Home > Articles > Programming > Python

  • Print
  • + Share This
This chapter is from the book

Example: superposition of sound waves

As discussed in SECTION 1.5, the simple audio model that we studied there needs to be embellished to create sound that resembles the sound produced by a musical instrument. Many different embellishments are possible; with functions, we can systematically apply them to produce sound waves that are far more complicated than the simple sine waves that we produced in SECTION 1.5. As an illustration of the effective use of functions to solve an interesting computational problem, we consider a program that has essentially the same functionality as playthattune.py (PROGRAM 1.5.8), but adds harmonic tones one octave above and one octave below each note to produce a more realistic sound.

Chords and harmonics

Notes like concert A have a pure sound that is not very musical, because the sounds that you are accustomed to hearing have many other components. The sound from a guitar string echoes off the wooden part of the instrument, the walls of the room that you are in, and so forth. You may think of such effects as modifying the basic sine wave. For example, most musical instruments produce harmonics (the same note in different octaves and not as loud), or you might play chords (multiple notes at the same time). To combine multiple sounds, we use superposition: simply add their waves together and rescale to make sure that all values stay between −1 and +1. As it turns out, when we superpose sine waves of different frequencies in this way, we can get arbitrarily complicated waves. Indeed, one of the triumphs of 19th-century mathematics was the development of the idea that any smooth periodic function can be expressed as a sum of sine and cosine waves, known as a Fourier series. This mathematical idea corresponds to the notion that we can create a large range of sounds with musical instruments or our vocal cords and that all sound consists of a composition of various oscillating curves. Any sound corresponds to a curve and any curve corresponds to a sound, so we can create arbitrarily complex curves with superposition.

Computing with sound waves

In SECTION 1.5, we saw how to represent sound waves by arrays of numbers that represent their values at the same sample points. Now, we will use such arrays as return values and arguments to functions to process such data. For example, the following function takes a frequency (in hertz) and a duration (in seconds) as arguments and returns a representation of a sound wave (more precisely, an array that contains values sampled from the specified wave at the standard 44,100 samples per second).

def tone(hz, duration, sps=44100):
    n = int(sps * duration)
    a = stdarray.create1D(n+1, 0.0)
    for i in range(n+1):
        a[i] = math.sin(2.0 * math.pi * i * hz / sps)
    return a

The size of the array returned depends on the duration: it contains about sps*duration floats (nearly half a million floats for 10 seconds). But we can now treat that array (the value returned from tone) as a single entity and compose code that processes sound waves, as we will soon see in PROGRAM 2.1.4.

Weighted superposition

Since we represent sound waves by arrays of numbers that represent their values at the same sample points, superposition is simple to implement: we add together their sample values at each sample point to produce the combined result. For greater control, we also specify a relative weight for each of the two waves to be superposed, with the following function:

def superpose(a, b, aWeight, bWeight):
    c = stdarray.create1D(len(a), 0.0)
    for i in range(len(a)):
        c[i] = aWeight*a[i] + bWeight*b[i]
    return c

(This code assumes that a[] and b[] are of the same length.) For example, if we have a sound represented by an array a[] that we want to have three times the effect of the sound represented by an array b[], we would call superpose(a, b, 0.75, 0.25). The figure at the top of the next page shows the use of two calls on this function to add harmonics to a tone (we superpose the harmonics, then superpose the result with the original tone, which has the effect of giving the original tone twice the weight of each harmonic). As long as the weights are positive and sum to 1, superpose() preserves our convention of keeping the values of all waves between −1 and +1.

PROGRAM 2.1.4 (playthattunedeluxe.py) is an implementation that applies these concepts to produce a more realistic sound than that produced by PROGRAM 1.5.8. To do so, it makes use of functions to divide the computation into four parts:

  • Given a frequency and duration, create a pure tone.
  • Given two sound waves and relative weights, superpose them.
  • Given a pitch and duration, create a note with harmonics.
  • Read and play a sequence of pitch/duration pairs from standard input.

Program 2.1.4 Play that tune (revisited) (playthattunedeluxe.py)

These tasks are all amenable to implementation as functions, which depend on one another. Each function is well defined and straightforward to implement. All of them (and stdaudio) represent sound as a series of discrete values kept in an array, corresponding to sampling a sound wave at 44,100 samples per second.

Up to this point, our use of functions has been somewhat of a notational convenience. For example, the control flow in PROGRAM 2.1.1, PROGRAM 2.1.2, and PROGRAM 2.1.3 is simple—each function is called in just one place in the code. By contrast, PROGRAM 2.1.4 is a convincing example of the effectiveness of defining functions to organize a computation because each function is called multiple times. For example, as illustrated in the figure below, the function note() calls the function tone() three times and the function superpose() twice. Without functions, we would need multiple copies of the code in tone() and superpose(); with functions, we can deal directly with concepts close to the application. Like loops, functions have a simple but profound effect: one sequence of statements (those in the function definition) is executed multiple times during the execution of our program—once for each time the function is called in the control flow in the global code.

FUNCTIONS ARE IMPORTANT BECAUSE THEY GIVE us the ability to extend the Python language within a program. Having implemented and debugged functions such as harmonic(), pdf(), cdf(), mean(), exchange(), shuffle(), isPrime(), superpose(), tone(), and note(), we can use them almost as if they were built into Python. The flexibility to do so opens up a whole new world of programming. Before, you were safe in thinking about a Python program as a sequence of statements. Now you need to think of a Python program as a set of functions that can call one another. The statement-to-statement control flow to which you have been accustomed is still present within functions, but programs have a higher-level control flow defined by function calls and returns. This ability enables you to think in terms of operations called for by the application, not just the operations that are built into Python.

Whenever you can clearly separate tasks within a computation, you should do so. The examples in this section (and the programs throughout the rest of the book) clearly illustrate the benefits of adhering to this maxim. With functions, we can

  • Divide a long sequence of statements into independent parts.
  • Reuse code without having to copy it.
  • Work with higher-level concepts (such as sound waves).

This point of view leads to code that is easier to understand, maintain, and debug compared to a long program composed solely of Python assignment, conditional, and loop statements. In the next section, we discuss the idea of using functions defined in other files, which again takes us to another level of programming.

  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020