superneedle's knitting adventures

Taking the magic out of the "Magic Formula"

This is the first in a series of posts explaining the use of the "magic formula", a simple mathematical equation which allows knitters to calculate evenly spaced shapings in varied situations. I'm putting this here as much for my own reference as anything else. I'll be expanding upon the varied uses of the MF as soon as I get the chance to put fingers to keyboard. I'm a busy boy, so it'll probably take me a while to write the complete series. Please be patient....

What exactly is the magic formula?
The “magic formula” is a nickname coined by Alles Hutchinson, an American machine knitting expert, for a mysterious sum she discovered being used frequently in 1970's Japanese knitting magazines. This strange, but useful equation allowed knitters to precisely calculate their garment shapings, eliminating guesswork or trial and error. Although not immediately clear as to why, it worked perfectly every time as if by magic, hence the name.

The “magic formula” is more correctly known as a “diophantine equation” and was invented by Diophantus, an ancient greek mathematician credited as the father of algebra. A diophantine equation is defined as an equation in which only whole number solutions are allowed. In other words, the answer to the sum can only be a whole number, no fractions or remainders are allowed. This is exactly what makes it so useful to us as knitters, bearing in mind that we can't work with fractions of stitches or rows, only whole ones.

There are 4 basic uses or expressions of the magic formula:

Expression 1a - used to evenly space shapings along a diagonal line, such as on a sleeve underarm, a raglan armhole or a v-neckline.

Expression 1b - same as above, but shapings always occur on even rows only. This is particularly useful for handknitters who prefer to work shapings with the RS of work facing. For this particular expression of the formula to work, there has to be at least two times the number of rows as stitches.

Expression 1

Expression 2 - used to evenly divide up groups of stitches for shaping, such as on a sloped shoulder or flared panel shapings on a sideways knit skirt. Shaping can only happen on alternate rows with this particular expression of the magic formula.

Expression 2

Expression 3 - used to evenly space shapings horizontally across a single row of knitting, such as in a circular yoked sweater, or to deliberately create fullness above a ribbed border.

Expression 3

Expression 4 - used to evenly space buttonholes or other design elements such as cable panels across a knit piece.

Expression 4

I'm working on diagrams to explain everything as clearly as possible. So stay tuned!

Progress...of sorts

Just to prove I actually do attempt some knitting every now and again, I make this small offering to the Knit-Gods:

It's a baby blanket [or soon will be] in Rowan's Cotton Glace, for a colleague who is giving birth to a bouncimg baby boy in July. Gertrude, my knitting assistant [a.k.a the Garter Carriage, if you have no idea what I'm rambling about, check out Sonja's excellent article here] is chugging away on my electronic machine...slowly....veeerrrry slowly. Even on a machine, this kind of knitting isn't much faster than knitting by hand. In theory, you can set up the Garter Carriage and go away and do something completely different, whilst the machine gets on with things and creates a beautiful textured fabric and then waits patiently for you to return when it has knit the required number of rows. The reality is slightly different. There have been myriad dropped stitches, frayed yarn and carriage jams due to Gertrude being a temperamental beast when confronted with inelastic plant fibres. It also seems that all Cotton Glace is not created equal - The green is comparatively softer, but seems to have been dyed unevenly having a slightly striped effect when knitted up. Needless to say, Gertrude loves this yarn, and hates the orange which is quite hard and much more shiny due to having been more tightly spun. Dropped stitches everywhere with this yarn. Nevertheless, I intend to persevere. Watch this space.

I also had the opportunity recently to induct a new member into my gang of knitting machines. The LK150 is classed as a hobby machine, due to it being made of plastic and being relatively cheap.  It's a mid-gauge machine and is intended to knit yarns in the Double Knitting and Aran category. Despite having used all kinds of knitting machines for years, the tension settings on it have me stumped. Does anyone have any info to share on tension settings [both carriage and tension mast]?  I haven't had much time to experiment yet, so if someone could offer a shortcut I would be most grateful.

Decreases

 

[click to see larger image in a new window]

From back to front:
streamlined decrease, standard decrease, wale deflection decrease, double decrease, bias decrease.   

When I get a moment, I’m going to collate all of the information in this post into a printable PDF, but until then please have patience and of course feel free to point out glaring technical errors.

Apologies for the lurid greenness my samples, it was what was on hand.  [I think the Hulk would be proud of me, though].  All of the samples in the photo above were knit on a standard [4.5mm needle spacing] gauge knitting machine for speed, but could be just as easily knit by hand.  The yarn was cheapo 4 ply cotton.

The other thing I should mention is that I used a 6-point transfer tool for all of the samples.  Experiment with the number of points/stitches for different effects.
All of the samples were knit to emulate the standard ‘decrease 1 st every alternate row’ raglan formula.  This is reflected in the written instructions.

The first two decreases are very well known and serve more as something to compare the later decreases to than anything else.

The Streamlined Decrease
I call this a streamlined decrease simply because of the sharp, clean line created by stitches cutting across stitches.  Hand knitters will be very, very familiar with this type of decrease, it being used in 90% of commercially available patterns.  An extra step, although not a very difficult one, is required for machine knitters to achieve the same effect.

By Machine:
*Counting from edge of work towards centre of bed, using a single-point transfer tool place the 7th st onto the 6th needle. 
Transfer all stitches towards centre to fill space left by empty needle. 
Return newly empty needle at edge to non-working position. 
Knit 2 rows.*
Repeat from *to*

By Hand:
[At beginning of row, leaning towards left]
*Knit 5, ssk, work to end.
Work one row straight*
Repeat from *to*

[At end of row, leaning towards right]
*Work to last 7 sts, K2tog, k5.
Work one row straight*
Repeat from*to*

The Standard Decrease
This decrease is very similar to the previous example, but slightly less graphic.

By Machine:
*Using a 6-point transfer tool, move edge 6 sts in by 1 needle.
Return empty needle to non-working position. 
Knit 2 rows.*
Repeat from *to*

By Hand:
[At beginning of row, leaning towards left]
*Knit 5, K2tog, work across row.
Work one row straight.*
Repeat from *to*

[At end of row, leaning towards right]
*Work to last 7 sts, ssk, k5.
Work one row straight.*
Repeat from*to*

The Wale Deflection Decrease
As you can see in the sample, the edge stitches are deflected by the decrease [hence the name] and bounce off at an angle.  This type of fashioning looks good in finer yarns and shows up best worked several stitches away from the edge [my default number is six regardless of gauge]. 

My Italian colleagues call this decrease “one at the edge and one on the inside”, which is an odd way of explaining that two different decreases are used on alternate shaping rows – once at the edge and then once several stitches inside the edge.
 
By Machine:
*Using a 6-point transfer tool, move edge 6 sts in by 1 needle.
Return empty needle to non-working position. 
Knit 2 rows [or as many rows as your pattern dictates]. 
Using a single point transfer tool, move edge st onto adjacent needle.
Knit 2 rows [or as many rows as your pattern dictates].*
Repeat from *to*

By Hand: 
[At beginning of row, leaning towards left]
*Knit 5, K2tog, work across row.
Work one row straight.
Ssk, work across row.
Work one row straight.*
Repeat from*to*

[At end of row, leaning towards right]
*Work to last 7 sts, ssk, K5.
Work one row straight.
Work to last 2 sts, K2tog.
Work one row straight.*
Repeat from*to*

The Double Decrease
This is the same classic fully-fashioned decrease often seen on commercially produced knits.  In industrial production, sweater manufacturers use this particular method for a very simple reason - to save time [time is money!].  By decreasing two [or more] stitches at a time, they can knit twice [or more] the amount of plain rows before having to stop again.  As a by-product, they also get a more pronounced effect to their shaping.  Compare the double decrease with the standard decrease and you’ll see what I mean.  When working with heavier yarns, you might want to reduce the number of plain stitches worked before and after the decrease for a more graceful look.

To work this decrease by hand, a cable needle is necessary, it functions as a temporary third needle allowing you knit pairs of non-adjacent stitches together.

By Machine:
*Using a 6-point transfer tool, move edge 6 sts towards centre by 2 needles.
Return both empty needles to non-working position. 
Knit 4 rows [this allows for a decrease rate of 1 st every 2 rows]*
Repeat from *to*

By Hand:
Back Double Decrease [at beginning of row, leaning towards left]:
**Knit 4, slip 2 sts onto cable needle and hold parallel to and behind left needle. *Insert right needle into first st on left needle and at the same time into first st on cable needle, knit these 2 sts tog; rep from * once more - 2 sts decreased.  Work 3 rows straight.**
Repeat from**to**

Front Double Decrease [at end of row, leaning towards right]:
**Knit to last 8 sts, slip 2 sts onto cable needle and hold cable needle parallel to and in front of the left needle.
*Insert right needle into first st on cable needle and at same time into first st on left needle, knit these 2 sts tog; rep from* once more - 2 sts decreased.
Knit remaining 4 sts on left needle. 
Work 3 rows straight.**
Repeat from **to**

The Bias Decrease
The bias decrease is a simple variation on the double decrease.  Adding an increase at the beginning and end of the row causes the decreases to move across the fabric, pulling the edge stitches into a bias slant.  Added to this, increasing into the first/last stitch has a cancelling out effect on the double decrease, reducing it to a single decrease.  It obviously follows that to maintain the standard raglan formula, the decrease has to be repeated every two rows and not every four rows as in the previous example.  This decrease looks especially good on raglans, creating a chevron effect where the bias stitches meet at the seamline.  It’s commonly found on classic fine-gauge knitwear along shoulder seams and sleeve caps.

The handknit instructions are particularly wordy, but please don’t be put off.  All you are doing is increasing one stitch to bring the double decrease back to a single decrease.  Try it, it sounds much harder than it is.

Incidentally the term of reference my Italian colleagues use for this is “increase one, decrease two”.  I think it sums up this little manoeuvre extremely well.

 
By Machine:
*Using a 6-point transfer tool, move edge 6 sts towards centre by 2 needles.
There are now two empty needles in working position at the edge.
Make stitch by picking up heel of edge st and placing onto adjacent empty needle. Return remaining empty needle to non-working position. [1 st decreased]
Knit 2 rows.*
Repeat from *to*

By Hand:
Bias Back Decrease: [at beginning of row, leaning towards left]
**Inc 1 into first st, knit 3, slip 2 sts onto cable needle and hold parallel to and behind left needle.
*Insert right needle into first st on left needle and at same time, into first st on cable needle, knit these 2 sts tog; rep from * once more.  [total of 1 st decreased during shaping]
Work 1 row straight.**
Repeat from **to**

Bias Front Decrease: [at end of row, leaning towards right]
**Knit to last 8 sts, slip 2 sts onto cable needle and hold cable needle parallel to and in front of the left needle.
*Insert right needle into first st on cable needle and at same time into first st on left needle, knit these 2 sts tog; rep from* once more. Knit 3, inc 1 into last st on left needle.  [total of 1 st decreased during shaping] 
Work 1 row straight.**
Repeat from **to**

Ps. A big shout-out to Domiknitrix, who I just noticed has listed me on her inspirations page!

Lust

I really, really, really, really want one of these...

Maybe now they can get to work on that knitting machine I mentioned...