source file: mills2.txt Date: Tue, 20 May 1997 16:03:28 +0200 Subject: Equal Divisions -- of What? From: PErlich David wrote, > Lost ya there. Doesn't "quantization" denote equal divisions? E.g., in what > sense are the two spaces in 1:1, 9:8, 5:4 equal, or based on equal divisions > of linear frequeny space, whatever that means? Strange as it might seem, > after years of picturing frequency space logarithmically, I can't get myself > to visualize it linearally - it doesn't seem to make any sense that way. > The short answer is that 1:1, 9:8, 5:4 means, say, 800, 900, and 1000 Hz (cycles per second), so that the "equal divisions" are of 100 Hz each. Couldn't be simpler. Picturing frequency space linearly is typical when working with Fourier transforms, etc. However, pitch space is not the same as frequency space. In fact, pitch space most closely resembles log-frequency space. An example where this is important is in distinguishing between _true_ or _physical_ white noise and _psychoacoustic_ white noise. The two are very different. White noise has equal power in any constant intervals in a linear frequency scale. To quote Manfred Schroeder, "Pink noise, also called 1/f noise, has equal power in octave frequency bands or any constant intervals on a _logarithmic_ frequency scale. This is a desirable attribute in many applications. For example, pink noise is a favorite test signal in hearing research and acoustics in general because it approximates many naturally occuring noises. Pink noise also has the approximate property of exciting equal-length portions of the basilar membrane in our inner ears to equal-amplitude vibrations, thus simulating a constant density of the acoustic nerve endings that report sounds to the brain. Pink noise is therefore the _psychoacoustic_ equivalent of white noise." David seems to be right that picturing frequency space logarithmically makes more sense. Indeed, it is more natural, given that nature gave us basilar membranes on which equal lengths correspond to equal portions of log-frequency space. So it is odd that David asks, >I'm working on the assumtion that a tuning which correlates to the way we and >our world are made would, *by virtue of that fact*, provide a more solid >foundation for more meaningful and profound compositions. That would seem to >be a foregone conclusion, at least through the window of my world-view. If >any of you disagrees with that assumption and can cite anything other than >purely circumstantial evidence against it, please shoot me down. Please! :-) with JI in mind as the most natural tuning. David, care to reconsider? Now for the long answer. In mathematics we have harmonic series as contrasted with arithmetic series as well as geometric series. The harmonic series is named after the musical harmonic series, since when looking at the length of a string, or wavelength of the sound wave, or repetition time of the waveform, the partials of a perfect string go as 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, . . . So that is the harmonic series as known to mathematicians. This certainly does not appear equally spaced in terms of human quantities like length and time. So what is? What is the musical equivalent of the mathematical arithmetic series 1, 2, 3, 4, 5, 6, 7, 8, 9, 10? The answer: the subharmonic series! Subharmonics are integer multiples of the string length, wavelength, or repetition time of the "fundamental." (Note that I am not ascribing any musical significance to this fact). Finally, consider the geometric series . .x^-9, x^-8, x^-7, x^-6, x^-5, x^-4, x^-3, x^-2, x^-1, x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, . . . where x>0 and x does not equal 1. Whether the units of measurement are string length or wavelength or time period, as above, or frequency, as in the 800:900:1000 Hz example, the result is the same: equal temperament! Conversely, if you take the log of the above series, you get the integers, so you can see that an equal temperament is equally spaced in log-frequency or even log-length or log-time space. Anyhow, it appears that from a physical point of view, the subharmonic series is what most people would describe as "equally spaced," while psychologically, since we hear pitch in log-frequency space, an equal temperament is the tuning most likely to be heard as equally spaced. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 20 May 1997 16:04 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06132; Tue, 20 May 1997 16:04:27 +0200 Date: Tue, 20 May 1997 16:04:27 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA06133 Received: (qmail 394 invoked from network); 20 May 1997 14:04:21 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 20 May 1997 14:04:21 -0000 Message-Id: <97051923410702/0005695065PK2EM@mcimail.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu