argument againsts intervals as frequency ratios

From a well-known textbook on this subject, it is written: "Octaves, 5th and 4ths correspond approximately to the frequency ratios 1:2, 2:3 and 3:4. But the ear is remarkably insensitive to frequency ratios between simultaneous and successive pure tones (Allen, 1967;Plomp, 1967; Plomp & Levelt, 1965). Western musical intervals are perceived linearly and categorically, and intervals are defined by the center and boundaries of the category – not by ratios such as 5:4 or 81:64. Intervals can vary in size by up to asemitone (e.g., a major 3rd ranges from 350 to 450 cents: Burns, 1999). Typicalintonations deviate systematically from frequency ratios (major 3rds larger than 4:5,8ves larger than 2:1) and the size of the deviation depends on register (Rosner,1999). The exact size of a performed interval is the result of a compromise between partially conflicting constraints (Terhardt, 1974a) such as roughness, temporalcontext, musical style, emotion, and melodic emphasis: intonation thus depends notonly on sensitivity to the musical surface but also on cultural knowledge (Burns,1999). Frequency ratios do not directly affect or determine intonation; instead, pureintonation minimizes roughness between harmonic complex tones (Hagerman &Sundberg, 1980; cf. Mathews & Pierce, 1980) – but most intonation is closer to equal temperament or Pythagorean than pure (Burns, 1999). These empirically based arguments cast doubt on ratio-based, abstract-mathematical theories of the natureand origins of scales."

"Octatonic">The exact size of a performed interval is the result of a compromise
between partially conflicting constraints (Terhardt, 1974a) such as
roughness, temporal context, musical style, emotion, and melodic
emphasis:

I will agree 110% on the roughness and temporal context parts of this. Ages ago I had a discussion with Bill Sethares and we both agreed temporal context is what keeps things like the initial noise in the striking of a piano key, striking of a guitar string, or attack/"sizzle" in a snare from being obscenely dissonant.

>"Frequency ratios do not directly affect or determine intonation;
instead, pureintonation minimizes roughness between harmonic complex
tones"
I don't know about the rest of you, but I only partly agree with this. For something like a saw wave with very strong overtones I whole-heartedly agree...but the fact is many instruments have only the first 3-6 overtones being of loudness approaching the root tone. Also note, even Sethares "derivation of 7-tone diatonic JI" assumes that the timbre has all overtones at almost as high an amplitude as the root tone: not exactly realistic for many instruments.
For instruments more concentrated on very low overtone loudness, the actual roughness between the root tones becomes more important than "aligning/minimizing-roughness-between higher overtones". Of course, 4/3 is much less rough than 5/4 with either type of instrument, but then again 4/3 spaces the root tones much further apart than 5/4 (along with being more periodic).

To me where this gets dicey is intervals like 13/11 vs. 8/7 where 13/11 has further apart root tones but 8/7 is more periodic. Depending on which instrument is used, IMVHO, the "less pure" 13/11 may actually sound less rough in many cases.

(Octa-tonic) It's funny you use the term roughness to describe what I consider periodicity. General question to all: I usually call roughness between root tones just plain "roughness" and roughness assuming fairly loud overtones that match the harmonic series "periodicity"...aren't those the proper terms for these phenomena?

-Michael

>"pure-intonation minimizes roughness between harmonic complex tones
(Hagerman &Sundberg, 1980; cf. Mathews & Pierce, 1980) – but
most intonation is closer to equal temperament or Pythagorean than pure
(Burns, 1999). These empirically based arguments cast doubt on
ratio-based, abstract-mathematic al theories of the natureand origins
of scales."

Pytharorean indeed. It still amazes me how Pythagorean tuning took off and evolved into mean-tone all the while Ptolemy's tuning systems died. Not that Pythagorean theory or the idea of building scales on estimated "circles of 5ths" is a bad idea but, IMVHO Ptolemy was the one with the smarter idea.

Look at http://www.tonalsoft.com/enc/d/diatonic-genus.aspx.
If you look at Ptolemy's "Didymus" system, it concentrates on optimizing periodicity and matches the first four notes of 7-tone JI diatonic scale perfectly. And his "tense diatonic" scale nails the last 3 tones of diatonic JI! Put them together and you get

24 : 27 : 30 : 32 Ptolemy's "Didymus" +

36 : 40 : 45 : 48 Ptolemy's "Tense Diatonic" =

24 : 27 : 30 : 32 : 36 : 40 : 45 : 48 (!!!7-tone DIATONIC JI!!!)

I dare any of you to argue that he didn't, between those two scales, derived 7-tone diatonic JI long before just intonation was invented...or to argue his systems are "not suitable for poly-phonic music" (which is, IMVHO, pure BS).

Then again, I agree with you that trying to achieve "pure intonation" as a consideration above almost all else is not an optimum approach. Meanwhile, to back up this stance (at least to a good extent), Ptolemy's Homalon IE "smooth" scale system merges on 7TET and provides near-optimum reduction of root-tone roughness while keeping a good degree of average periodicity.
As I've said far too many times, his Homalon system is my favorite scale system around and works beautifully in heavy polyphony (both in harmony and melody, I've found) for almost any type of instrument except those with very strong upper overtones (in which case strict JI IMVHO takes the lead).

-Michael

--- In tuning@yahoogroups.com, "octatonic10" <octatonic10@...> wrote:
>
> From a well-known textbook on this subject, it is written: >"Octaves, 5th and 4ths correspond approximately to the frequency >ratios 1:2, 2:3 and 3:4.

This is simply deceptive. The octave corresponds EXACTLY to the frequency ratio, and not only are the fifth and fourth of 12-tET extremely close (less than two cents) to the frequency ratios, it is overwhelmingly historically documented that that is exactly where they come from. Not to mention untold numbers of guitarists and other instrumentalists tuning from the flageolet tones.

>But the ear is remarkably insensitive to frequency ratios between >simultaneous and successive pure tones (Allen, 1967;Plomp, 1967; >Plomp & Levelt, 1965).

"The ear"?! Speak for yourself. And what do you mean, "pure tones"? SINE WAVES?! Hahahaha! What of the world's music is performed with sine waves? What quackery.

>Western musical intervals are perceived linearly and categorically, >and intervals are defined by the center and boundaries of the >category o?= not by ratios such as 5:4 or 81:64. Intervals can vary in >size by up to a semitone (e.g., a major 3rd ranges from 350 to 450 >cents: Burns, 1999).

I guess anyone who is such a braying jackass as to make such grand proclamations about "the ear" would be incapable of considering "the whole world", not just their little corner of the West.

> but most intonation is closer to equal temperament or Pythagorean >than pure (Burns, 1999). These empirically based arguments cast >doubt on ratio-based, abstract-mathematical theories of the >natureand origins of scales."
>

This is obscene- how can you get more "abstract-mathematical" than exponents of the square root of two? The harmonic series is "abstract"? No it is not, it is physics.

"Empirically-based"? More like moronic scoffing at the majority of the world's music.

I love your passion! You're probably right: Different theories for different types of music.... I didn't mean to belittle other cultures. No music or culture is universal, but being open-minded and respectful of other cultures is. Sincerely, Kelly Johnson

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "octatonic10" <octatonic10@> wrote:
> >
> > From a well-known textbook on this subject, it is written: >"Octaves, 5th and 4ths correspond approximately to the frequency >ratios 1:2, 2:3 and 3:4.
>
> This is simply deceptive. The octave corresponds EXACTLY to the frequency ratio, and not only are the fifth and fourth of 12-tET extremely close (less than two cents) to the frequency ratios, it is overwhelmingly historically documented that that is exactly where they come from. Not to mention untold numbers of guitarists and other instrumentalists tuning from the flageolet tones.
>
>
> >But the ear is remarkably insensitive to frequency ratios between >simultaneous and successive pure tones (Allen, 1967;Plomp, 1967; >Plomp & Levelt, 1965).
>
> "The ear"?! Speak for yourself. And what do you mean, "pure tones"? SINE WAVES?! Hahahaha! What of the world's music is performed with sine waves? What quackery.
>
> >Western musical intervals are perceived linearly and categorically, >and intervals are defined by the center and boundaries of the >category o?= not by ratios such as 5:4 or 81:64. Intervals can vary in >size by up to a semitone (e.g., a major 3rd ranges from 350 to 450 >cents: Burns, 1999).
>
> I guess anyone who is such a braying jackass as to make such grand proclamations about "the ear" would be incapable of considering "the whole world", not just their little corner of the West.
>
> > but most intonation is closer to equal temperament or Pythagorean >than pure (Burns, 1999). These empirically based arguments cast >doubt on ratio-based, abstract-mathematical theories of the >natureand origins of scales."
> >
>
> This is obscene- how can you get more "abstract-mathematical" than exponents of the square root of two? The harmonic series is "abstract"? No it is not, it is physics.
>
> "Empirically-based"? More like moronic scoffing at the majority of the world's music.
>