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Acoustic instruments don’t allow for the same measured precision as a computer program might in achieving the exact effect that Shepard posited, but it certainly is possible to utilize string instruments to create this illusion, and there is definitely ample room for creative exploration.

Here is a clear, fun, and engaging example of the effect being explored by composer Alba S. Torremocha in a piece for string quartet titled “Shepard Roots.”

Alba S. Torremocha

The consistency of timbre in the string quartet and the instruments’ ability to glissando between notes makes Modal Scarf Amazon Forest by VIDA VIDA eIQoi7M
particularly well-suited for exploring the concept.

Here’s a funny version done with slide whistles.

Try to emulate Michael Bach’s stepwise program with two acoustic instruments playing a chromatic scale in octaves. The upper voice should start forte and decrescendo as it ascends, and the lower voice should start pianissimo or even niente and crescendo as it ascends. As they pass each other in amplitude, they’ll pass off the listener’s attention to one another to keep the illusion alive in the ear.

It’s harder to emulate the glissando version acoustically, since you’ll have to account for dissonance if the octaves aren’t totally lined up, but try it nonetheless! Some things to keep in mind as you listen back:

Again one of the reasons the loop beginning again is not noticeable in Bach’s program is on account of the computer’s ability to execute precisely measured frequency and amplitude. In the program’s case, each pitch can almost be thought of as one tone with the first partial of the overtone series increasing and decreasing as desired.

If you don’t have access to acoustic instruments, you might begin exploring the concept in MIDI with a digital string library . Here are some instructions for how make a Shepard Tone in a DAW .

Beyond attempting to simulate the effect with acoustic means, this can be a great jumping off point to explore composing with glissandi, dynamic balance, and shifting volume within an ensemble .

Learn more about how toharness the elegant might of a string quartet or the entire string section of a symphonic orchestra. Join Orchestration for Strings today and get six weeks of 1-on-1 pro mentorship, guidance and feedback on your work. It’s like having a personal trainer, but for music!

Learn more about how toharness the elegant might of a string quartet or the entire string section of a symphonic orchestra. Join Orchestration for Strings today and get six weeks of 1-on-1 pro mentorship, guidance and feedback on your work. It’s like having a personal trainer, but for music!

Note that these membership predicates are polymorphic on sorts, not on kinds. This is because to be syntactically well-formed the argument term must be of the right kind, namely the connected component containing the sort being tested. Thus a membership at the kind level is either trivially true or a syntactic error. Also, the presence of the system truth values is required for the predicates to be meaningful, so they are only added to modules that import the module TRUTH-VALUE (which is included by default, as part of BOOL , unless the user specifies otherwise).

Advisory. In fact, the symbol Universal does not denote a real sort: it is instead a place holder for parsing purposes that is given an interpretation by the polymorphic attribute (see Section 4.4.4 ). The concrete effect of the interpretation of Universal is the instantiation in each connected component of the operators with one or more Universal arguments.

The natural numbers module NAT provides a Peano-like specification of the natural numbers with an explicit successor function, while at the same time providing efficient built-in operators thanks to the iter theory (see Section 4.4.2 ) and an efficient binary representation of unbounded natural numbers arithmetic using the GNU GMP library.

The natural numbers sort hierarchy has top sort Nat and (disjoint) subsorts Zero and NzNat . The sort Nat is generated from the constant 0 (of sort Zero ) and the successor operator s_ .

fmodNATis protectingBOOL. sortsZeroNzNatNat. subsortZeroNzNat<Nat. ***constructors op0:->Zero[ctor]. ops_:Nat->NzNat[ctoriterspecial(...)].

Having 0 and successor as constructors means that you can define functions on the natural numbers by matching into the successor notation; for example:

fmodFACTORIALis protectingNAT. op_!:Nat->NzNat. varN:Nat. eq0!=1. eq(sN)!=(sN)*N!. endfm

Try entering this module into Maude and then entering the commands

Maude>red100!. Maude>red1000!.

(The results are omitted; the first has 158 digits and the second 2568 digits.)

Natural numbers can be input, and by default will be output, in normal decimal notation; however 42 is just syntactic sugar for s_^42(0) . The command set print number on/off controls whether or not decimal notation is used by the pretty printer. Thus executing the command set print number off will cause numbers to be printed using iteration notation.

Most of the usual arithmetic operators are provided in NAT . They are not defined algebraically but could be given an algebraic definition by the user if desired, for example for theorem proving purposes.

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Your course, your style.

Every instructor has a different style of teaching. In course structures, you can set up your course based to your teaching style.

About course structures

Even if you're new to online instruction, course structures enable you to create a course in a short amount of time. Use a course structure's course menu links, instructions, and content examples to jump-start your course organization and create a meaningful learning experience for students.

After you complete three basic steps, your course will be ready for students.

Select a course structure

Before you begin, you can export or archive your course before you add a course structure to ensure you have an original version.

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A selected course structure's content is added to your course and doesn't replace existing menu items and content. You can delete unnecessary items.

Choose a course structure on the Teaching Style page.

If the Select Course Structure section isn't visible, your institution has disabled it.

Or, choose a course structure in the pop-up Quick Setup Guide . If your institution has enabled it, the pop-up window appears each time you enter your course until you disable it at the bottom of the window. You can also access the pop-up window using these steps:

If you browse and choose a different course structure from the list after selecting Use this Structure and opt to Include content example , the Include content examples for phrase doesn't update until a new course structure is applied.

Add only one structure to a course. Adding more than one course structure in the same course can result in duplicate menu items and content. To prevent this, create or request an empty course to experiment with additional course structures.

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Including a course structure's content examples populates your course with items appropriate to the course structure type. The content examples are intended to be edited for your own use and are designed to inspire both experienced and novice Blackboard Learn instructors.

If you want to include the course structure's content examples, select the Include content examples check box. If you have content in your course, the course structure's content appears also.

Create content

After you choose a course structure, you can customize your course. The next list provides the essential tasks for preparing your course for students.

Change Edit Mode to ON to reveal editing features.

Organize the course menu

You can rename, reorder, delete, hide, and add course menu links as needed. Deleting a content area link from the course menu is a quick way to delete an entire area as well as the items within it. However, the content area and all items within it are permanently deleted. This action is final. If you're unsure, hide the content area instead.

If a tool link on the course menu doesn't work, the tool isn't enabled. You can delete the link or ask your institution to enable it.

If you did not include content examples, the course structure's course menu links provide empty areas where you can create content.

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Create content items, tool links, and files

After your course menu is organized, you can upload files from your computer, edit existing content, and create new content and tool links. Using a wide variety of content types and tools provides a rich, interactive learning experience for your students.

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Edit, move, copy, and delete content

When you add a course structure with content examples to your existing course, the content examples appear in addition to the existing content and are unavailable to students. You can edit, move, copy, or delete any of the content.

As you are building your course, if you're still developing some content or don't want students to see it, make it unavailable.

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Preview your course as a student

As an instructor, you want to be confident that your course is well-designed and functions as intended—before your students see it. Use student preview to review the course content and validate the course behaviors, such as those that control the availability of course content or require a particular interaction from the student to be triggered.

More on previewing your course as a student

To revert, select Existing Menu in the course structures' list and select Use This Structure . No changes are made to your course.

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