Talk:Just intonation

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Wikipedia:WikiProject Tunings, Temperaments, and Scales This article is part of the WikiProject Tunings, Temperaments, and Scales to improve Wikipedia's articles related to musical tunings, temperaments, and scales.
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Contents

[edit] Update external link to JIN

you have a link on this page to the Just Intonation Network

effective immediately, the URL for the Just Intonation Network has moved

Old URL: www.dnai.com/~jinetwk

New URL: www.justintonation.net


Please update you link.

thank you.

--DBD (David B. Doty--Just Intonation Network)

Done - incidentally David, if you're still around here - you could have edited the page yourself. Wikipedia can be edited by anyone. See Wikipedia:Welcome, newcomers if you're interested. --Camembert (24 November 2004)

[edit] Key of examples

Not that there's anything wrong with it, but is there any reason for the examples being changed from C major to F major? Just curious. --Camembert (22 August 2003)

[edit] Outline

My proposed outline:

  1. introduction: Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. Another way of considering just intonation is as being based on lower members of the harmonic series. Any interval tuned in this way is called a just interval. Intervals used are then capable of greater consonance and greater dissonance, however ratios of extrodinarily large numbers, such as 1024:927, are rarely purposefully included just tunings.
  2. Why JI, Why ET
    1. JI is good
      1. "A fifth isn't a fifth unless its just"-Lou Harrison
    3. Why isn't just intonation used much?
      1. Circle of fifths: Loking at the Circle of fifths, it appears that if one where to stack enough perfect fifths, one would eventually (after twelve fifths) reach an octave of the original pitch, and this is true of equal tempered fifths. However, no matter how just perfect fifths are stacked, one never repeats a pitch, and modulation through the circle of fifths is impossible. The distance between the seventh octave and the twelfth fifth is called a pythagorean comma.
      2. Wolf tone: When one composes music, of course, one rarely uses an infinite set of pitches, in what Lou Harrison calls the Free Style or extended just intonation. Rather one selects a finite set of pitches or a scale with a finite number, such as the diatonic scale below. Even if one creates a just "chromatic" scale with all the usual twelve tones, one is not able to modulate because of wolf intervals. The diatonic scale below allows a minor tone to occur next to a semitone which produces the awkward ratio 32/27 for Bb/G.
  3. Just tunings
    1. Limit: Composers often impose a limit on how complex the ratios used are: for example, a composer may write in "7-limit JI", meaning that no prime number larger than 7 features in the ratios they use. Under this scheme, the ratio 10:7, for example, would be permitted, but 11:7 would not be, as all non-prime numbers are octaves of, or mathematically and tonally related to, lower primes (example: 12 is an octave of 6, while 9 is a multiple of 3).
    2. Diatonic Scale: It is possible to tune the familiar diatonic scale or chromatic scale in just intonation but many other justly tuned scales have also been used.
  4. JI Composers: include Glenn Branca, Arnold Dreyblatt, Kyle Gann, Lou Harrison, Ben Johnston, Harry Partch, Terry Riley, LaMonte Young, James Tenney, Pauline Oliveros, Stuart Dempster, and Elodie Lauten.
  5. conclusion

http://www.musicmavericks.org/features/essay_justintonation.html

Hyacinth (30 January 2004)

[edit] Just tuning

I was going to merge the content below from Just tuning, but which "one possible scheme of implementing just intonation frequencies" does the table show? Hyacinth 10:29, 1 Apr 2005 (UTC)
It shows the normally used just intonation scale - I don't think that it has a special name. It can be constructed by 3 triads of 4:5:6 ratio that link to each other, e.g. F-A-C, C-E-G, G-B-D will make the scale of C. Yes, this should definitely be included. (3 April 2005)
Please Wikipedia:Sign your posts on talk pages. Thanks. Hyacinth 22:03, 3 Apr 2005 (UTC)

I am no expert, and I've not done any Wikipedia changes either, so forgive me if I'm wrong in what I'm doing (content) or how I'm doing it (method), but the main text gives 6/5 as a minor third, and I currently disagree.

Scholes' Oxford Companion to Music, eighth edition, in the section on intervals, says a minor third is a semitone below a major third, ie 15/16 * 5/4 = 75/64 and not 6/5 as stated in the main text. The Oxford Companion to Music also states that by going up a semitone an interval becomes an augmented interval and so a major tone (a second) would become an augmented second as follows: 9/8 * 16/15 = 6/5. Thus 6/5 is an augmented second, and 75/64 is a minor third. Ivan Urwin

There are many semitones. A minor third is a chromatic semitone (25/24) smaller than a major third.
I can also give you a reductio ad absurdum for your reasoning. If 6/5 is an augmented second, then 5/4 * 6/5 = 3/2 is not a perfect fifth, but a doubly augmented fourth. Then if 4/3 is a perfect fourth, 4/3 * 3/2 = 2/1 is not an octave, but an augmented seventh. —Keenan Pepper 00:32, 14 April 2006 (UTC)
Why not split the difference, and call a semitone the twelfth root of 2, or 1.059463...? right between 16/15 at 1.066667 and 25/24 at 1.041667. Even better, one could call a minor third 300 cents, or the fourth root of 2, again smack between those silly over-simplified integer ratios. Of course I'm kidding; thanks, KP! Ivan, I'm not familiar with your Scholes reference; how completely does it treat the differences between just tunings and various temperaments? I'm guessing that's where the oversimplification may lie. Just plain Bill 01:49, 14 April 2006 (UTC)
Okay, I think I missed a key word in the Scholes Oxford Guide to Music text, approximately like this ...
If an inverval be chromatically increased a semitone, it becomes augmented.
I am looking at this as a mathematician and so the musical terminology throws me somewhat: dividing by 5 being called thirds and dividing by 3 being called fifths, etc. If I were to rewrite the Scholes text as shown below and use 25/24 as a definitiion for a 'chromatic semitone' as per Keenan Pepper's remarks, then I'd agree.
If an inverval be increased by a chromatic semitone, it becomes augmented.
The way this arose was me looking at the ratios with my mathematical background. Prime factorisation of integers is unique. The only primes less than 10 are 2,3,5, and 7. I gather that 7 is used for the 'blue' note in blues, and that most western music just uses or approximates ratios based on 2, 3 and 5. With 2 being used to determine octaves and with notes an octave apart being named similarly, that brings practical ratios down to just determining the power of 3 and the power of 5. I was making a 2 dimensional table of the intervals, and putting names to the numbers with the help of a borrowed book, but it appears the complex terminology for simple mathematics got the better of me.
I am happy to drop my remarks and delete all this in a few days time (including Keenan's and Bill's remarks), but I'll give you chance to read it and object before I do. Maybe some moderator will do that. Maybe you two are the moderators. Whatever. Anyway, thanks guys.
Ivan Urwin
You're mostly right about the primes. See Limit (music).
The labeling of intervals as "seconds", "thirds", etc. corresponds to the number of steps they map to in 7-per-octave equal temperament. 3/2 maps to four steps, so it's a "fifth", 5/4 maps to two steps, so it's a "third", 16/15 maps to one step, so it's a "second", and 25/24 maps to zero steps, so it's a kind of "unison" or "prime". Intervals separated by 25/24 have the same name, for example the 6/5 "minor third" and the 5/4 "major third".
Conversations on talk pages are usually never deleted, only archived when they become too long, so don't worry about that. —Keenan Pepper 05:04, 15 April 2006 (UTC)
I'm lost here (not a moderator, by the way, just another netizen) when you speak of "determining the power of 3 and the power of 5." The interval of a fifth is just the musical pitch space between the first and fifth notes of a scale. Because I happen to be used to vibrating strings, a perfect fifth being a 3/2 frequency ratio now seems as obvious to me as the fact that x^2+y^2=1 makes a unit circle, just a matter of familiarity. I'm equally happy to talk about it or to send it to oblivion as you suggest.
We are lucky to have folks around like Keenan Pepper who can quickly point out the discrepancy in types of semitone, for example. Just plain Bill 03:56, 15 April 2006 (UTC)
This is the sort of table I had. You can see that going up a major tone consists of going right couple of cells, so looking at the entry 10/9 (minor tone), I could quickly see that a mojor tone higher than that would be a third, and similarly a major tone higher than a semitone would be a minor third.


Power of 3 (fifths) -3 -2 -1 0 1 2 3
Power of 5 (thirds)
-3 128/125
-2 256/225 128/75 32/25 48/25
-1 64/45 16/15 (semitone) 8/5 6/5 (minor third) 9/5
0 32/27 16/9 4/3 (fourth) 1/1 3/2 (fifth) 9/8 (major tone) 27/16
1 40/27 10/9 (minor tone) 5/3 (sixth) 5/4 (third) 15/8 (seventh) 45/32 135/128
2 25/18 25/24 25/16 75/64 225/128
3 125/64

The powers of two in the table just bring the ratios to within an octave. Clearly one could add more ratios to the table. I have just included it for illustration.

Ivan Urwin

That helps me make more sense of it. Thanks, and also to Keenan for pointing out the [tonality diamond]] of Harry Partch. Until I sit with this some more, I have nothing really useful to add... cheers, Just plain Bill 14:46, 16 April 2006 (UTC)

[edit] content for merge:

Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. This table shows one possible scheme of implementing just intonation frequencies.

Just tuning frequencies of all notes in each key based on A = 440 Hz when in the key of C. The just intonation scale ratios of 24:27:30:32:36:40:45 are used and each key note has the same frequency in the scales with +/- 1 sharp or flat.

Note that the 6th note in a key changes frequency by a ratio of 81/80 when it becomes the 2nd of the key with one more sharp or one less flat. All other notes retain the same frequency. In C all frequencies are an exact number of Hertz.

In just intonation incidentals tuning must be worked out on a case by case basis. Often the minor third and minor seventh take the ratios 28 and 42 when the tonic is taken as 24, so that in C the tuning for Eb and Bb would be 308 Hz and 462 Hz. These frequencies allow dominant seventh chords with frequency ratios of 4:5:6:7.

For frequencies in other octaves repeatedly double or halve the tabulated figures.

There is a difference between Gb and F# which amounts to a ratio of 312 / 219 = 1.0136433 as discovered by Pythagoras.

Key \ Note C Db D Eb E F Gb G Ab A Bb B
Gb (6b) 278.123 309.026 347.654 370.831 417.185 463.539 494.442
Db (5b) 260.741 278.123 312.889 347.654 370.831 417.185 463.539
Ab (4b) 260.741 278.123 312.889 347.654 391.111 417.185 469.333
Eb (3b) 260.741 293.333 312.889 352 391.111 417.185 469.333
Bb (2b) 264 293.333 312.889 352 391.111 440 469.333
F (1b) 264 293.333 330 352 396 440 469.333
C (0) 264 297 330 352 396 440 495
G (1#) 264 297 330 371.25 396 445.5 495
D (2#) 278.438 297 334.125 371.25 396 445.5 495
A (3#) 278.438 297 334.125 371.25 417.656 445.5 501.188
E (4#) 278.438 313.242 334.125 375.891 417.656 445.5 501.188
B (5#) 281.918 313.242 334.125 375.891 417.656 469.863 501.188
F# (6#) 281.918 313.242 352.397 375.891 422.877 469.863 501.188
Key / Note C C# D D# E F F# G G# A A# B
Equitempered 261.626 277.183 293.665 311.127 329.628 349.228 369.994 391.995 415.305 440.000 466.164 493.883

[edit] JI/ horn deletion

Hello, you noted that natural horns play far from just intonation, but the lead from the article says

In music, Just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series.

If natural horn players always modify pitches other than the key note, the deletion makes sense (my small exposure to them suggests they use the natural notes) but if it is because the 7th and 11th harmonics don't fit in the diatonic pattern it doesn't because these are rational intervals and members of the same harmonic series, and the same notes played from the trumpet marine. --Mireut 23:37, 4 February 2006 (UTC)

Actually, I think the whole section (as it was) is rather silly (especially given that there are only two instruments, one of which is rather bizarre). There are thousands of instruments capable of just intonation not mentioned here, many quite conventional. The question is more whether or not they are ever asked to.
However, on the Natural Horn specifically, I think I'd have to argue with you. It produces a harmonic series quite easily, and quite naturally. To produce other notes, yes, there is a stopping technique that is used to flatten pitches. The seventh harmonic is actually used a fair bit, though you're right that the 11th is hardly ever used (Benjamin Britten's Serenade is a fun exception). However, in its standard technique, the harmonic tones which are used (1,2,3,4,5,6,(7),8,9,10,12,(14),15,16) are indeed just.
You might argue that if this is true, the natural horn can only be played just in one key then. This is also true. A quick study of natural-horn writing will attest to this. They were only asked to play in one key at a time, and the out of key notes were expected to be dull and out of tune. I don't think you can rightly argue that the Natural Horn is not a just-intonation instrument. Rainwarrior 00:08, 5 February 2006 (UTC)

According to my understanding just intonation means that each note is in a perfect ratio with the preceding note. In the natural horn, each note is in a perfect ratio with the fundamental of the horn itself, not with each other note. This is not very clearly stated in the article, I agree, but it's rather complicated and I'm not sure how to re-word it. The natural horn is a little more like Pythagorean intonation, although it's not quite that either. I hope this helps. Makemi 00:02, 5 February 2006 (UTC)

Pythagorean tuning is a tuning system of a series of fiths. This is quite different from what the horn does. If you compared a pythagorean diatonic scale to the natural horn's you'd get quite a few differences. On C: C (same: 1/1), D (same: 9/8), E (different: 81/64 vs 5/4), F (different: 4/3 vs irrational), G (same: 3/2), A (different: 27/16 vs irrational), B (different: 243/128 vs 15/8), C' (same: 2/1). Rainwarrior 00:24, 5 February 2006 (UTC)
Actually, I think the whole section (as it was) is rather silly (especially given that there are only two instruments, one of which is rather bizarre)
Sure but then why favor bagpipes? --Mireut 00:22, 5 February 2006 (UTC)
I think this is also silly. That information probably belongs on the bagpipes page. Rainwarrior 00:24, 5 February 2006 (UTC)

Oops, sorry, I reverted without looking at the talk page. My bad. —Keenan Pepper 02:45, 5 February 2006 (UTC)

Who is the "you" in Mireut's first message in this section? What section of the article are we discussing? Hyacinth 09:14, 5 February 2006 (UTC)
"You" was Makemi who had just removed "Natural Horn" from the "instruments that play naturally..." section, which was later reverted by Keenan Pepper. Rainwarrior 18:00, 5 February 2006 (UTC)

Ok, I was totally wrong that the natural horn is anything like Pythagorean tuning. I maintain that it certainly isn't Just Intonation as I have studied it. Firstly, I think the definition on the main page is a bit unclear. According to J. Murray Barbour, Tuning and Temperament: A Historical Survey, (East Lansing: Michigan State College Press, 1953), 5.

A tuning is a system wherein all the note relationships can be expressed in ratios between integers. A temperament is a modification of a tuning which requires radical numbers to express some or all of the ratios between notes.

Thus the leading definition on this page is somewhat misleading. Just intonation must be between each and every note in a musical piece, within its own context. Now, granted a certain amount of bending is possible with natural horn, although there is disagreement about whether this is historically correct. Nonetheless, although a horn in C can play the notes in the diatonic scale of C in just intonation in relation to C, it cannot, for instance, play anything like just intonation in the key of G (without making it a horn in G), or, say, within a piece in C between a G and a D. Sorry, I don't have numbers to back me up on this. Er, I also think it's kind of a silly section to have. I didn't get involved in this project at first because frankly, the whole subject makes my head hurt, but I'm pretty darn sure the natural horn is pretty far from just intonation. Makemi 19:46, 5 February 2006 (UTC)

But the harmonic series, as played by the natural horn, does have integer relationships between "each and every note". —Keenan Pepper 20:03, 5 February 2006 (UTC)
It doesn't. It certainly does play stopped notes now and then which are essentially irrational. (Incidentally, this example about playing in G on a C horn is not very good, becauce G shares a lot of tones in common with the C harmonic series). However, I don't think this is enough to say that it isn't a just-intonation instrument. All of its important notes are just.
To speak of the classical horn repertoire, however, does not necessarily denote the capabilities of the instrument. Classical composers did not complain and argue about tuning systems, so far as I know. The fact remains that the natural horn very easily plays notes in the harmonic series, and only with difficulty is made to play anything else. It is a just-intonation instrument that has been hijacked occasionally to play with well tempered musicians. It's a really unique instrument that way.
There are other instruments which are quite retuneable (keyboards, harps, kotos), and others that can be made to play in JI with careful playing technique (guitars, saxophones, trombones, violins)... should they be considered for this list as well? Or should we only include those which are traditionally played in JI? (In which case, we should probably add a Veena, Sitar, and Tambura in there for starters.) Rainwarrior 05:23, 6 February 2006 (UTC)
How about just delete this list as well as the unsubstantiated bagpipe section and the second paragraph from indian music. The article is negative enough already. --Mireut 14:39, 6 February 2006 (UTC)
I've been planning a bit of cleanup to this page in a week or two. Indian music I think deserves at least a mention, given that it is a well known example of JI that is still practiced. If there is already a page devoted to its tuning, we can just link that (but I haven't found one on Wikipedia yet). If not, a summary diagram of its scale systems I think is worthwhile (just the swaras, and not the srutis which (correct me if I am mistaken) is a very dated theoretical psychoacoustics idea). The just intonation of bagpipes can potentially be folded into the actual bagpipes page, and then a link can be placed in the JI-instruments section we are currently discussing.
You wrote "irrational?" in your edit summary. To answer this question: Every JI interval is rational, meaning it can be represented as A/B where A and B are both integers. There is a fine amount of variation in pitch that can be produced by stopping the horn, and it is unlikely (unless the note is held for a long duration, which is not typical for stopped notes in classical music) that they player will have enough time to tune a just interval on a stopped note. Thus I would say the tuning of a stopped note is probably going to be irrational (meaning it could be anywhere, not just on an integer-ratio boundary). Rainwarrior 17:43, 6 February 2006 (UTC)
"... a well known example of JI that is still practiced." Any a cappella group, string quartet, or for that matter, any orchestra worth its salt "still practices" just intonation where appropriate. As mentioned in the article, it may be actually more difficult for the first two to do equal temperament. That is my naive belief. Just plain Bill 18:05, 6 February 2006 (UTC)
Small a-capella groups do tend to justify quite a bit (I've never heard a good barbershop that didn't). String quartets, sometimes (I saw the St. Lawrence String Quartet justify a lot of their playing once, it was quite exciting, but I've heard as many quartets justify as not). Orchestras hardly ever. Even in professional orchestras, the tuning gets pretty hairy a lot of the time. Choral music has the same problem, confounded by the lack of training in tuning and the standard use of a piano for rehearsal. (I actually wrote the part of the article you're referring to, so I tend to agree with what you're saying.)
Indian music, though, I think makes a stronger case than most, because its tuning system is fixed, and rigorously implemented. It is not left to intuition, and there isn't argument about the right way to tune a note. Finally, this music is crafted with this specific just tuning system in mind, which cannot be said for the string quartet, or for most western music.
At any rate, your comments are something to keep in mind for a future update. Rainwarrior 18:49, 6 February 2006 (UTC)

[edit] Instruments that play naturally in just intonation

Someone added pedal steel guitar, but I removed it. If we list pedal steel guitar we should also list trombone, violin, viola, cello, bass, theremin... The natural horn and the tromba marina are special in that they only produce pitches from the harmonic series by design. —Keenan Pepper 00:04, 21 April 2006 (UTC)

I'm not sure how to add this, and I am not going to wreck the page to make a point, but I play barbershop music with some friends on trombone because its unique combination of tonal quality and range along with pitch flexibility make it a natural fit for SATB vocal music; in fact, "ringing" a chord is pretty easy when you get used to doing it. The standard positions on the slide are not justly tuned but adjusting them by ear as needed is pretty easy.Es330td 14:30, 9 May 2006 (UTC)

I completely agree that the trombone is very suitable for playing in just intonation, but I still don't think we should list it for the reason above; the Trombone has no "natural" intonation... it is simply a "free" instrument that can play in any tuning, like any of the fretless stringed instruments, the voice, etc... - Rainwarrior 15:19, 9 May 2006 (UTC)

[edit] What about synthesizers??/I don't get the vocal example

In response to "The human voice is the most pitch-flexible instrument in common use."

Might I add, that a (digital) synthesizer can easily vary pitch given a proper controller, and of course return to it's original tuning. I'm not talking about Moogs or Mellotrons here, mind you.

I'm not sure I understand why a vocalist wouldn't be subject to the same sorts of tuning constraints as a violinist playing glissando. (sore throat from extended vocal sessions, etc.)

I could easily tell my synth to resonate sounds microtonally, at specific frequencies, but I'm not exactly sure what I'd sing if someone told me "sing a note at 490 Hz, and then over the next quarter note modulate to 245 Hz evenly please."

So I don't see how any instrument other than a synthesizer could really qualify to such rigid criteria.

Furthermore, I don't find my voice flexible _at all_, and I think most non-vocalists would have the same problem. I probably don't sing in tune for more than a two-octave range. And given that I am not a proficient vocalist, I don't understand the pitch stability argument either. Futhermore, in the musicals that I've seen, the vocalists have a long warm up scale... Of which the primary purpose is to find the proper tuning for the vocalists. Actually the more I think about it, the more odd the claim of perfect pitch being a property of the human voice seems to be. (and I can't make glissando sounds like a violin, either)

If anyone wants to explain what they mean by those claims, I would like to discuss, perhaps I am misunderstanding the scope and precision of your points.

[edit] Added a sound example

I added an example sound clip comparing just intonation to equal temperament, because I felt it desperately needed one. The poor reader may well look up and down the page and think to himself, "This is good and all, but what does just intonation sound like?" This morning, that person was me. I had already heard a clip or two comparing them some time ago, and was trying to find one again. I found one, but wasn't satisfied, so I made my own and I put it up here. But I couldn't figure out a good place to put it, so I ended up writing a bit about the difference from equal temperament below the opening paragraph. I don't think this is the best that could be done, which is why I'm posting about it here... my additions could probably be worked into the article more smoothly, but I do think it's important to contrast with equal temperament early in the article (since the first question likely to pop in anybody's head is "So what makes this different?"). - furrykef (Talk at me) 15:46, 29 August 2006 (UTC)

I think sound examples are great to have. Thanks for making one. I have suggestions though; the first is not to use a piano sample. Piano tones are produced by three strings which are never exactly in tune, leaving a detuned phasing sound effect regardless of whether you are using just intonation or not. Instead, I would suggest a reed instrument sound.
I think the most dramatic difference is in the comparison of the 7-limit dominant seventh chord with its equal tempered counterpart. How about an example like this: an ET 7th chord played, brief pause, then a sustained ET 7th gradually tuned into a 7-limit 7th, brief pause, then the 7-limit 7th along. What do you think? - Rainwarrior 00:08, 24 November 2006 (UTC)

[edit] Timing of equal temperament

Surely equal temperament became dominant before the 20th century?

I don't think so. I think well temperament was dominant in the 19th century. - furrykef (Talk at me) 01:12, 30 September 2006 (UTC)
I think it depends on locality and instrument, as well as historian. "Well temperament" is a modern concept collecting a wide range of sources. - Mireut 13:28, 30 September 2006 (UTC)

[edit] Indian tuning note

Editor's note from one who practices North Indian Classical Music and studies Just Intonation: 27:16 is not correct in Indian tuning. 5:3 is indeed the correct value. Dha (the major 6th of the scale) is the Ga (major third) of Ma (the perfect fourth) represented mathematically, this is shown as:

4/3 X 5/4 = 20/12 = 5/3

Dha is not the Re (major second) of Pa (the fifth)

9/8 X 3/2 = 27/16

This is a common mistake made by inexpert bamboo flute makers, and results in dha sounding quite sharp. The "wolf tone" that results from using 9:8 and 5:3 is a non issue because the music is melodic and not harmonic; i.e. notes are played one at a time. —Preceding unsigned comment added by 59.144.100.14 (talk)

[edit] "Western Instrument Builders"

Doesn't this section seem to be an excuse to plug one builder's instrument? There are myriad guitar builders and makers of other instruments that have made numerous attempts, many successful, at building justly intoned and microtonal instruments throughout Europe and America. Surely a section titles "Western Instrument Builders" should be put on hold until a fairly representative list can be compiled. Wikipedia is often used as a marketing tool by contributors. Don't let a scholarly entry become an advertisement. —The preceding unsigned comment was added by 168.8.249.165 (talk • contribs) 14:35, 26 February 2007 (UTC)

Yep. Looks like spam to me. - Rainwarrior 16:36, 26 February 2007 (UTC)

[edit] Ratios and fractions

On this page, and Interval (music) also, the term ratio is used but immediately followed by examples written as fractions. I know it is common and accepted in music theory to use either form, 3:2 or 3/2 for example. But perhaps these two pages (and probably others) should explain that first? It looks wrong to say "ratio such as 1024/927" as this page does.

Also, it seems to me that the very first paragraph is confusing about the difference between frequency and frequency ratio. A ratio by itself, like 1024/927 (the first example given, written as a fraction!), doesn't give you the specific frequencies of the two notes, right? The frequency ratio stands for any two notes whose frequencies of that ratio, no? Could this be explained more clearly in the first paragraph? Pfly 21:54, 15 March 2007 (UTC)

Well, I inserted a brief note about the colon vs fraction style (3:2 vs 3/2). I think this is useful in avoiding confusion among people who are not very familiar with the topic, especially since the page is full of both styles without an explanation. I also took out the link to rational numbers. The first sentence already says "whole number ratios". Music theory can be mathematically intimidating, even for people like me who have been at it for years. I figure the opening sentences ought to keep mathematical references simple.
The same logic nearly led me to remove the bit about the two notes being members of the same Harmonic series (music), as potentially confusing, needlessly, for people new to the topic. Even for me (and I like to think I am at least reasonably familiar with the topic!), it took me some time reading the harmonic series page before I realized how this fit in. The idea, I think, is that for any given fundamental pitch there is a harmonic series (even if the page on harmonic series tends to use the phrase "the harmonic series" a lot, as if there is only one). If the two notes of an interval are rooted on fundamentals that are not in a just ratio to one another, the interval is not just. But it took me a few minutes to figure that out -- perhaps it is too much information for the second opening sentence? Pfly 07:02, 19 March 2007 (UTC)
Perhaps it could be explained more clearly with an example. If two notes have the frequency ratio 5:4, then they represent the fifth and fourth harmonics of the same overtone series (specifically, the fundamental would be two octaves below the lower note of the interval). But you're right, this doesn't need to go in the intro. —Wahoofive (talk) 16:02, 19 March 2007 (UTC)

[edit] Twelve tone scale

I changed the title of "the twelve tone scale" to "a five-limit twelve-tone scale" primarily because the word "the" implies that this scales is the only twelve tone scale. Originally it said "most common", which I toned down to "one way"; I don't think it's really altogether that common (though I have seen more or less this scale in a few places; Partch's tonality diamond comes to mind), and if any just-intonation version of the twelve tone scale is "most" common it would be the Pythagorean. Is the objection to the term "a" or "five-limit" or both? I added the five-limit label because I thought it was more descriptive.

I would not object to calling this section "the twelve tone scale" if it included one or two other examples of just tunings of twelve tone scales; in this case "the twelve tone scale" becomes a less specific entity to which the various alternatives belong. However, with only one example, I think it sends the wrong message.

Secondly, I don't think recourse to harmonics is useful. The harmonic series exists because its frequencies are all in integer relationships and can therefore physically reinforce each other; it is correct to say that the harmonic series is stable because its tones have integer relationships, but not to say that integer relationships are stable because they appear in the harmonic series. One is the cause of the other. The tones used in a JI scale need not have harmonics; chords of pure sine waves under just intonation have the same stability of sound that other tones do. There are a large variety of instruments whose tones do not contain integer harmonics as well (and "just" tuning on these instruments becomes a bit of an art, sometimes you do want to adjust more to the warped harmonics, some times you want to be closer to integer relationships, but it depends on the situation). - Rainwarrior 03:35, 4 July 2007 (UTC)

The section talks about "a" tuning for "the" scale, which is quite appropriate. If you want more than one tuning, feel free to add one (but remember that the article is about "just" tuning).
Harmonics are the real reason for using a just tuning. It makes two notes having some common harmonics. Of course expressing this in integer numbers is an idealisation, because many real instruments do not produce exact harmonics. Even so, then the tuning is adjusted to the harmonics, not to the integers.
−Woodstone 07:54, 4 July 2007 (UTC)
I don't quite agree with you on the nature of harmonics versus just intonation, but I can see your point and it's valid enough; I'll leave it alone. I'll add a little bit of information about the Pythagorean scale and try to make this "a" or "the" business a little more explicit in the article (after which I won't feel the need to reduce the section to "a"). - Rainwarrior 03:22, 6 July 2007 (UTC)
And sorry if my reversions seemed "blunt" but I felt I had good reason for making the amendations I did, and you had undone them without any argumentative edit summaries. With a comment like "indicate some background of why this way" I don't understand why my changes are being undone, so I feel like reinstating them. I wasn't trying to be rude or stubborn, I just didn't see any reason provided why I shouldn't have made those changes. Sorry. - Rainwarrior 04:18, 6 July 2007 (UTC)
I have added a table for the pythagorean tuning. And sorry as well. I had not seen your earlier comments here, when I did my "blunt" reversal, so I thought your reversal to be blunt. Glad to cooperate. We can prettify the table if desired. −Woodstone 12:36, 6 July 2007 (UTC)
That's a good addition. The table makes it easier to compare them. - Rainwarrior 04:00, 7 July 2007 (UTC)

[edit] Tuning samples

I just noticed and listened to the tuning samples. The first two, I suppose, are pure sine waves and give a very clear impression of the differences. The third one, sounds like a live recording of a piano. It is not clear how much of the difference is caused by mistuning and how much by an intentional different tuning system. Could someone convert the third example to a sine wave? −Woodstone 12:29, 13 October 2007 (UTC)

It's not a live recording of a piano, it's a sample of a piano. No mistuning is possible unless I calculated the playback ratios incorrectly, which I don't think I did. - furrykef (Talk at me) 23:36, 13 October 2007 (UTC)
Nevertheless, because it's a sampled piano, it contains the non-harmonicity inherent in pianno strings, disturbing hearing the tuning differences in isolation. Sines are better suited for that. −Woodstone 09:45, 14 October 2007 (UTC)
Wouldn't a sawtooth wave be better than a sine wave? —Tamfang 07:50, 20 October 2007 (UTC)
For some purposes, "pure" sinusoids might be best, but for general listening I believe you are right. Even if band-limited, sawtooth, square, triangle will all have extra harmonic content to help with perception of intervals. __Just plain Bill 22:24, 20 October 2007 (UTC)
I think a filtered saw wave is more or less ideal for this purpose. - Rainwarrior 02:49, 24 October 2007 (UTC)
Is that a term of art? There are many ways to filter a wave. —Tamfang 06:47, 28 October 2007 (UTC)
Well, I propose this. Let's just have two sound clips in the article, one with a sawtooth wave and another playing the same thing with the piano sample. I see your point about the purity of the harmonics, but I still like having the piano example because it's more like what the listener will hear in practice... most music isn't played with either sine or sawtooth waves. ;) Of course, I can expand the piano version to contain more, so you could hear the individual notes in succession before each combined interval. - furrykef (Talk at me) 19:40, 24 October 2007 (UTC)
In very rough terms, a sine sounds like a flute, a sawtooth like a violin, and a square-wave like a clarinet. In practice, a piano note can never be sustained the way winds or strings can do. I like the filtered sawtooth idea the best; using a piano tone to demonstrate just intonation just seems wrong to me, but that's my personal bias showing. ;-) __Just plain Bill 22:04, 24 October 2007 (UTC)
Again, I tend to agree with Bill here. The tone of a piano is inherently flawed, harmonically; in addition to the detuned beating of three strings, the overtone series is skewed by inharmonicity (e.g. we get the stretched octave phenomenon). It is not an instrument that should be used to teach just intonation (it is not a good instrument to teach proper tuning on in general). I said that the saw wave is ideal because it has a very regular and full harmonic series (unlike square or triangles which are missing even harmonics, and sines which have none). - Rainwarrior 03:30, 25 October 2007 (UTC)
Yes, but I think there is still a need to illustrate the difference in practice as well as in theory. Somebody listening to sawtooth examples can easily be left wondering if it would make a difference with real instruments. In fact, it was the very process of creating the sound clip with the piano sample that fully convinced me that just intonation is worth pursuing in the first place! - furrykef (Talk at me) 10:47, 25 October 2007 (UTC)
I don't know why you think synthesizers aren't used "in practice", but I'd point out that there are plenty of instruments that do not have fundamentally flawed harmonic series' that could be used (brass or woodwinds, for instance). - Rainwarrior 16:25, 25 October 2007 (UTC)
Synthesizers are used in practice, yes, but usually not pure sawtooth waves unless you're making chip tunes or you're going for effect. I think I misunderstood what was meant by "filtered"... so I presume you mean something like the typical output of a subtractive synthesizer (minus any chorus effects and whatnot, of course)? - furrykef (Talk at me) 12:11, 27 October 2007 (UTC)
"... individual notes in succession before each combined interval" is a fine thing, I believe. With a reasonably realistic envelope, a sawtooth won't sound all that strange, especially since synthesized sound is a big part of modern musical experience. __Just plain Bill 16:15, 25 October 2007 (UTC)

It seems we never got an answer to what wave form the first two examples in the article are made of. Anyone knows? Or can check? −Woodstone 20:23, 25 October 2007 (UTC)

Those are definitely sine waves. - furrykef (Talk at me) 12:11, 27 October 2007 (UTC)

[edit] Just Intonation Propaganda

First off, I'm a huge fan of JI. I think the possible sonorities that can be produced are just so much richer than a bland ET world. I wish more people paid attention to subtleties of tuning.

That said, I'm also a realist. A few of the claims on this page, however, are not realistic. Claims like most a cappella vocal ensembles naturally gravitate toward JI and that string quartets accidentally stumble upon JI as stated later in the article -- these are simply not backed up by any evidence. Numerous acoustical studies (as can be seen in many articles from places like the Journal of the Acoustical Society of America to various journals of music perception) demonstrate that performers and listeners are in general able to accept much wider discrepancies in tuning than the JI community is willing to admit. (Usually +/- 20 cents or so is heard as perfectly acceptable within musically syntactically appropriate contexts, e.g., leading tones get detuned, etc.) While it is true that some performers and ensembles explicitly try to lock in on particular tunings, in the practice of the _vast_ majority of musicians (and even the vast majority of professional ensembles), expressive tuning is simply a fact of life.

Similarly, toward the end of the article there is an implicit claim that ET is harder to tune than JI. Again, there is simply no evidence that I have seen in the literature to support such an assertion. While it does require, say, a much more complex system to tune ET on a piano or harpsichord with any precision than many meantone or just tunings, that's not relevant to the subsequent statement that performers have to suppress urges to move toward JI, which would only apply to variable pitch instruments. On variable pitched instruments, it is equally difficult to attain almost any tuning with great precision unless the performer specifically trains to approximate a certain tuning (JI, ET, or otherwise). The one exception that the claims in the article may be hinting at is that while playing/singing a long sustained sonority, it is probably easier to "lock in" on something close to JI -- if one is listening properly -- but even those tendencies become fairly weak when playing/singing in a large ensemble with many people on a part.

In sum, there are a few times in this article where we get JI propaganda instead of the facts that have been established through empirical tuning studies. I wish the world worked this way, and indeed I bought into the propaganda myself for a few years... but then I read enough studies to convince me that despite thousands of years of theorists (starting with the Pythagoreans) claiming that JI is somehow more natural or something, the evidence implies that microtuning inflections are hugely variable in practice. Except in the case where performers are consciously trying to "lock in" on a sustained vertical sonority, there just seems to be no evidence that JI is this ideal thing that we naturally fall into. 65.96.183.164 (talk) 00:41, 16 March 2008 (UTC)

I agree that "those tendencies become fairly weak when playing/singing in a large ensemble with many people on a part." No argument that it is far more difficult to get a tight performance out of a large group, but the case is different with e.g. barbershop quartets or other close-harmony a cappella groups, or with well-schooled string quartets. Once someone hears the difference, nothing is stopping them from more naturally gravitating to JI rather than than to some temperament or other.
I'll look askance at academic "studies" that claim +/- 20 cents is "perfectly acceptable," if such exist. Question their methodology, inquire what simplifications they made to make the problem tractable to "measurement." Matching monophonic pitches "pretty closely" outside of a musical context is one thing, tuning chords on the fly is quite another. It's pretty easy to hear the difference between JI and ET with those chords; again, in the context of a smaller group that are actually listening to one another. People who may not explicitly name the difference may still easily perceive it, even if subliminally, as a difference in quality of sound.
All this is pure supposition, based on nothing more than my own common sense, so I'll leave it on the talk page. I'll be happy to read and comment on relevant on-line references regarding human pitch perception, but haven't got the time to go chasing them. __Just plain Bill (talk) 16:24, 8 June 2008 (UTC)

[edit] Just major and Just minor

In many contexts one finds two sets of ratios given -- a just major and a just minor set.

This corresponds (I think) to Mozart's instruction and practice (see http://tonalsoft.com/monzo/55edo/55edo.htm ) of working with a 20-tone scale -- roughly speaking all 'black notes' are split into two with the sharps being flatter than the flats.

In pragmatic terms this corresponds to the fact that in common era western music one commonly finds in the same piece of music a g sharp and an a flat (eg Beethoven last sonata 2nd movement 1st page). A just minor makes the a flat sound right and the g sharp wrong, whereas a just major has the opposite effect.

So coming to the point -- can we have a table like the following? Im putting in the ratios as I understand them

C scale

Note Ratio
c 1
c# 25/24
d_ 16/15
d 9/8
d# 75/64??
e_ 6/5
e 5/4
f 4/3
f# 45/32
g_ 64/45
g 3/2
g# 25/16
a_ 8/5
a 5/3
a# 125/72??
b_ 16/9
b 15/8
c 2/1

Please pardon me if this is complete layman-speak :-)

Rpm13 (talk) 17:24, 16 May 2008 (UTC)

[edit] cleanup needed here

this wiki is going to need some cleaning up, as it makes statements as if they are "givens" when they are not. For example, in the section on Pythagorean tuning, the statement about "simple ratios" doesn't make sense, for it is patently not the complexity of the ratio that creates dissonance or even distance from Just Intonation. A number far more complex than any Pythagorean interval, some irrational for example, could describe an interval some miniscule distance from a Just major third. Let's say this immensely complex number describes an interval only .001 cents from 5/4. Obviously this is going to sound more "Just" than a far simpler ratio say 8 cents away. Directly equating simplicity of number and purity of musical intervals doesn't make sense, and speaking as if this equivalence is real only serves to further marginalize the Just Intonation community.

Frank Zamjatin (talk) 13:09, 5 June 2008 (UTC)

It sounds to me like your concern is with some specific statements being unsourced. (Making sense or being true isn't nearly as important.) So do you think it makes sense for you to tag these statements made as "givens" with {{fact}} or something of that flavor? If the statement can be reliably sourced then it remains. If it can't be, then it must be removed. --Ds13 (talk) 18:07, 5 June 2008 (UTC)

But there needs to be some common sense- for example, you could "reliably source" the "fact" that mink don't suffer when they're killed to be made into fur coats. No kidding, check out a 1950's World Book encyclopedia, 1959? if I recall correctly. You can find sources for all kinds of obviously nutty ideas and indications of having failed 7th grade math classes, but it seems to me that it would be common sense and common courtesy to indicate both a source and the fact that it is a "point of view". Calmly stating that for example 1.25000123 is further away from 1.25 than 1.258 is because "it's a more complex number" (which is the kind of thing the statement in question clearly implies) needs some qualification. Frank Zamjatin (talk) 15:41, 11 June 2008 (UTC)

Fair enough. I still suggest jumping in and tagging specific statements. To get started, I just tagged one that sounds POV: "only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure" and one that, as you say, is a bit dubious numerically: "Major thirds, for instance, receive the rather unstable interval of 81/64, well sharp of the preferred 5/4". Let's see what happens. --Ds13 (talk) 16:18, 11 June 2008 (UTC)