source file: mills2.txt Date: Sat, 23 Nov 1996 07:22:54 -0800 Subject: Post from Brian McLaren From: John Chalmers From: mclaren Subject: non-octave scales and octave equivalence -- With typical insight, Paul Erlich made a particularly interesting comment about non-octave scales in topic 6 of digest 797. He wrote: "The point is that if a note comes close enough to an octave or a multiple octave, it will sound equivalent, especially in the case of harmonic partials. For example, an interval of 33 Pierce steps exhibits equivalence, even though it is a very different pitch class in the tritave scheme. Even when the even partials are removed, I believe the virtual pitch sensation is not very octave-specific." My ears agree with Paul Erlich's here. Removing odd or even partials doesn't seem to affect my perception of the Bohlen-Pierce scale. However, Paul E. did not mention whether he was talking about "an interval of 33 Pierce steps" *melodically* or *harmonically.* That is, sounding the interval as a vertical dyad or as two sequential notes one after the other. Now, my experience is that this makes a *huge* difference in the perception of octave equivalence in non-octave scales. My ears hear sequential (melodic) intervals as being octave equivalent even if they are significantly off from the octave--upwards of 30 or 40 cents in many cases, especially if the interval is a multiple of an octave--say, 2 octaves, 3 octaves, etc. However, the range of detuning within which my ears will accept an interval as octave equivalent is much smaller when the interval is a vertical dyad (harmonic): somewhere in the range of 0-18 cents. As a concrete example, take the Bohlen-Pierce scale. Play melodically the interval of 8 scale steps; if you play the melodic interval reasonably quickly, you'll find that your ear accepts it as a melodic octave. But if you sound that 8-step interval as a vertical dyad, it will not sound like an octave at all since the interval is 1170.4338 cents, outside the acceptable vertical range for octave equivalence (except at very low fundamental frequencies). This brings up an interesting point with regard to non-octave scales: as most of you know, Enrique Moreno has a very different conception of non-octave scales than Gary Morrison or Your Humble E-Mail Correspondent. Enrique believes that it is pointless and meaningless to try to assign to the intervals of non-octave scales familiar categories such as "third" or "fifth" or "octave." Instead, Enrique suggests that we accept the intervals of non-octave scales on their own merits, rather than misguidedly trying to jam them into familiar but conceptually and musically limiting categories. This view has merit. It recognizes the fact that non-octave scales sound different in a basic way from octave = 2.0 scales; Gary Morrison has described non-octave scales as sounding like "the musical equivalent of thick rich chocolate milk shakes" and this is true--there's something unutterably exotic and gorgeously alien about most non-octave scales. They all share a very sultry foreign "sound" which renders, say, the 12th root of 3, the 15th root of 3 and the 13thr oot of 3 and the 25th root of 5 and the 37th root of 31 much more akin to one another in "sound" or what Ivor Darreg called "mood" than any trivial considerations of audible octave equivalence. On the other hand, there are problems with Enrique's view of non-octave scale. For one thing, there exist infinitely many non-octave scales which are audibly identical to familiar octave = 2.0 divisions of the octave. For example: I defy anyone to tell the difference audibly between 12-TET and the 51st root of 19, or the 105th root of 431, or the 114th root of 727, the 122nd root of 1153, or the 126th root of 1453. THe difference between a 2/1 and the equivalent interval in each of these "non-octave" scales is less than 1/3 cent-- you *cannot* hear the difference between these intonations and 12. There exist infinitely many non-octave scales audibly identical (not close, *identical* to the ears, with a 2/1 less than 0.1 cents off from 1200 cents) to 13-TET, 14-TET, 15-TET, and so on. This being the case, we are forced to recognize that for a significant sub-class of non-octave Nth root of K scales, there is *no audible difference whatsoever* between these and some N-TET octave = 2.0 scale. This being the case, it would obviously be perverse in a tuning audibly identical to 12-TET to try to describe the intervals in exotic Nth root of K terms rather than in terms of the familiar fifth, major and minor third, fourth, major and minor second, and so on. Thus the situation for non-octave scales is more complicated than anyone has mentioned to date. On the one hand, listeners will tend to hear intervals in these scales *very* differently melodically than harmonically if the interval is slightly off from a familiar interval. On the other hand, there exist a large class of non-octave scales which sound audibly *identical* to familiar dvisions of the octave. Lastly, there's the question: In a given Nth root of K non-octave scale, what is the most consonant interval? That is, what is the interval which takes the musical and acoustic place of the 2:1 octave in ordinary divisions of the octave with harmonic series timbres? -- There is no simple answer to this question. A superficial answer is: obviously, if we're talking about the Nth root of K, then K is the most consonant interval in all cases. This is sometimes true, and sometimes clearly false. In the 13th root of 3, the 3:1 ratio is clearly the primary consonant interval. It functions musically in the same way that a 2:1 does. If you "double" pitches at an interval of 13 scale steps in the 13th root of 3, you'll get much the same result as when you double pitches at an interval of 12 scale steps in 12/oct. In the 21st root of 17, however, the interval of 21 scale steps is not nearly as great a point of acoustic rest as the interval of 3 scale steps. Moreover, all Ks are not created equal. Intervals which are low members of the harmonic series multplied by small integers tend to sound more consonant than Ks which are high members of the harmonic series. Thus , an interval of 17:1 sounds less consonant than 6:1 since 6:1 is 3:1 times 2, while 17 is relatively far up the harmonic series. Even this statement must be qualified, for the harmonic series exhibits the property that consonance decreases as one climbs the harmonic series, then suddenly it begins to increase as one climbs further, then consonance decreases again, then it suddenly increases, and so on. For example: 2, 3, 4, 5, 6 are highly consonant. 7 is less so, intermediate in fact between consonance and dissonance; 8, 9, 10 are highly consonant, 11 is much less consonant; 12 is highly consonant; 13 is relatively dissonant; 14, as a multple of 7, is intermediate in consonance; 15, 16 are highly consonant; 17 is relatively dissonant; 18 is highly consonant; 19 is quite consonant, ditto 20 and 21; but 22 and 23 are relatively dissonant; 24, 25 are highly consonant; 26 is dissonant, 27 is extremely consonant; 28 intermediate; 29 is dissonant...and so on. Thus the particular K is question must be considered, in addition to the issue of whether the Nth root of K scale contains an interval interval within the N:1 span that sounds more consonant than N:1. One last point is that the absolute size of the musical interval in question is very important. Paul Erlich mentioned that an interval of 33 scale-steps of the Bohlen-Pierce scale sounds like an interval of 4 octaves. However, this interval comes out to 4828.0396 cents, 28 cents away from 4 octaves. The ear doesn't tend to notice this discrepancy for very large intervals because the two notes are so greatly separated from one another than there is little opportunity for the harmonics of the lower and the upper note to beat with one another. Most acoustic timbres exhibit very little energy above the 16th harmonic, and the 16th harmonic is the fundamental of a pitch 4 octaves above the base note of a dyad. Thus, while an interval of 33 scale steps in the Bohlen-Pierce scale is about as far away from the octave as an interval of 8 scale steps (28.039 cents for the former as opposed to 29.567 cents for the latter), 8 scale-steps in the 13th root of 3 sounds very far from octave equivalence while 33 scale-steps sounds reasonably close to octave equivalence because many harmonics of both notes fall within the critical band in the case of the 8-step interval while almost no harmonics of both notes fall within the same critical band in the case of the 33-step interval. In short, octave equivalence and the question of which intervals will most tend to function and sound as points of acoustic and musical rest in intervals formed from the notes of non-octave scales are issues more complex than anyone on this forum appears to have suggested. -- Paul Erlich goes on to write that "In the case of inharmonic partials, octave equivalence may play less of a role, but still exists, and is less demanding as to intonation." Both my experiments with additive synthesis inharmonic timbres in Csound and William Sethares' experiments with resynthesized Fourier-analyzed timbres with stretched partials strongly contradict this statement. In particular William Sethares has a set of instrument timbres resynthesized with all harmonics stretched so that the octave is a ratio of 2.1 instead of 2.0, etc. Playing a vertical octave dyad with such timbres produces unbearable dissonance; but playing a vertical octave whose ratio is 2.1 rather than 2.0 produces the familiar sensation of octave equivalence. So the evidence *strongly* indicates that 2:1 octave equivalence goes away when the timbre becomes inharmonic, and this is confirmed by William Sethares' mathematical procedure for finding scale pitches from an inharmonic timbre. --mclaren Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 23 Nov 1996 18:34 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05277; Sat, 23 Nov 1996 18:35:39 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA05269 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA12668; Sat, 23 Nov 1996 09:33:58 -0800 Date: Sat, 23 Nov 1996 09:33:58 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu