_______ 2. Thus Have I Heard PDF Print E-mail
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Saturday, 31 October 2009 10:38

2. Thus Have I Heard

 

Acoustic Observations

 

The following will present a few features of acoustic theory salient to this discussion. No attempt is made to cover this topic in detail and interested readers may consult the references listed in the bibliography for further reading.

 

We begin with a pure tone and demonstrate consonance inherent within the ‘non-linearity of the ear’, that is, our perceptions curve around a median range. The amplitude of a pure tone varies the perception of harmonics we hear in a sound.[1] The pitch of a tone appears to shift upwards as its loudness increases and more harmonics are observed.

 

For demonstration purposes a simple patch has been created to play pure sine tones. It plays these tones in relation to a “global_pitch_center” in cycles/second or Hz, that can be adjusted as required. Instead of using MIDI notes, ratios [converted to decimals] are sent to a “poly~” sound module.

 

Upon trying various pitch centers, harmonics can be heard prominently at the octave and fifth and perhaps even beyond [given some patience, ear training and the right combination of amplitude and frequency].

 

These harmonic overtones are related to the amplitude of the fundamental in their relative strength but moreover, in terms of their perceived pitch the relationship is constant  i.e. if the fundamental is perceived to be slightly higher in pitch due to an increase in amplitude, the harmonics become more noticeable and proportionately shift upwards. This property demonstrates a natural consonance inherent within the perception of a pure tone itself.

 

Theory has it, if we play two tones together we begin to hear difference and summation tones. This table appears in Physics and Sound of Music [Rigden, 84]:

 

Interval

f1

f2

f2-f1

2f2-f1

3f1-2f2

Octave

f1

2f1

f1

 

 

Fifth

f1

3/2f1

1/2f1

1/2f1

 

Fourth

f1

4/3f1

1/3f1

2/3f1

1/3 f1

Major third

f1

5/4f1

1/4f1

3/4f1

1/2 f1

Major sixth

f1

5/3f1

2/3f1

1/3f1

 

Major third

f1

6/5f1

1/5f1

4/5f1

3/5 f1

 

The reader is challenged to verify these relationships using the tools provided.

 

I can offer only this brief summary of combination tones as they inform our discussion of consonance. The reason I cannot provide more detail is that personally I have great difficulty in hearing them. This may be because my audio equipment is not of sufficient quality, or that I cannot generate the sound at a high enough volume to create the effect.

 

Notice that according to the chart, if we play a perfect fourth at high enough volume and frequency, we should hear a missing fundamental, or the ‘fourth’ two octaves below [1/3]. Do you hear this? I’m not sure that I do. It may be interference with the sound of my fan in the background, or the humming refrigerator, or the neighbour’s dog barking in the background that make me sometimes believe that I do.

 

Honestly, I don’t really care to. I don’t find pure sine tones appealing to listen to for long as they can become somewhat tedious. Of pedagogical value no doubt, but the hearing of these difference tones and fundamentals I would equate to being taken into a desert and being told to stand there until one can see ‘the mirage’. After standing in the dust and heat long enough, you start telling your guide that ‘yes, you do see them’, when in fact you’re simply too hot and thirsty to care anymore. 

 

In future demonstrations, we will employ examples using “groove~” and “sig~”, a simple form of granular synthesis [2] to produce complex tones. This alleviates aural fatigue, and with a bit of creativity we can toy with granular variables to introduce compositional processes operating at the micro-meso level [20-500 ms] for interest.[3] It is also important for another reason: complex tones are less subject to changes in pitch as a result of changes in sound level.[4]

 

The sine tone “micro_beep” module will always be available for use in testing of precise relationships when needed. Any of our tuning configurations will work with it too. Any sound producing module capable of accepting floats could be configured to work with this system.

 

Fire up the granular “etheramin” to play two pitches, holding one constant and allowing the other to slide freely. This little patch allows us to set the interval and the range that the free pitch can bend.

 

            max patch: etheramin

 

Setting the bend range to roughly (0.95 - 1.05) tune the unison and explore the space around it. Experiment and notice at what point you begin to hear two sounds. I’m not telling my answers. This is called the ear’s ‘limit of detection’.

 

Now try settings of (about 1.95, 2.05) and listen to the octave and the space around it. Close your eyes and tune the octave. Did you get the right answer? I sure as heck didn’t! Close, but not quite.[5] Seems to me there’s more ‘room’ at the top of the octave, no?[6]

 

Next, tune the octave below (.49 .51). A phenomena called beats, the result of the phase cancellation of waves guides our ears in being able to tune these intervals.[7] I found it much easier to be in tune this time. How about you?

 

There is much to learn here. We see that there are perceptual tastes and preferences with regard to the intonation of various interval sizes and relationships. The amount of sliding and bending room we have to play with influences our decisions. This little experiment helps one appreciate how difficult it is to be ‘in tune’ or play with perfect intonation.  It seems hopeless for acoustic instruments [as they are understood today] to be capable of producing these precise ratios.

 

Test this: tune an octave blind-folded and leave it untouched for a while. Now type in the number 2.0 and compare. Which can you tolerate longer? There is certainly an element of satisfaction in having access to this kind of precision.

 

Now for some fun: set the etheramin to (0.5, 2.0)

Do you draw the same picture that I do? [8]

 

I do not believe that limitations of traditional performance practice or the fluctuating tastes of the ear should be used as a criteria by which the usefulness of this method is to be evaluated. If our music doesn’t fit our instruments, then we are compelled to find a way to make the instruments fit the music. We can always retreat if it doesn’t work out. If it does, well, then there is the exciting possibility of something new, rich and strange developing that will alter the way humans perceive and interact with music.

 

The next example demonstrates the harmonic series, the series of integer multiples  above the fundamental, or global pitch center. The MIDI keyboard has been adapted so that D# above middle C or MIDI note 63 has been designated as 1, E as 2, F 3, etc. This arrangement has been chosen for reasons of symmetry that will become obvious later, and because the pattern of integers in the series falls nicely across the keyboard. Notice that  1, 4, 8, 16 the octaves (2 being the exception), are all black notes, the odd numbers 3 5 7 9 are white notes, the pattern changes with 10 12 14 on the white notes, and 11 and 13, the prime pair C# and D# respectively.[9] ‘D’ is left undefined, producing division by 0 and other nasty nonsense. In this patch, it triggers a ‘bang’.

 

Obviously, one would want to develop a more logical pattern for ease of navigation if one were to build an instrument based on this series.

 

When playing the adapted keyboard, the series of tones we hear historically has been called the ‘harmonic series’. I will be calling it the ‘dominant harmonic series’ or ‘P series’ for reasons that I will explain as we continue the argument. It can be notated like this:

 

 

 

The pitches we hear in this sequence are integer multiples of their fundamental. This can be expressed as:

 

            P = f*1, f*2, f*3, f*4 …

 

 

One can hear some absolutely beautiful elements of this series that we do not currently have in common practice music. Particularly the tones 7, 11, 13. Some elements are familiar – the perfectly intoned major triad between (3) 4 5 6, the major triad built on the dominant (3) found at (6) 9 12 15 [this time (3 4 5) * 3],  yet there are also some combinations that may sound strange to our ears, perhaps out of tune! Yet I have found that if one takes time to familiarize themselves with the sound of these relationships, one comes to a new appreciation. It feels to me that no matter what combination of integer relationships I choose, the sound feels as though it ‘floats within its own ocean’.[10]

 

Before we leave this all too brief exposure to harmonic  concord and purity, let us present a quick comparison: The relationship (3) 4 5 6 as previously mentioned creates a major triad. This can also be reduced to 1/1, 5/4, 3/2 or 1.0, 1.25, 1.5. Now listen to the triad produced by the equivalent relationship converted to equal temperament [ET]. The similarity cannot be denied, yet there is a quality of harshness involved in the ET version that is not present in the pure harmonic version.

 

Now if you’re ready for a shock, test the difference between 4 5 6 7 and its equal temperament version. Notice that the dominant seventh chord which we traditionally considered as being a dissonance that requires resolution, in the harmonic series  has a quality of consonance and stability that does not demand any subsequent movement. Certainly not as consonant as the octave, perfect fifth or major 4 5 6 triad, but the ear accepts it as having an interior logic and harmony not found in the very turbulent ET version of the chord.

next 3. Oh, the Injustice! 



[1] Science of Music.

[2] Roads, Curtis. Microsound

[3] I use a simple bit of code based on an example posted by Andrew Benson, that I affectionately call, the ‘winds of chaos’.

[4] Rossing, p.113

[5] You get better with practice.

[6] would be interesting research project to invite musicians and non-musicians to tune these intervals and compare.

[7] Rigden, 74.

[8] It helps if you’ve got a tablet to draw with. My sarcastic mantra invented to avert impending carpal tunnel syndrome: ‘I love clicking the mouse.’ When in the world are we going to come up with something better? No offense mouse-lovers, I got beefs with the standard typewriter keyboard too! Just don’t get me started…

[9] I gather most working in intonation put tape on their keyboards [as long as keyboards persist in their current form, that is ; ]. Reminds me of when I was a child and wrote the note name letters in permanent black marker on the plastic keys of my first Bontempi organ. Once I had learned the names I wanted to rub them out but they wouldn’t come off. I tried to cover them up with tape, working in reverse as to how I do today [and the way I would personally recommend others work] but much to my chagrin, the names of the notes kept showing through!

[10] try this to hear a familiar tune the way the aliens intended: 9 10 8 4 6

Last Updated on Tuesday, 25 May 2010 22:16