white key to the next white key as they pass over that augmented second.
Because melodic minor changes when descending, we'll now ascend the first octave and then descend the second. ➚ ➚ ➚ ➚ ➚ ➚ ➚ ➘ ➘ ➘ ➘ ➘ ➘ ➘
f♯natural minor
f♯
g♯
a
b
c♯
d
e
f♯
e
d
c♯
b
a
g♯
f♯
f♯harmonic minor
f♯
g♯
a
b
c♯
d
e♯
f♯
e♯
d
c♯
b
a
g♯
f♯
f♯melodic minor
2
4
1
3
f
g♯
a
c
c♯
d♯
e♯
f♯
e♮
d♮
c♯
b
a
g♯
f♯
4
2
1
4
c♯ natural minor
c♯
d♯
e
f♯
g♯
a
b
c♯
b
a
g♯
f♯
e
d♯
c♯
c♯ harmonic minor
c♯
d♯
e
f♯
g♯
a
b♯
c♯
b♯
a
g♯
f♯
e
d♯
c♯
c♯ melodic minor
c♯
d♯
e
f♯
g♯
a♯
b♯
c♯
b♮
a♮
g♯
f♯
e
d♯
c♯
g♯ natural minor
g♯
a♯
b
c♯
d♯
e
f♯
g♯
f♯
e
d♯
c♯
b
a♯
g♯
g♯ harmonic minor
g♯
a♯
b
c♯
d♯
e
f𝄪
g♯
f𝄪
e
d♯
c♯
b
a♯
g♯
g♯ melodic minor
g♯
a♯
b
c♯
d♯
e♯
f𝄪
g♯
f♯
e
d♯
c♯
b
a♯
g♯
Notice that each of these ascending and descending scales is either a BRIDGE, WAVE , or HOOF , and the top turnaround of every scale becomes a part of the descending scale that follows. In the case of f♯ ascending melodic minor, the bridge isn't as broken as it appears once you add more octaves.
Notice also that the turnarounds for f♯ melodic minor and c♯ melodic minor require us to skip a finger. This should have felt weird, and that is because:
Our third rule of scale fingering: Unless turning at f♯ or c♯ melodic minor, never skip a finger.
1. Never place a first finger on a black key. 2. Always prioritize crosses under a black key to a white key. 3. Unless turning at f♯ or c♯ melodic minor, never skip a finger.
