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The narrow tubes get sewn into a secondary wide tube, with specific seam shift (or edge shift), depending on desired outcome. If I want a quilt top of parallelograms, I would probably match points to points (so seam shift zero), but for a quilt top of rows of triangles, I make seam shift non-zero, as in the photo below.
Then I sew together the two outside strips, matching the beginnings (left side of the photo) at seam allowance from the side edges. The seam is straight but the pre-tube twists around during sewing. When done sewing, the result is a (non-twisted) primary wide tube. (The tip angle of parallelograms/triangles is visible at the left edge of the tube.)
I first sew strips into a pre-tube (strip set) at desired offset angle. This angle determines the tip of the triangles and parallelograms. I may pre-trim the strips at that angle, or simply offset them. I prefer offsetting rather than pre-cutting so I do not have to deal with bias (yet).
Watch my YouTube videos https://www.youtube.com/channel/UCeJQNp4f-_9MALkUwr-HNfg for what to do next. If I want a top of parallelograms, I now make a cut perpendicular to the bottom edge reaching to the top edge, sew together the right and left edges, and I'd be done. If I want instead a top of all-over triangles, I could now cut similarly as for the first cut, but with a rotary cutter, through all the other swaths to create rows of triangles with slanted ends. Doing that would cause me to have to do something time-consuming to the slanted ends. So instead I first do some slitting (cutting that does not reach the outer edges), do only one more seam, then extend the slits into full cuts, to get secondary narrow tubes, which then get cut across into rows of triangles with perpendicular sides. Or better still, the secondary narrow tubes get sewn together into a ternary wide tube, which then gets cut with one perpendicular cut to make one top of all-over triangles.
The basic outline as above of moving from (pieced) strips to pre-tubes to wide tubes to narrow tubes can get iterated several more times, for many different outcomes.
The primary wide tube gets cut parallel to the straight tube edge into primary narrow tubes of specific widths. The widths depend determine the slants of parallelograms, or if making rows of triangles,
they determine the other angles of triangles (tip angle was chosen at the beginning)
By design of seam/edge shift, there is at least double seam allowance through the centers of all parallelograms, so that after I cut the secondary wide tube through those swaths, there is enough room for seam allowance for triangles rows. So, I cut through one of those swaths, one layer of the tube only, and the result is a parallelogram as below.
The how of tube piecing
by Irena Swanson
Bargello quilts and trip-around-the-world quilts are often made with what I call primary narrow tubes, but one seam is undone in each tube before sewing the resulting rectangles together. So the usage of tubes is only temporary. I made several trips-around-the-world in that way in the 1990s and early 2000s. The first book on using such temporary tubes is Trip Around the World Quilt by Eleanor Burns, published in 1988. Two early books using temporary tubes in bargello quilt making are Bargello Tapestry Quilts by Marilyn Doheny and Bargello Quilts by Margie Edie, both published in 1994.
Seminole patchwork makes intricate patterns with strip piecing. The Seminole Native Americans of the Everglades area invented the method probably near the end of the 19th century to make exquisite patterns on their clothing. In general this method is not done with tubes and so there are corners to trim away and discard. Tube piecing can make Seminole patterns without discards. As far as I know, the first book on Seminole patchwork is The Seminole Patchwork Book by Cheryl Greider Bradkin, published in 1978 by Yours Truly. Another early publication is Seminole Patchwork Patterns by Lassie Wittman, self-published in 1979, followed by the book The Complete Book of Seminole Patchwork: From Traditional Methods to Contemporary Uses by Beverly Rush and Lassie Whitman, published in 1982 by Madrona Pub. Two other early books are Guide to Seminole Patchwork by Nancy Devlin, published in 1980 by Starshine Stitchery Press, and Strip patchwork: Quick and easy patchwork using the Seminole technique by Taimi Dudley, published in 1980 by Van Nostrand Reinhold.
In early 2018 I discovered Barbara Johannah. Every quilter should know Barbara Johannah! She was the one who introduced strip piecing and many other streamlined methods, based on Seminole piecing and Ernest Haight's whole-cloth piecing. Her first book was Quick Quilting. Make a Quilt This Weekend, published in 1976, followed by The Quick Quiltmaking Handbook , and Continuous Curve Quilting, both in 1980, Half Square Triangles. Exploring Design in 1987, and Barbara Johannah's Crystal Piecing in 1993. If you look at her books, do not be discouraged by her promotion of half-inch seam allowance in the earlier books. Johannah's and Height's contributions to efficient quilt making are invaluable!
In April 2017 I discovered Anita Hallock and her fast patch method. Her book Fast Patch, A Treasury of Strip Quilt Projects was published in 1989 by the Chilton Book Company. The book has a huge selection of strip-pieced 90-degree and 45-degree blocks (four-patch, nine-patch, half-square triangles, hourglass blocks, Ohio stars, sawtooth borders, feathered stars, shoofly, pinwheels, ...) . She does not use tubes, but uses an otherwise similar system of speeding up, streamlining, making big patterns, and cutting them down to further interesting patterns. Her web page is at http://www.pacinfo.com/~hallock/, with more information on the book at http://www.pacinfo.com/~hallock/treasury.html .
I discovered Rita Hutchens and her tubular strip piecing method in August 2015. She uses primary wide and primary narrow tubes as well. Her book Totally Tubular Quilts was published in 2003 by C&T Publishing, her web page is at http://www.ritahutchens.com/index.html, and she has several YouTube videos at https://www.youtube.com/user/RitaHutchens.
There is a related method called strip tubing, developed by Daniella Stout and Georgette Dell'Orco. Check out https://www.cozyquilt.com/Features/StripTubing.aspx at the Cozy Quilt store in San Diego, which has links to their books. Strip tubing creates small-circumference strip tubes from which one cuts individual blocks that are then handled further.
I next trim the construction precisely at the chosen angle. (Now there is bias along the edge.)