Difference Tone Training
a demonstration adapted from Titchener's Experimental Psychology (1901-1905), vol. I, part 1, pp. 39-46
This demonstration recreates an example of introspective training from E.B. Titchener's laboratory manual of 1901-1905. The purpose is to prompt thought about the prospects of introspective training as a means of improving the quality of introspective reports about conscious experience. The demonstration requires speakers or headphones, and a high-speed internet connection is recommended.
Titchener and most of his contemporaries (circa 1900) regarded introspection as the principal method of psychology. Nevertheless, they felt that the introspective reports of untrained subjects are prone to error, even when the reports are restricted to current or very recently past conscious experience. (I share this view: Schwitzgebel & Gordon, 2000; Schwitzgebel 2002a&b, 2004, in preparation-c.) They consequently recommended extensive training in introspective procedures. Wilhelm Wundt, Titchener's teacher and founder of the introspectionist school of psychology, is reputed not to have admitted introspective data from observers with fewer than 10,000 trials of experience in introspective report (Boring 1953).
To assess the merits of classical introspective training techniques, it would be desirable to train people as the introspective psychologists recommended. Titchener's vast, four part laboratory manual (1901-1905) details the methods. While it is not practical to attempt all the procedures Titchener describes, we may easily run through one example. Doing so will help reveal both the promise and the problems of classical introspective training.
The procedure here will approximately follow the procedure described by Titchener (1901-1905), vol. I, part 1, pp. 39-46. The reader will be trained to discern difference tones. Although I will raise some theoretical issues about the nature of introspection and introspective training, this is not intended to be a discursive essay on those topics (or any other). Rather, it is a practical demonstration and an invitation to reflection. Questions are raised but not answered. Readers looking for extended theoretical discussion are directed to the essays listed at the end of this article, especially "Introspective Training Apprehensively Defended" (Schwitzgebel 2004).
What a Difference Tone Is
When two pure tones are played together, it sometimes sounds as though a third tone is also present. For example, a person simultaneously hearing two loud tones of 800 and 1000 hertz may also report hearing another, quieter tone at about the pitch one would expect from a 200 hz stimulus. More generally, if two generating tones (or primary tones) of frequency U (for the upper tone) and L (for the lower tone) are played together, people will under some conditions report hearing a difference tone (or first difference tone) that sounds like a tone presented at frequency U - L, despite the fact that waveform analysis may show no sound wave at that frequency.
Similarly, people sometimes report hearing, in addition to or instead of the first difference tone, a second difference tone at approximately the pitch one would expect from a stimulus of frequency 2L - U (600 hz, if the generating tones are 800 and1000 hz). Titchener also describes a third difference tone at 3L - U and a summation tone at U + L. Other investigators report tones at 3L - 2U, 4L - 3U, 2U - 2L, 3U - 3L, and other frequencies. As a class these are known as combination tones. In stimuli with prominent overtones (overtones have a frequency that is an integer multiple of the fundamental frequency of the stimulus, such as 2U, 3U, 4U, etc.), people sometimes report hearing combination tones arising from the overtones.
It is generally thought that combination tones are heard as a result of "non-linearities" (failures of simple additivity) in the ear's physical response to sound.
Adjusting Your Speaker Volume
To prepare yourself to hear difference tones, please increase your speaker or headphone volume to the maximum comfortable setting while listening to the following stimulus: volume test. (If this stimulus is taking a long time to download, cancel the download and follow the directions in the next section.)
The stimulus should be somewhat unpleasant but not painfully loud. Listen to it several times to be sure that the volume is as high as possible without causing pain. I have noticed in previous versions of this demonstration that people typically do not initially turn the volume high enough.
If your speakers produce distortion at the required volume level, decrease the volume and move one ear close to one speaker, or get headphones, then perform the volume test again. Overdriven amplifiers produce tones much like the combination tones produced by the ear. Perhaps it does not matter, from an introspective perspective, whether the tones of interest are produced by the amplifiers or by the ear -- since ear and air are equally outside the brain -- but it seems preferable to avoid this complication.
Directions for People with Slow Internet Connections
If the test stimulus took more than a few seconds to download, you will need to download all the stimuli together as a package, since it is important to access them quickly during the training and tests. Download this zipfile into an empty temporary directory, then unzip it. Instead of clicking on the links in this document, open the files associated with the link. (So, for example, if the link is "http:// ... [whatever] .../T01.wav", play the sound file "T01.wav" from your temporary directory.) In Internet Explorer or Mozilla Firefox, when you move the cursor over a link (without clicking), you can see the filename near the bottom of your screen, if the "status bar" is not hidden. Unfortunately, the zipfile is large (30 MB) and may take several hours to download.
Optional Pre-Test and Post-Test
If you wish to take it, an optional pre- and post-test is available in Appendix A below. Titchener's original procedure does not involve such a pre- and post-test.
Titchener's Training, Part One
We will now follow approximately the procedure described by Titchener (1901-1905), vol. I, part 1, pp. 39-46. The most noteworthy differences between the present procedure and Titchener's are: the optional pre-test and post-test; a more consistent reliance on the multiple-choice format; the addition of pauses for methodological reflection; and (of course) the presentation of the tones through the computer. Appendix B contains a further discussion of some of the procedural details.
Throughout the procedure, note the features of your auditory experience as much as is practicable without detracting too much from the central task of attempting to discern the difference tones. For example: Do the tones seem to change pitch or loudness in the course of presentation? Do you hear more than one combination tone at a time? Does the loudness, or the very presence, of a combination tone vary with attention? Do the properties of the tone vary depending on the other tones that are heard or attended to? Is there a “fusion” of tones more for some intervals than for others? These were some of the questions that interested Titchener and other early introspective psychologists, and Titchener generally encouraged his introspective observers to report such details. (You may forget to reflect on these matters once you are absorbed in the training, but Titchener collected from his students written introspective reports that were expected to contain at least a few such details.)
Now listen to the two tones together. In this sample, the g3 plays for 1 second, then b3 joins it for five seconds.
You should hear at least one tone distinctly lower than either of the two generating tones. The first difference tone, on which this training will focus, is two octaves below the g3 at g1. (From here forward, the term 'difference tone' should be taken to mean 'first difference tone' unless otherwise specified.) It may sound a bit like background hum in your speakers of the sort that you are accustomed to ignore. Some people, especially those listening at lower speaker volumes, describe this difference tone as considerably quieter than either of the generating tones; others find it every bit as loud and salient as the generating tones or even more so. Since the tone is a perfect two octaves below the lower generator, it may also harmoniously "fuse" with that generator to some extent. Titchener recommends that you listen to the stimulus several times, until you are "entirely satisfied with [your] introspections" (1901-1905, vol. I, part 1, p. 41).
If you did not hear the difference tone, try either increasing the volume on your speakers, switching to headphones, or reducing background noise. If you still do not hear the difference tone, you might try listening to g1 at a lower volume, so that its loudness more closely matches that of the expected difference tone; or you might listen to the stimulus first at a low volume (which may make the difference tone inaudible), then at a higher volume, listening for the addition of a lower pitched tone. If you still feel unable to hear the difference tone, try listening to the remedial stimuli below (not adapted from Titchener), which most people (but not all) find to contain obvious difference tones. If you heard the difference tone in g3+b3 and you wish to abide closely by Titchener's presentation, you should skip the remedial tones for now, though of course you may return to them after the training procedure is complete.
Optional Interlude: Remedial Stimuli
Remedial difference tone sample 1. Listen to these two harmonica notes played separately: 7draw, 8draw; then together: 78draw. Can you hear a faint tone at this pitch: diff? (This example and the stimuli are from Pat Missin's website, http://www.patmissin.com/ffaq/q26.html , accessed December 2002.)
Remedial difference tone sample 2. Here a tone of 1000 hz is played for two seconds, then it is joined by another tone of 1000 hz that slowly rises to 2000 hz. The difference tone is a low note that slowly rises to the pitch of the 1000 hz tone: rising. The second difference tone may also be fairly salient as a tone declining in pitch about two-thirds of the way through the stimulus. (You may also hear another declining tone just before the second difference tone becomes salient, at 3L - 2U, and you may notice a few fluttering "beats" as the rising tone first joins the steady, 1000 hz tone.)
Remedial difference tone sample 3. Here g3 and b3 are played together, starting very quietly and rising in intensity to the normal level. You may hear the low difference tone g1 come in as the intensity increases: g3+b3amp. High tones, like a ringing in the ears, may also be salient near the end of the stimulus. (These may be due to aural harmonics, summation tones, and/or amplifier distortion.)
If you heard a difference tone in any of the remedial stimuli, but not in the g3+b3, please return to that interval again, now that you have a better sense of what you're looking for. If you still hear no difference tone, you may not hear any in the remainder of this demonstration.
Titchener's Training, Part Two
Hopefully, you heard a difference tone in g3+b3 above. Now listen again to the stimulus and then to 1A, 1B, and 1C. Determine which of these tones is identical to the difference tone. You may play the stimulus and the options several times if necessary. After you have settled on a choice, compare it with g1 to verify. (Appendix C also contains the answer, but looking there risks spoiling the training by revealing the correct responses to questions you have not yet tried.) (If your browser brings up separate windows for each sound file, you may wish to place the windows next to each other so that you can move quickly between the sounds presented, thereby minimizing tonal memory requirements. Close the windows once you move on to the next part of the training.)
If you have chosen incorrectly, listen to the tones and intervals again until you feel that you can properly identify the pitch of the difference tone. Proceed to (2) either when you are confident that you can identify the pitch of the difference tone or when you judge that further effort would be of little use.
Throughout the training and post-test you may wish to take occasional breaks, to "cleanse the palate" as it were.
(2.) The second trial proceeds much as the first. This time, play the tones g3 and d4 (the fourth D above middle-C), first separately, then together, listening for the difference tone. In this case, the difference tone is one octave below the lower generating tone, at g2. (This also happens to be the pitch of the second difference tone.)
As before, if you chose incorrectly, listen to the stimulus and choices again until you feel you can properly discern the difference tone and identify its pitch. If you are struggling, you may wish to go back to the optional section containing the remedial stimuli, if you have not already listened to those stimuli.
Interlude for Methodological Reflection
Let's pause to consider a few methodological questions before proceeding, as a way into thinking about some of the core issues around the idea of "introspective training".
(i.) What were you attending to in these experiments? Were you attending to the world -- that is, to the sounds (assuming sounds are the sort of thing that exist in the world)? Or were you attending to your ("inner" or "conscious") experience? Titchener regarded introspection as a kind of attention to experience. Does that seem to you to be what is going on in these samples? Or does it seem to you that attention passes through the experience, as it were, to lodge on the outward object that is being perceived, the sounds coming from your computer? Philosophers such as Fred Dretske (1995) and Michael Tye (2003) have argued that sensory experience cannot be an object of attention, that any attempt to attend to sensory experience necessarily results, instead, in attention to the object sensed. While I concur with Titchener on this point and disagree with Dretske and Tye (see Schwitzgebel in preparation-b), I concede that in cases of this sort it can seem as though attempts to attend to auditory experience collapse into attention to sounds in one's environment.
Introspective reflection on experience differs from sensory attention to one's environment at least in its goals, and in what would constitute error. Suppose a tone is present in your experience due to tinnitus or "ringing in the ears", rather than as a result of external stimulus. Suppose also that you do not realize that tinnitus is the cause. If your aim in reporting the tone is introspective, you have made no mistake; if your aim is sensory -- to describe external sounds or to describe what would be heard by a normal perceiver in the same external conditions -- reporting the tone is a mistake. In so far as it is possible to distinguish cognitive processes by appeal to their goals, then, introspective and sensory processes differ. I believe attention-consuming processes are better individuated by appeal to goals than by attempting to locate a single, particular (and in some cases chimerical) "object" of attention, but this is contentious and we cannot enter the issue farther here.
(ii.) Suppose that in either (1) or (2) you guessed incorrectly but then became convinced of the correct answer after further listening. Is it possible, in these cases, that you made an introspective error about your own experience? A number of philosophers (e.g., Shoemaker 1963; Rorty 1970) have rejected the possibility of genuine introspective error. Infallibilist views of this sort can be intuitively appealing. How could one possibly be wrong, it is sometimes rhetorically asked, about one's own ongoing sensory experience -- about whether one is in pain or not, is having a visual experience of "redness" or not, is experiencing a tightness in the chest or not? Reflection on examples of this sort have led some to the extreme position of thinking that one can never -- or only very rarely and in unusual circumstances -- be in error about even the smallest aspect of one's currently ongoing sensory experience. I find such a view implausible, and I invite you to consider its plausibility for yourself in the present case.
To be fair to infallibilism, however, let me point out a couple of moves possible in its defense. The first is to point out that error may arise from failure of memory: Although you may apprehend your tonal experience perfectly as it occurs, by the time it comes to choosing a reference tone a split second later, your memory may have faded enough to create an error. Another strategy is to deny that the purported error is "error" at all: The experience is in each case exactly as described, shifting in line with your shifting judgments about it. People will differ in their sense of whether such responses can fully explain people's apparent mistakes in this demonstration.
It is not unreasonable, of course, to suppose that a developing knowledge of difference tones may affect one's auditory experience to some extent. One may accept the view that auditory experience prior to training is different than auditory experience after training without necessarily supposing (as in the previous paragraph) that our experience always marches in step with our judgments about it.
(iii.) Finally, let me share with you an unresolved confusion I have about my own experience. Listening to the second stimulus, g3+d4, I find myself feeling that the difference tone is a bit higher than the "correct" g2, although the g2 is a closer match than the next higher pitched option. This impression has survived repeated presentations of the stimuli. At first I was tempted to think that the presence of the higher tones might create a context in which the auditory sensation of the g2 difference tone is higher in pitch than a g2 sounded alone. But of course the "g2" difference tone does not exist apart from the generating tones. It is not some independent thing to be affected by them. Perhaps, then, rather than saying that the pitch of the difference tone is raised by the presence of the other tones, it would be better simply to say that the pitch of the difference tone is a bit higher than that of g2? But consider this sound: g3+d4(+g2), which is the g3 and the d4, with a bit of g2 added as a third generating tone. It doesn't seem as though my auditory experience contains both the true g2 and the difference tone, heard separately, or that it contains any roughness of the sort one would expect listening to two tones of slightly different frequency played together. Maybe the g2 is shifted slightly upward in my experience to match the difference tone or the difference tone is shifted slightly downward to the g2? Or maybe, despite the repeated attempts and my subjective feeling of confidence, my introspective judgment about that difference tone is simply erroneous?
Titchener's Training, Part Three
Below you will find the intervals from the minor 6th through the octave, along with the associated (first) difference tones and three test tones for verification. Proceed as before, listening several times to the stimulus and options if you don't find the answer immediately obvious. Don't be surprised, however, if some of the difference tones seem to be inaudible.
Feel free to adjust the volume on your computer, but bear in mind that any adjustment done now will invalidate the post-test unless you then restore the volume to its original setting.
More Methodological Reflections
(iv.) If you can't discern a difference tone in one of the stimuli, does it follow that there is no such tone in your experience? At least three possibilities beckon: (a.) The difference tone is genuinely a distinct part of your experience, but you have not (yet) succeeded in discerning it. (b.) The difference tone contributes in some way to your overall experience, but not in such a way as to be separately discernible, even in principle. (c.) The difference tone is is no respect part of your experience. I see no straightforward way to distinguish between these three options. This creates a serious difficulty for introspective methodologies. How are we to know when to accept a denial of an aspect of experience?
(v.) Even if you guess correctly on a particular stimulus, it doesn't follow that the difference tone is a separately discernible part of your experience. You may pick out the right answer simply because that tone somehow accords with the experience of the stimulus in a way the others do not, or you might be gaining a sense of the pattern of rising pitch in the difference tones and thus know where to expect one. You may feel you have a good sense of the basis of your selection from among the alternatives, but should we trust people's feelings on such matters?
(vi.) If errors and indecision are due to failures of tonal memory over the brief interval between hearing the stimulus and hearing the response options, rather than difficulty in apprehending the auditory experience itself, it seems natural to suppose that similar failures of tonal memory would emerge in the task of picking one tone out of a chord of three. To explore that possibility, try the following test. A tone will be presented for one second. Then it will be joined by both a higher and a lower tone for five seconds. The lower tone will be 0.5% of the intensity of the other two (23 decibels less intense), and its pitch will not be in the vicinity of either the first or the second difference tones of the two higher tones. Try to determine the pitch of the lower tone. Chord. ChA, ChB, ChC. Feel free to play the chord several times, listening for something in the range of the three options. Key. If you find this test easier than the difference tone tests, tonal memory may not be the issue.
(vii.) Suppose you reliably guess the "wrong" tone for one of the intervals, and after repeated verifications, you still feel that that is the genuine pitch of the difference tone for you. Can you convince yourself that you could be mistaken in this? Or does that idea just seem preposterous?
Titchener's Training, Part Four
Now try the intervals from the tri-tone down to the semitone (excluding the major third, which was presented in (1)). Do not listen to the key until you have chosen from among the three options. The difference tones in this series are lower and farther apart, so the options will be a major whole tone removed from each other, rather than a semitone as in the previous series.
Titchener's Training, Part Five
So far, the pitch and the order of presentation of the stimuli has followed Titchener. At this point in the text, however, Titchener recommends that his introspectors-in-training practice hearing the difference tone when the sound intensity is slight, when the generating tones differ substantially in intensity, and when their duration is short. He does not specify any particular tone intervals, sound intensities, or durations. We will practice briefly with three stimuli in each category, varying the frequency of the generating tones within Titchener's scale. In these exercises, as in the preceding ones, you may wish to listen to the stimulus more than once before guessing.
Here are three stimuli in which the intensity of the higher of the generating tones is 30% of the intensity of the lower generator. As in previous practice, if you guess incorrectly in these exercises, listen to the stimuli and alternatives again.
Now try three stimuli in which both generating tones are at 30% of the standard intensity. If you're having trouble hearing a difference tone at first, try bringing the speaker a little closer to your ears. Then resume normal position and assess whether you can still hear it.
Here are three stimuli in which the duration of the generating tones is short. You will probably have to listen to the stimuli several times.
Titchener continues training with the second difference tone, the third difference tone, and the summation tone, but rather than introducing an extensive training series for any of these, I will provide three examples of the second difference tone, then conclude the demonstration.
(22.) g3+f-sharp4. 22A, 22B, 22C. Key. This is the same stimulus as in (6). Some people find the low second difference tone more salient than the first difference tone, which is similar in pitch to the lower generating tone. Titchener again recommends that the experiment be repeated "until [the observer] is satisfied with his introspections" (1901-1905, vol. I, part 1, p. 43). Can you hear both the first and the second difference tones? If so, can you hear both at once, or only one at a time?
If you feel that you can hear the second difference tones in any of the stimuli, consider this: Are second difference tones qualitatively any different from first difference tones, apart from differences in pitch?
For fun, listen to this song. Can you hear this tune in the difference tones? You may also find it interesting to determine how many combination tones you can hear in what was presented above as the second remedial stimulus, a 1000 hz tone joined by another tone rising from 1000 to 2000 hz: rising. (The audibility of combination tones in this range is discussed in Plomp 1965, 1976.)
If you took the optional pre-test, return now to Appendix A and follow the instructions for the post-test. Even if you didn't take the pre-test, you may want to take the post-test if you wish to hear some challenging stimuli.
Let's suppose that you felt more competent in discerning combination tones at the end of the training than at the beginning and furthermore that this feeling of competence is well justified. The question then arises: What exactly have you learned?
Titchener would say that you have improved your ability to introspect your auditory experience. Perhaps it is also correct to say that you have been musically trained, trained perceptually, to respond in certain ways to environmental stimuli. It would seem that there should be a difference between these two types of training: In introspective training, you learn to discern features of your own experience, while in perceptual training you learn to discern features of the environment. Yet here, the two types of training may seem to collapse together, as noted in reflection (ii) above. Were you attending to the environment or to your ("inner") auditory experiences in the training and test? Psychoacousticians and music teachers sometimes train people in "analytic listening" -- that is, in picking individual tones out of a complex stimulus. They don't typically frame such a task as introspective, but one may wonder: Is the training procedure above really different in kind from that? Is "introspecting auditory experience" any different from listening carefully?
In reflection (ii) I mentioned the different implicit goals of introspection and perception, but this may seem a tepid way of distinguishing two processes: If I kick a can to get it out of the way or if I execute the exact same motion to demonstrate proper kicking technique, in a sense I have engaged in two different activities, but in another quite robust sense I have done the same thing in the two cases, only with different ends in mind. Perhaps the Titchenerian student of introspection and the person aiming to improve her analytical listening skills similarly engage in identical processes, only with different ends in mind? I think there must be substantial truth in that thought. It seems too easy to slip back and forth between one way of thinking about the training procedure and another, or even simply to do the tasks with no specific framing goal of either sort decisively in mind.
This is not as bad for Titchener as it might seem. We can still distinguish introspection in the robust sense -- introspection as involving a deliberate effort to reach a judgment explicitly about the nature of one's sensory experience, regardless of how things stand in the world beyond -- from ordinary perception unencumbered by such introspective goals. The existence of ambiguous or in-betweenish cases in which its not clear whether one is introspecting does not undermine the existence of at least occasional instances of introspection in the robust sense. To the extent you drifted from the robust, explicit consideration of your experience, lapsing into something more like making ordinary judgments about the world, you were less than ideal as a Titchenerian subject. Titchener repeatedly cautions against this kind of lapse (calling it "stimulus error" or "R-error"). Careful listening is part of what you are doing as you go through these stimuli -- of course! -- but when you are at your Titchenerian best, robustly introspecting, reaching explicit and deliberate conclusions specifically about your experience, listening is only a part of your activity.
Even if you completely neglected to approach the training tasks in a robustly introspective way, you may still have learned something useful for Titchenerian introspection. Imagine a musically and introspectively untrained subject entering Titchener's laboratory and confronted with a complex sound -- an interval of a third struck upon a piano, say. Asked to describe what her auditory experience is like upon hearing that sound, that person may be limited to making a few very general remarks. Auditory experience is difficult to parse and to conceptualize. To be a really good Titchenerian subject, one needs a vocabulary for discussing aspects of one's experience and practice parsing things in those terms. Musical training, including training in discerning difference tones, provides a vocabulary that can be employed for explicitly introspective ends, even if the training is not itself explicitly introspective. This may be as valuable an aspect of the training procedure adapted above, for Titchener's purposes, as any practice you may have had in robustly Titchenerian introspection.
There were many reasons for the death of classical introspective training: Gestalt psychologists raised legitimate concerns and objections that were somewhat overplayed; the classical introspective psychology research program yielded few socially valuable results and bogged down in sterile debates; behaviorism and functional psychology offered exciting and seemingly much more useful research prospects. But also, most people found introspective training techniques of this sort tiresome, as has been emphasized by Titchener's student and the great historian of psychology, (the unfortunately but in this context aptly named) E.G. Boring (1953). Maybe an hour or an hour and a half with this demonstration is okay, but can you imagine enduring 10,000 introspective trials? Still, if we wish seriously to study conscious experience, it would seem that some researchers, at least, should develop skill at parsing particular aspects of their experience and recognizing features of it -- and that this skill should be recognized as a kind of introspective acumen. The time is ripe for a limited resuscitation of the techniques of classical introspective training.
For their helpful comments and discussion, I would like to thank Jörg Buchholz, Kirk Gable, Mike Gordon, David Kirchner, Felipe Leon, Jessica Ludescher, Pat Missin, Larry Rosenblum, Frank Russo, Josh Rust, Colleen Ryan, Sandy Ryan, and Jeff Yim.
Appendix A: Optional Pre- and Post-Test
To help you assess whether you have learned anything from the training procedure, I have provided a short optional pre-test and post-test. The questions on this test are intended to be difficult to answer without training, so do not be discouraged if you get them wrong. Also, feel free at any time to skip forward to the training section.
Here is a sample of the pre-test procedure. First, listen to this Sample Test Stimulus twice. Then decide this: Did you hear in the test stimulus a tone of approximately this pitch: Sample Option A? You may listen to this option twice, if necessary. Did you hear a tone of approximately this pitch: Sample Option B? You may also listen to this option twice if necessary. In answering these questions, do not go back and listen to the test stimulus again. (In this example, you should have answered "no" to both questions.)
All the test stimuli consist of one tone played for one second, joined by a second, higher tone for five seconds. The options will always tones be lower in pitch than either of the two primary tones in the stimulus. If you hear those tones at all in the test stimulus, they will likely seem to be much quieter than the two primary tones.
You may hear in the test stimulus a tone at either of the two options, at both, or at neither. Do not guess randomly. Since you will only be listening to three test stimuli in the pretest, a couple of lucky guesses could lead to a false impression of competence. Most untrained listeners will not be able reliably to discern a lower tone in any of the test stimuli.
To start the pretest, choose one of the test stimuli from Group I below, listening for a quiet tone lower in pitch than either of the two primary tones. Play the chosen test stimulus twice, then listen to the two associated options without returning to the test stimulus.
Group I (choose only one of the three test stimuli):
Test Stimulus 1. Try not to be too distracted by the fluttering "beats". Did you hear a tone of approximately the pitch of Option 1A? Option 1B? (Option 1A is very low, and you may not hear it at low or intermediate speaker volumes.)
Once you have decided whether you heard a tone at Option A, Option B, both, or neither, listen to whichever of the following "keys" matches the test stimulus you chose: Key for Test Stimulus 1; Key for Test Stimulus 2; Key for Test Stimulus 3. The key plays the "right" answer. If a moderately practiced ear would discern one of the two option tones in the test stimulus, the key will play that tone. If it would discern both tones, the key will play both; if neither, the key will be silent. Please note now which of the three stimuli you chose and whether you were successfully able to discern the lower tone or tones, if any. Do not listen to the other two test stimuli in Group I: You will be asked to listen to them as part of the post-test.
Follow the same procedure for the next two groups of test stimuli. Chose one test stimulus from each group, listen to it twice, then quickly play the options and determine whether you heard a tone at one, both, or neither of the two pitches. Finally, listen to the key. Bear in mind that you may hear a combination tone that is not among the two options (in which case the correct answer is "neither".)
Group II (choose only one):
Group III (choose only one):
A few people, even without any specific training in hearing difference tones, find the pre-test easy. If you are among those people, try the following more challenging stimuli. If you found the pre-test challenging or impossible, skip these stimuli. You may listen to them after the training if you wish.
I have not constructed a more challenging post-test, but you may return to these three stimuli after the training to determine whether you feel any more comfortable with them.
Since the perceived loudness of difference tones varies substantially with relatively small changes in the intensity of the generating tones, it is important that the stimuli of the post-test not be more intense than those of the pre-test. If you turned your speakers up during training, turn them back down. If you are in the habit of pressing the headphones against your ears to hear better, drop that habit if it was not already in place during the pre-test. Improved performance on the post-test should not be attributable to increased stimulus intensity.
If you have picked up a habit such as humming, you may wish to drop that habit. The same might apply to more "internal" habits, such as silently maintaining an auditory image of the difference tone (hearing it as one might hear a tune "in one's head") or attempting consciously to determine where a difference tone may be expected -- although it is difficult to draw a line between cognitive habits acquired during training that should be regarded as legitimate features of introspective expertise and those that should be regarded as non-introspective crutches.
Stimuli will vary considerably in pitch and interval, and the keys may be selected from any of several possible combination tones, to disrupt simple pattern recognition strategies.
The same stimuli are presented below that were presented in the pre-test. Listen first to the stimuli that you did not choose for the pre-test. As in the pre-test, listen to each stimulus only twice and do not listen again to the stimulus while attempting to answer the question. You may also listen to the options twice if you wish. Recall that one option may key, or both or neither. Do not play the key until you have decided whether you hear a combination tone at either, both, or none of the frequencies. As in the pre-test, do not guess randomly. You may listen to the stimuli you selected in the pre-test after you have listened to all the other stimuli.
If you wish, you may go back and listen again to any tests you failed. Can you pick out the combination tones now that you know what they are?
You may want to try also the following more challenging stimuli.
Appendix B: Procedural Details
Standard, contemporary, psychophysical procedures cannot be followed in this presentation. It is not, for example, possible to control properly for background noise, speaker distortion, or sound pressure level. Despite these shortcomings the training can produce interesting results. The main purpose, recall, is not to establish any particular psychophysical claim but rather to prompt thought about the nature and prospects of introspective training.
All the tones presented are simple sine waves, without overtones, generated by Cool Edit 2000 Lite, version 1.1. Durations are accurate to .03 seconds. Titchener states that combination tones are easier to hear without the distraction of overtones (1901-1905, vol. 1, part 2, p. 72), and recommended Quincke's tubes in part for this reason.
The training procedure follows Titchener in using a just intonation scale based in C-major (with the A above middle C at 440 hertz) for the generating tones, rather than the musically standard equal temperament scale. In just intonation, the ratios of the frequencies of the different notes are relatively simple fractions, but the transposition of a song into a different key is not possible without retuning. Titchener suggests that combination tones are easier to hear the more closely the generators approach just intonation vibration ratios (ibid.). Titchener (1901-1905), vol. I, part 1, p. 32, lists the just intonations for the natural tones. To resolve ambiguities regarding the intonation of sharps and flats, I referred to Rossing, Moore, and Wheeler (2002).
In Titchener's notation, middle C is 'c1', the C above that is 'c2', and so forth (omitting the superscript). One octave below middle C is lower-case 'c'; one octave below that, capital 'C'; then 'C1' and so forth (omitting the subscript). Notes other than C take the format of the C below them (thus, the A above middle C is 'a1' and the A below it is 'a'). Titchener's scale and notation are described in vol. I, part 1, pp. 32-33. (Thus, Titchener's c1 is approximately the standard C4 in contemporary equal temperament notations.)
Much of the training uses g3 as the lower generator. Since I abide by Titchener's just intonation scale for C-major, the tri-tone is a bit sharp (in (8)) and the major whole tone a bit flat (in (11)) relative to the standard just intonation vibration ratios for those intervals.
When a multiple choice is offered, the tones presented are either separated by exactly a semitone (for tones over 900 hz) or exactly a major whole tone (for tones below 900 hz), in just intonation, although that typically takes the tones slightly off Titchener's scale. (Note that in just intonation, two semitones (16/15 ratio) do not constitute a major whole tone (9/8 ratio), nor do two major whole tones constitute a major 3rd (5/4 ratio).) If the pitch of the predicted difference tone does not precisely match that of the tones in Titchener's scale, the exact predicted difference tone is nonetheless presented. The predicted difference tone is equally likely to be the highest, the lowest, or the middle tone (except when confusion might result from approximating the pitch of another salient combination tone), and the order of the tones is randomized.
Appendix C: Key
1. g3 (1584 hz) + b3 (1980 hz). Key: B (396 hz). Distractors: A (445.5 hz), C (352 hz).
2. g3 (1584) + d4 (2376). Key: C (792). Distractors: A (704), B (891).
3. g3 (1584) + e-flat4 (2534.4). Key: B (950.4). Distractors: A (835.31), C (891).
4. g3 (1584) + e4 (2640). Key: A (1056). Distractors: B (1126.4), C (990).
5. g3 (1584) + f4 (2816). Key: C (1232). Distractors: A (1155), B (1314.13).
6. g3 (1584) + f-sharp4 (2970). Key: C (1386). Distractors: A (1299.38), B (1218.16).
Chord. g3 (1584) + f3 (2816) + e2 (660) 0.5% intensity. Key: C (660). Distractors: A (586.67), B (742.5).
8. g3 (1584) + d-flat4 (2252.8). Key: C (668.8). Distractors: A (594.49), B (752.4). (2227.5 is the true tri-tone.)
9. g3 (1584) + c4 (2112). Key: A (528). Distractors: B (469.33), C (594).
10. g3 (1584) + b-flat3 (1900.8). Key: A (316.8). Distractors: B (400.95), C (356.4).
11. g3 (1584) + a3 (1760). Key: A (176). Distractors: B (198), C (156.44). (1782 is the true major 2nd.)
12. g3 (1584) + a-flat3 (1689.6). Key: B (105.6). Distractors: A (93.87), C (83.44).
13. e3 (1320) + b3 (1980) 30% intensity. Key: B (660). Distractors: A (835.31), C (742.5).
14. a4 (3520) + b4 (3960) 30%. Key: A (440). Distractors: B (391.11), C (495).
15. f3 (1408) + a-flat3 (1689.6) 30%. Key: B (281.6). Distractors: A (250.31), C (316.8).
16. f3 (1408) 30% + a3 (1760) 30%. Key: C (352). Distractors: A (312.89), B (396).
17. c4 (2112) 30% + f-sharp4 (2970) 30%. Key: C (858). Distractors: A (965.25), B (1085.91).
18. d3 (1188) 30% + b3 (1980) 30%. Key: C (792). Distractors: A (891), B (704).
19. d4 (2376) + f4 (2816). Key: B (440). Distractors: A (391.11), C (347.65).
20. e3 (1320) + c4 (2112). Key: C (792). Distractors: A (891), B (1002.38).
21. a-flat3 (1689.6) + f4 (2816). Key: A (1126.4). Distractors: B (1056), C (1201.49).
22. g3 (1584) + f-sharp4 (2970). Key: B (198). Distractors: A (176), C (222.75).
23. d4 (2376) + c5 (4224). Key: A (528). Distractors: B (469.33), C (417.19). Key2: E (1848). Distractors: D (1971.2), F (2102.61).
24. c4 (2112) + a-flat4 (3379.2). Key: C (844.8). Distractors: A (950.4), B (1069.2). Key2: E (1267.2). Distractors: D (1188), F (1113.75).
Test 1. d2 (594) + e-flat2 (633.6). Key: A (39.6) (1st difference tone). Distractor: B (176). (2nd difference tone: 554.4.)
Test 2. e2 (660) + c4 (2112). Neither response keys. A (220), B (352).
Test 3. e2 (660) + d3 (1188). Both responses key. A (132) (2nd difference tone), B (528) (1st difference tone).
Test 4. f2 (704) + a3 (1760). Key: A (352) (3rd difference tone (3L - U)). Distractor: B (528). (1st difference tone: 1056.)
Test 5. a2 (880) + c4 (2112). Key: A (528) (3rd difference tone). Distractor: B (66). (1st difference tone: 1232.)
Test 6. d3 (1320) + e3 (1408). Neither response keys. A (594), B (330). (1st difference tone: 88 (this tone may be quite salient); 2nd difference tone: 1232).
Test 7. a3 (1760) + g4 (3168) 30%. Key: B (352) (2nd difference tone). Distractor: A (528). (1st difference tone: 1408.)
Test 8. b3 (1980) + a4 (3520). Key: A (440) (2nd difference tone). Distractor: B (297). (1st difference tone: 1540.)
Test 9. g4 (3168) + f5 (5632) 50%. Key: B (704) (2nd difference tone). Distractor: A (330). (1st difference tone: 2464.)
Test 10. c3 (1056) + b3 (1980) 8%. Both responses key. A (924) (1st difference tone). B (132) (2nd difference tone).
Test 11. c2 (528) + f3 (1408). Key: B (880) (1st difference tone). Distractor: A (792).
Test 12. c1 (264) + e2 (660). Key: B (924) (summation tone). Distractor: A (1408). Some authors (not Titchener) believe that the summation tone is impossible to hear unless it is the product of amplifier distortion.
Note: For the test stimuli, the distractor is chosen so as not to be within a major 3rd of any combination tone of the form nU + mL, when n and m are integers from -5 to 5. (Exceptions are Test 11, where the distractor is a major 2nd lower than the key and a major 2nd higher than the fourth difference tone (4L - U), and Test 12, where due to the moderate pitch of the summation tone and the low frequency of the generators, it is impossible to avoid being within a major 3rd of some of the more obscure combinations, such as 3U - 2L.)
Here are the generator tones for the difference tone song (Scarborough Fair):
c3 + d3 (1056 + 1188) (major whole tone). Diff: c (132).
e2 + g2 (660 + 792) (minor 3rd). Diff: c (132).
g2 + b2 (792 + 990) (major 3rd). Diff: g (198).
c3 + e-flat3 (slightly flat) (1056 + 1254) (minor 3rd). Diff: g (198).
b2 + d3 (990 + 1188) (minor 3rd). Diff: g (198).
d2 + f-sharp2 (594 + 742.5) (major 3rd). Diff: d (148.5).
e-flat2 + g2 (633.6 + 792) (minor 3rd). Diff: e-flat (158.4).
g2 + b-flat2 (slightly flat) (792 + 940.5) (minor 3rd). Diff: d (148.5).
c2 + e2 (528 + 660) (major 3rd). Diff: c (132).
Stickney and Englert (1975) presents a similar example.
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