Talk:Interval (music)

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The neutrality of this article is disputed.
Please see the discussion on the talk page.

[edit] Minor scale, minor intervals

I don't like the verbiage "in minor scales the minor intervals are introduced" because for one thing it creates the false impression that minor scales are minor because they have minor intervals. THis is not the case, minor scales are minor because they have a minor third. Moreover, the major scale has within itself a number of minor intervals, just not between the tonic and any other note. I like the pictures though, so I'm a bit loathe to go changing anything.JFQ

Indeed the major and minor scales have exactly the same number of major and minor intervals as do all the seven modes. -- User:Karl Palmen

The minor scale introduces minor intervals from the tonic, maybe? I've never understood why it's a minor second and not a diminished second -- after all, it doesn't occur in the minor scale. -- Tarquin

It's because only flatted perfect intervals (and occasionally the flatted minor seventh) are called diminished. Major intervals invert to minor intervals, diminished intervals invert to augmented intervals, and perfect intervals invert to perfect intervals.

And there's a minor second between the second and the third in a natural minor scale.

And it's minor because it's got that dark, um, minor quality. yeah. Oy, this stuff is so subjective. - JFQ

Minor scales differ from Major scales only in which note is the tonic. Both use exactly the same set of notes. When this set has no sharps or flats, one gets A minor or C major. - User:Karl Palmen

Correction: the difference between minor and major scales is whether they have a minor or major third above the tonic. Major scales include Lydian and Myxolydian as well as Ionian/modern major, minor scales include Dorian and Phrygian as well as Aeolian/modern minor (natural minor); besides there are the harmonic and melodic minor scales which use different scale material from natural minor (a minor on the white keys of the keyboard).

In a broader sense, minor and major scales can include any modal scales, no matter what their scale material, according to whether they have a major or minor third above the tonic. This includes all sorts of Eastern and Indian scales, for instance. frodolives

This does not contradict what I have said above. Indeed all seven modes use the same set of notes and differ only in which of these is the tonic or first note. - User:Karl Palmen

[edit] Consonance and dissonance

The "concordant and discordant intervals" bit should be moved to a separate article that deals with the subjects or consonance and dissonance at a more general musical level (bring in chords, talk about the history of dissonance, etc.).

We don't have any articles on consonance or dissonance, and I'm not sure what to call a page like that. It seems silly to make an article called "dissonance", becuase that implies that there should be a similar "consonance" article. Getting redirected to the exact opposite thing would be confusing. -- Merphant

I guess "consonance and dissonance" would be the way to go. --Camembert

The bit about discordant intervals being the basis for suspensions is confusing (or confused) - one of the most common suspensions is that of the 4th dissolving into the 3rd, and it's dissonant strictly in the harmonic context - the 4th in itself is a concordant interval.

Personally I see no problem with having an article each on consonance and dissonance, with a link to the respective opposite. But then I haven't been around here for very long.

Also be careful not to confuse "consonant-dissonant" with "concordant- discordant" - I'm not entirely sure what exactly the difference is but I am very sure there is one. <<sigh- off to do some research>>frodolives

As far as I know, "concord" and "consonance" are synonyms, but I could be wrong, and I'd be interested to hear of a distinction (maybe there was some sort of distinction made by Renaissance theorists?).

The bit on concords and discords is certainly simplisitic and needs work (if not on this page, somewhere else). We've got to be careful though, because the purported relative consonance of intervals has varied from theorist to theorist, and also, as you suggest, because the perceived disonance of an interval is very much dependent on its context. I think it might be better to go into detail on this on consonance and dissonance (created since the above talk, the first is just a stub, the second is itself a bit simplistic). --Camembert

By the way, I still think one page on consonance and dissonance is better than a page on each, as I don't see how it's possible to talk about one without reference to the other. --Camembert

[edit] Rewrite

This article, as it stands, I feel, is not only unclear about the various "diatonic intervals" discussed, because it does not reference them to other ways of labelling intervals, and is thus heavily biased towards tonal music. This is part of what led to the confusing, "in minor scales the minor intervals are introduced." I have recently added a few sentences to the beginning (about measuring intervals in post-tonal theory) that changed the article from being entirely non-NPOV, but I hope to do more.Hyacinth 23:35, 27 Dec 2003 (UTC)

I just did a complete rewrite. I hope that my version is, most importantly, more consice, but I also hope that it is more clear and makes it easier to understand and compare the different intervals. I attempted to keep every subject and all pertinent information from the original article, while keeping out discussions of things such as dissonance (and resolution and...), which are properly dealt with and understood in their own articles. Hopefully I have done this switch (is it a move?) appropriately.Hyacinth 10:20, 10 Jan 2004 (UTC)

Original article: http://en.wikipedia.org/w/wiki.phtml?title=Interval_%28music%29&oldid=2124198

[edit] Source

source for formula definitions: Basic Atonal Theory by John Rahn.

[edit] Too many theoretical orientations in one entry

I have just made some changes to this very technical and complex entry. Frankly, I doubt that the entry as it is presently structured and filled out would be much use to anyone who is not already a skilled music theorist. I would recommend splitting it (or having parallel entries, cross-referenced) so that for example discussion of intervals in a purely traditional Western tonal context could be "quarantined", to achieve some semblance of simplicity and comprehensibility. --Noetica 06:21, 5 Feb 2005 (UTC)

I agree. IMHO, those unfamiliar with the subject will find this quite difficult. I am half convinced that it will be hard to refactor with so many threads. Marty Heyman 1:19PM 14 Feb 2005 (PST)

Well, I'd be happy to initiate a separate "quarantined" treatment of intervals in the Western tonal tradition myself. But I'd need recommendations about how to name it and fit it into the existing nexus of articles. (Still new here!) --Noetica 22:36, 13 Feb 2005 (UTC)

See: Wikipedia:Requests for comment#Article content disputes.

This is a common complaint: In my opinion most articles on Wikipedia are useless for those who are not already experts. But I do not see any reason to require all music articles to be comprehensible to the least educated reader and not requiring it of, for example, scientific or mathematical articles.

I feel strongly that there should be a comparative presentation of similar intervals in various theoretical or tuning systems, somewhere. Whether it is at Interval (music) or "Comparison of musical intervals" or some other title I do not know. There also should be "here are all the ratios", "here are all the integers", "here are all the diatonic intervals", etc., presentations. Hyacinth 17:28, 9 Mar 2005 (UTC)

Hyacinth, I don't know that it's a dispute so much as an exploration of ways we might all work together for improvement. Of course it's clear to me that you've done a lot of excellent work here, and elsewhere in music articles. Such fine detail! I may have some more things to say, myself. But I'm not sure how much time I can devote to this, having had slabs of painstaking work summarily (and quite obviously without thought, I say) dismissed in a couple of places. One has only so much time. Let's hope there is some good dialogue as a result of your registering this article for comment as you did. --Noetica 00:45, 10 Mar 2005 (UTC)

Agreed, this is not a dispute, and if RfC doesn't work I'll simply post a notice to some music editors talk pages. Hyacinth 02:03, 10 Mar 2005 (UTC)

[edit] Wahoofive's editing

Wahoofive, I appreciate much in your recent bold rewrite. Something had to be done. I think it is regrettable that you did not respect the fact that Hyacinth, who has had a significant part in developing the article, had issued an RFC for the article. In such circumstances we may reasonably view your action as high-handed, to say the least.

There are some incorrect or questionable details of content in your work, and some expository infelicities. To illustrate, let's look at your Interval qualities section:

Modern musicians usually define the interval quality based on the number of semitones between the two notes, roughly equivalent to the number of piano keys (black or white) between them. This quality is affected by the clef, key signature, and any accidentals applied to one or both of the notes. Note that the number of semitones does not by itself determine the interval name; the number of the interval must be determined from its notation (or inferred from its diatonic usage) first.

Why modern musicians, generally? This is Eurocentric, if it is left unqualified. And it is music theorists who do such defining, isn't it? Many musicians wouldn't know a fifth from a fire engine. Then there is, for the beginner, the possibility of confusing the two meanings of number, here. Last, the beginner may have trouble finding your meaning in the material after the semicolon.

Unison, fourth, fifth, octave. These intervals can be described as perfect, augmented, or (except for unison) diminished. Perfect fourths are five semitones, perfect fifths are six semitones, perfect octaves are twelve semitones. Perfect unisons are the same note, so are zero semitones apart. In each case, an augmented interval is one semitone larger, a diminished interval one semitone smaller. The word "perfect" derives from the presence of these intervals in the overtone series.

Described may be misleading, since the beginner may get the fleeting but distracting impression that these descriptions are interchangeable. The meaning we need is further qualified. Perhaps something like this would be better: Each of these intervals occurs in three varieties,... Then, note that the exclusion of the diminished unison is controversial (since the augmented octave arguably has it as its inversion). Better to say, therefore, possibly excluding the unison. Then, better to say a perfect unison occurs between notes at the same pitch, which in the context covers all that is required for a continuation to so it is zero semitones (for logical and syntactic consistency with treatment of the other intervals, which should also be presented in the singular). Then, it would be better to use closer (or perhaps narrower) and wider instead of smaller and larger, and you should say closer or wider than what, to help the beginner, who may have lost the thread after pondering over the perfect unison. So: An augmented interval is one semitone wider than its perfect equivalent; and a diminished interval is one semitone closer than its perfect equivalent. And your explanation of the term perfect is open to question, since it is not only perfect intervals that occur in the overtone series. That whole matter might bring in many irrelevancies.

Second, third, sixth, seventh. These intervals can be described as major, minor, augmented, or diminished.
- Major seconds are two semitones, also called a whole step, minor seconds are one semitone, also called a half step.
- Major thirds are four semitones, minor thirds are three semitones.
- Major sixths are nine semitones, minor sixths are eight semitones.
- Major sevenths are eleven semitones, minor sevenths are ten semitones.
- In each case, the augmented interval is one semitone larger than the major interval, and the diminished interval one semitone smaller than the minor interval. These intervals are rarely used; the only ones which occur with any regularity in common-practice music are the augmented sixth and the diminished seventh.

Similar strictures apply here as above: regarding describe, and all the rest. And then, what do you mean by are rarely used? Rarely used how: horizontally or vertically? The augmented second occurs very commonly, both melodically and harmonically. When we consider the vertical relations introduced by unessential notes like chromatic passing notes (or passing tones) in one part against a stationary second part, many of the other intervals occur. The same may be said for melodic occurrence of the allegedly rarely used intervals, like the augmented unison between C4 and C#4 in the melodic progression C4-C#4-D4. Even the diminished fourth occurs frequently (and though this is not said to be rarely used in the rewrite, it may be worth noting, to make a sort of a fortiori case). Using E4 and Ab4, for example: melodically in C4-E4-Ab4-G4-F4 and the like; harmonically in versions of the dominant 13th, like C3 Bb3 E4 Ab4 (in f minor). Why is nothing said about this matter of the relative frequency of occurrence for the first group of intervals (unison, fourth, fifth, octave)? The rarity of the augmented octave, both vertically and horizontally, might with profit have been noted. And then, why do you explain the term perfect, above, but leave the user mystified about the terms major and minor as applied to intervals?

It is possible to have doubly-diminished and doubly-augmented intervals, but these are quite rare.

The term perfect third is very occasionally used by early music experts to denote the major third interval when tuned to an exact 5:4 frequency ratio.

For grammatical propriety, the wording should be when it is tuned to. But there is a case for excluding this information from here anyway, if you are aiming to assist beginners. It belongs to a different and quite recherché theoretical schema.

Octave equivalency describes the perception that octaves are the same note, that the same notes repeat throughout the pitch range. Thus C and C', C5 and C3, and C and any C any number of octaves above or below, are all the same note or pitch class.

It doesn't describe it. It is a term used for the phenomenon. And, while Hyacinth, Opus33, Antandrus, Camembert, Stirling Newberry, you, and I all understand what is meant when you write octaves are the same note, the beginner may be totally confused by it! They have just been told that an octave is an interval between notes; now octaves are notes!

Much more could be said; but I have worked merely to illustrate the complexity of the task. I don't hold tight to some of the criticisms I make above. I just want to show that it is no easy matter to present basic music theory to neophytes, and this is why discussion is imperative first of all – especially when discussion has been explicitly called for, through a properly established channel. --Noetica 09:22, 12 Mar 2005 (UTC)

I apologize for not going through proper channels. I couldn't think of any way to discuss such a major change on this Talk page. Is there a scratch section where I could have presented this as a draft for comment?

Your editorial criticism is mostly valid. Please feel free to improve my work, of course -- unless my offense is so serious that you might revert to the previous version. Some of the sections you critiqued, such as "octave equivalency," were lifted verbatim from the previous version, btw.

The issue with the previous version is that it presented conventional interval names (such as perfect fifth) alongside specialist jargon such as frequency ratios, cents, and Allen Forte's interval-class sets. Even early-music experts, who talk happily for hours about tuning systems and syntonic commas and the difference between 10:9 and 9:8 whole steps, still refer to perfect fifths and major seconds -- that is the standard language for describing intervals. Therefore it should be described in detail first, before the more-specialized terminology.

As for non-Western music, it would be great to get some input on that. I'd speculate that most probably use simple-proportion tuning (e.g. 5:4) for the same intervals, but don't have to temper them since their music doesn't modulate or become chromatic. Much of the difference will just be language. But see slendro and pelog, and pseudo-octave. Wahoofive 17:19, 12 Mar 2005 (UTC)

I don't doubt that your work was well intended, Wahoofive. And I don't say that it worsens things! As for lifting material verbatim from the earlier version, perhaps it is fair to criticise your work for the content of that material too, because in a major rewrite such as you claim to be doing you may be taken to endorse that content if you keep it.

Others can advise you better than I can about "scratch pages" and the like.

I completely agree with you concerning all that specialist jargon, and the mixing of theoretical orientations. I drew attention to it in the first place.

I still think it would be good to isolate intervals in traditional Western tonal theory, and to treat them thoroughly first (no, with some general common introduction preceding it, because the general notion of the interval needs to be presented lucidly as a preliminary to everything else). That traditional Western system is hard enough to give a proper treament, without complications from outside the system.

So far the article does not treat the category diatonic interval accurately. In the traditional system, a diatonic interval is one that occurs in some or other major or harmonic minor scale. Its proper opposite is chromatic interval (a term that does not occur in the article at all, now or before the rewrite). The augmented unison, third, sixth, and octave are all chromatic intervals, but they are classified among the diatonic intervals in the article. Getting this all right is not easy. Even the New Grove does a poor job. It has, in the article Chromatic: "An interval is said to be chromatic if it is not part of a diatonic scale (e.g. F–F#, B–Eb)." This definition is misleading because it fails to exclude the melodic minor scale. [Finally fixed MUCH later (shame on me): O, also B-Eb does belong to a diatonic scale (C harmonic minor)! Shame on you, New Grove! – Noetica 22:02, 23 November 2006 (UTC)] A beginner might think that the interval Bb-B is not chromatic because each of the components can be found somewhere in either the ascending or the descending form of c melodic minor. Do a quick Google search on "chromatic interval" if you want to see bizarre "explanations". I found some gems here, where not one of the contributors gets it right:

www.ibreathemusic.com/forums/ archive/index.php/t-4818.html

That's the sort of confusion we have an opportunity to sort out. However, having contributed what clarification I have so far, I shall be playing no substantial part from now on, because I have decided to withdraw from involvement in Wikipedia soon. I'll put a note about this on my User page shortly. --Noetica 23:09, 12 Mar 2005 (UTC)

I'm not sure what you mean by "it would be good to isolate intervals in Western theory first." Isn't that what I've done? Or maybe you mean we should move the "labelling intervals" section lower down (or even eliminate it), which would be fine with me.

I don't see why the distinction between "diatonic" and "chromatic" intervals is important. Why does it matter whether diminished fourths appear in some diatonic scale or not? Maybe the reason these reference books don't treat it well is that the topic isn't that central to music theory. I'm not happy with the name "diatonic intervals" for the second section and would be happy to hear of suggested alternatives.

--Wahoofive 23:27, 12 Mar 2005 (UTC)

I didn't say you didn't do it! I simply intended that, however things work out in the end, after any re-working that is to come, it would be good to have things structured that way. Nothing I said contradicts this. Perhaps a separate article is the way to achieve such a separation. Perhaps several separate sub-articles, with a short general article at the hub.

You may not think that the distinction between "diatonic" and "chromatic" intervals is important; but it is nevertheless a traditional part of the content the article is intended to cover. The confusion evident in discussions such as the one I drew to your attention is evidence that it is a generally ill-covered topic, and one that could be sorted out here. But as things stand, the article is quite misleading in that area (since it wrongly classifies some intervals). Better not to add to the confusion!

The reason some reference works get such things wrong is simply bad thinking, bad expression, and bad editing, in an area in which fewer and fewer people are competent (and the evidence for this last is easy to find!). One might as well say "who cares about the Greek chromatic tetrachords?" Well, an encyclopedia that addresses such things must care! And Wikipedia is such an encyclopedia. --Noetica 00:09, 13 Mar 2005 (UTC)

I haven't read all of the above discussion, but looking at the difference, User:Wahoofive, I think your edit lost a lot of information. Unless you have an immediate proposal for another location for that information, I suggest a revert. For example, all explination of simple intervals was removed, now they are just mentioned and linked to. In my opinion this isn't the kind of trimming that is useful for the lay reader. Wikipedia:Explain jargon.

As another example, almost all of the information below is lost:

Unison: The ratio of 1:1 is a unison (specifically, a perfect unison; there are other variants), two notes at the same pitch. In integer notation it is a 0 and is also zero cents. It is the simplest and most consonant of intervals.

I'm fine with people boldly reinterpreting, rewriting, and clarify what I rewrote in the first place, but I don't think that removing the information is clarifying. Hyacinth 07:32, 13 Mar 2005 (UTC)

The edit did ad some useful information. For example:

"In musical set theory intervals are numbered according to the number of half steps, from 0 to 11; see integer notation. This system ignores enharmonic intervals and is only used for atonal music. A similar system is interval classes, which are similar but consider each interval and its inversion to be equivalent, and use only the smaller number; thus the largest interval class is 6."

That first sentence is not explained (only implied), and should be kept no matter what. This system is not, however, used only for atonal music. Next sentence is good. Hyacinth 07:37, 13 Mar 2005 (UTC)

I reverted, added the above information, and just reorganized the Labelling intervals section to clarify and address Wahoofive and Noetica's concerns. Now, instead of this overwhelming unbroken stream of unexplained ways in which intervals may be described, the ways are broken into theory and use. Please comment and revise. Hyacinth 08:22, 13 Mar 2005 (UTC)

Jargon that may need to be better explained includes consonance and dissonance (currently explained as "stable and unstable" respectively), such as Wahoofive's text:

"These terms are relative to the usage in different compositional styles. In atonal music all intervals (or interval classes) are considered equally consonant. In 16th-century usage, perfect fifths and octaves, and major and minor thirds and sixths were considered consonant, and all other intervals dissonant. In the common practice period, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously thought to be dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16th-century practice continued to be taught to beginning musicians throughout this period."

Enharmonic intervals, Wahoofive's text:

"Two intervals with different names which have the same number of semitones are called enharmonic. For example, a diminished fifth and an augmented fourth both have six semitones. This particular interval is also called the tritone since it is three whole steps."

I don't think it is necessary to explain the reason's and differences behind just intonation and equal temperment in detail here, and I think their qualities are nicely described here (if not explicitly). Hyacinth 08:40, 13 Mar 2005 (UTC)

I propose three changes to the article as it now stands:

Rename "diatonic theory" to something more reflective of the fact that the vast majority of musicians consider this standard terminology for describing intervals
Move this section above "just intonation" for the same reason (I don't have an issue with the ratios being first, but David Cope and Hindemith are of interest only to specialists)
Re-introduce the graphics from my version.

I also have a strong objection to the way "consonant" and "dissonant" are used in the list of intervals. To refer to perfect fifths and fourths as both equally consonant is to introduce a physicist's bias, since the majority of musicians wouldn't use the terms in that way. Hindemith's theories are almost a historical curiosity now; I don't see any evidence that they've gained traction in music teaching in general.

--Wahoofive 04:25, 14 Mar 2005 (UTC)

Hindemith and Cope are of interest only to specialists, but their theories (interval root, etc) are not. More importantly, this article was not structured according to Hindemith's theories, which I have only a passing acquaintance with. Hyacinth 20:49, 20 Mar 2005 (UTC)

I think changing the title of the section "Diatonic theory" to something which reflects many musicians limited knowledge of that theory as the only theory is uninformative and not NPOV. Hyacinth 20:49, 20 Mar 2005 (UTC)

As you can see, I moved Cope and Hindemith down into the "consonant and dissonant intervals" section. Cope definitely belongs there, since "interval strength" is a concept related to consonance and dissonance, but the "interval root" section maybe doesn't. It didn't really belong under "just intonation" where it was before, though, since this theory applies to equal-tempered intervals as well. Perhaps it belongs as a sub-section of "Inversion," since that seems related somehow.

I'll concede your position on the equal importance of the diatonic names of intervals, but the phrase "diatonic theory" still seems a little abstract to me. --Wahoofive 21:05, 20 Mar 2005 (UTC)

[edit] Generations of intervals

The following seems to belong in its own article describing whatever theory it is:

The intervals can be divided into five "generations", which correspond to negative powers of two:
Zeroth generation (1+2⁻⁰): P1, P8.
First generation (1+2⁻¹): P4, P5.
Second generation (1+2⁻²): M3, m3, M6, m6.
Third generation (1+2⁻³): M2, m7.
Fourth generation (1+2⁻⁴): m2, M7, π.

Each successive generation is more dissonant than the previous one.

Here is the derivation of each generation from the previous one: Start with the octave's ratio, 2:1. Multiply each of its two numbers by two, yielding 4:2. Then stick the missing number in the middle, which gives 4:3:2. This breaks up into a pair of ratios — 4:3 and 3:2. The minor one is 4:3 and the major one is 3:2. These are the perfect fourth and the perfect fifth, respectively, and they are the first generation.

Now take the perfect fifth's ratio, 3:2. Multiply each of its two numbers by two to obtain 6:4. Then stick the missing number in the middle, yielding 6:5:4. This breaks up into a pair of ratios — 6:5 and 5:4. The minor one is 6:5 and the major one is 5:4. These are the minor third and major third, respectively. Their inversions are 5:3 and 8:5, which are the major sixth and the minor sixth, respectively. So these are the second generation: M3, m3, M6, m6.

Now take the major third's ratio, 5:4. Multiply each of its two numbers by two, which yields 10:8. Then stick the missing number in the middle, giving 10:9:8. This breaks up into a pair of ratios — 10:9 and 9:8. The minor one is 10:9 and the major one is 9:8. Both of these are whole-tones, i.e. major seconds. The inversion of 9:8 is 16:9, a minor seventh. So these are the third generation: M2, m7.

Now take the whole-tone's ratio, 9:8. Multiply each of its two numbers by two, giving 18:16. Then stick the missing number in the middle to obtain 18:17:16. This breaks up into a pair of ratios — 18:17 and 17:16. The minor one is 18:17 and the major one is 17:16. Both of these are semitones, i.e. minor seconds. The inversion of 18:17 is 17:9, a major seventh. The last interval is the tritone. The tritone is ideally equal to the square root of two, which is irrational, but can be approximated by adding a semitone to a perfect fourth:

${4 \over 3} \times {17 \over 16} = {1 \times 17 \over 3 \times 4} = {17 \over 12}$

${4 \over 3} \times {18 \over 17} = {4 \times 6 \over 1 \times 17} = {24 \over 17},$

which is the inversion of 17:12. 24:17 has the same denominator as 18:17, and 17:12 has the same numerator as 17:16. So these are the intervals of the fourth generation: m2, M7, π.

Agree. No clue what theory that is, though. --Wahoofive 04:05, 14 Mar 2005 (UTC)

[edit] Diatonic

I do not see how "nomenclature" is less abstract than theory, it certainly is less readily understandable. "Diatonic theory" is also terminology actually used, as in: Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1930190808. Hyacinth 23:30, 21 Mar 2005 (UTC)

[edit] Just vs Equal

I think that anything which is applied to equal tempered intervals only when those are considered as the just intervals they approximate should go in a just section where there is one. Hyacinth 23:32, 21 Mar 2005 (UTC)

The article in its current state is organized with naming conventions first, then analytical tools. That's why I moved interval root and strength to other places on the page. --Wahoofive 00:46, 22 Mar 2005 (UTC)

[edit] Outline

I propose that we create an outline we agree on, and then use that to guide collaboration on the article. Hyacinth 10:24, 27 Mar 2005 (UTC)

The following is the current outline:

1 Labelling intervals
 1.1 Frequency Ratios
 1.2 Interval number and quality
  1.2.1 Shorthand notation
 1.3 Names used in atonal theory
  1.3.1 Ordered and unordered pitch and pitch class intervals
  1.3.2 Interval cycles
 1.4 Cents
2 Comparison of different interval naming systems
3 Consonant and dissonant intervals
4 Inversion
 4.1 Interval roots
5 Other intervals
6 Sources
7 External Links

The only problem I see at first glance is that atonal theory is marked off as special. What is the corresponding description for the intervals used in atonal theory: what to atonal theory "interval numiber and quality" is to diatonic theory. The latter being privileged by being assumed. Hyacinth 10:39, 27 Mar 2005 (UTC)

Maybe "Interval numbers and interval classes"? I'd advocate subsection titles which reflect the naming systems, rather than what kind of music the names apply to, since the latter is pretty flexible. Allen Forte has applied set-class analysis to tonal music as well.

Should "Comparison of naming systems" be a subsection of "Labelling intervals" rather than a top-level section? And does "Cents" belong there? It's a measure of interval rather than a name, but maybe that's an irrelevant distinction.

Actually, let me back off and ask if you're suggesting making more fundamental changes in the outline, as suggested by your first sentence. My two items are just minor tweaks. I'm pretty happy with it as is, but I'd like to hear any concerns.

We could have a subsection at the beginning of (1) expanding on narrow and wide, vertical/horizontal, etc., if there's more to say about it.

We do need to make room for non-Western naming systems; I'm sure India (at least) has one.

One section that needs more attention is the "consonant and dissonant" section because it assumes simultaneous (rather than melodic) intervals. Melodic intervals have their own dissonance rules in some contexts. "Inversion" needs to address this also. --Wahoofive 22:09, 27 Mar 2005 (UTC)

Regarding cents, whether they are a label or a measure I think is mostly an issue of semantics. We could change Labelling intervals to "Labelling and measuring intervals" but it is common to refer to an interval by its cents alone ("an interval of 200 cents"). I think the consonance and dissonance section is a long as it should be, see consonance and dissonance. Hyacinth 00:55, 28 Mar 2005 (UTC)

I made some changes. Here is the outline now (I'm not sure whether to update one outline to save space or list each outline draft for clarity and continuity):

1 Frequency Ratios
2 Interval number and quality
 2.1 Shorthand notation
3 Pitch class intervals
 3.1 Ordered and unordered pitch and pitch class intervals
4 Cents
5 Comparison of different interval naming systems
6 Consonant and dissonant intervals
7 Inversion
 7.1 Interval roots
8 Interval cycles
9 Other intervals
10 Sources
11 External links

I like the new organization. It's not so much that C&D has to be expanded but it needs clarification about simultaneous vs. melodic intervals. I've got a couple of other minor adds that I'll put in. Oh, and better to keep the record of discussion. We can always archive it. Text storage is cheap. --Wahoofive 05:01, 28 Mar 2005 (UTC)

[edit] Pitch class notation

Do we need the mathematical notation in the set-theory stuff? It seems kind of pretentious to say you subtract two numbers, then use a formula to show

DIFFERENCE = MINUEND - SUBTRAHEND
INVERSE = SUBTRAHEND - MINUEND
ABSOLUTE DIFFERENCE = | MINUEND - SUBTRAHEND |

The distinction between <> brackets and () parentheses is difficult to see, too. It's much more clearly explained on the musical set theory page. I think we could just summarize it in text only. --Wahoofive 05:08, 28 Mar 2005 (UTC)

But its not mentioned on musical set theory. Hyacinth 05:52, 29 Mar 2005 (UTC)

All I meant was the different types of brackets are explained at Musical_set_theory#The_set_and_set_types. Thank you for removing the formulas. --Wahoofive 17:04, 29 Mar 2005 (UTC)

[edit] Tables

I've been making some tables over at User:Hyacinth/Interval tables and I would welcome comments. Hyacinth 08:24, 28 Mar 2005 (UTC)

I prefer the second version of the perfect/major/augmented table -- easier to read. Maybe an additional column with compound intervals?
I oppose any table simply dividing intervals into "consonant" and "dissonant" -- there's no theoretical basis or historical consensus on this. (To be sure, it illustrates Helmholtz's theory, but that doesn't strike me as important enough to merit a table.) The perfect fourth has been a bone of contention among theorists for years, since for practical purposes it's been considered dissonant for 600 years (in the West), yet it is a simple ratio. Bringing in non-Western music makes it even murkier.
- The box going from "most consonant" to "most dissonant," however, might be a useful tool for illustrating Hindemith's schema (if the arbitrary dividing line between "consonant" and "dissonant" is removed).
Of course I approve of the "comparison" box, since I made it. Maybe should include interval class (max 6) as an additional column. BTW, although you don't have to use the table template I did, I strongly encourage "CELLSPACING=0" on all tables for aesthetic purposes.
I'm not sure why you divided that box into separate lines. I hope this isn't part of a plan to revert to having separate subsections on each interval. They each have separate pages, which I'm fine with.
- What might be useful, though, is to have a list of enharmonics for each interval.
- I don't know if it's practical to have a comparison of different ratios, or to compare meantone and other tunings on this page. Maybe just a mention in passing, with references to the Tuning and various Temperament pages.

--Wahoofive 17:24, 28 Mar 2005 (UTC)

There is theoretical basis and a modicum of consensus on consonance and dissonance, with even the twelve tone and atonal text books discuss intervals and simultaneities degree's of tension. The amount of disagreement only argues for more specific tables. What does cellspacing=0 do? Hyacinth 01:47, 29 Mar 2005 (UTC)

cellspacing=0 makes a single line between cells instead of a double line. Try it in the sandbox. Degrees of tension, or a table showing a continuum from consonance to dissonance, is fine with me; I'm only opposed to a boundary saying "this list of intervals is consonant and this other list is dissonant," unless it's attributed to a particular theorist. --Wahoofive 17:01, 29 Mar 2005 (UTC)

I don't like the consonant-dissonance thing too much. The only intervals that I consider really dissonant are the half-step (minor 2nd) and the tritone; the major seventh can also be dissonant (being a half-step less than an octive). In terms of the full-step (major 2nd), I don't consider this a dissonant interval when used in a major key, but consider it dissonant when used in a minor key (and the fact that minor penatonic scales drop the initial full-step seems to reinforce this). What makes an interval sound dissonant to a listener has as much to do with what key the listener thinks the song is in as much as the interval itself. Samboy 08:02, 9 Apr 2005 (UTC)

This and related articles are not about your or my own personal perceptions of consonance and dissonance. Hyacinth 17:25, 10 Apr 2005 (UTC)

[edit] Removed

(by Wahoofive)

It is important to note that while intervals may be named by their harmonic functions, for instance, a major second, may be described by a ratio, cent, or integer, not every interval described by these more general terms may be described with the harmonic function name. For instance, all major seconds (in twelve tone equal temperament) are 200 cents, but not every interval of 200 cents is a major second. See: enharmonic.
simple and compound

Hyacinth 06:13, 11 Apr 2005 (UTC)

[edit] Unison

Please forgive my ignorance on this topic, but is everyone sure that a unison is actually called an interval, when there is clearly no interval at all?--Light current 22:49, 5 November 2005 (UTC)

Is zero a number? By the way, Light current, don't forget to sign your talk-page posts with ~~~~ —Wahoofive (talk) 19:52, 5 November 2005 (UTC)

Do bears C**p in the woods? Now come on, give me a proper answer!--Light current 22:53, 5 November 2005 (UTC)

I guess I was being too subtle. Unison is an interval in exactly the same sense that zero is a number: you wouldn't use it to count anything, but it's essential as a starting point and an indicator of no change. —Wahoofive (talk) 04:01, 6 November 2005 (UTC)

In fact, I can pursue the analogy further. An interval is a difference between notes. A difference between two numbers can certainly be zero. —Wahoofive (talk) 04:05, 6 November 2005 (UTC)

[edit] Examples of intervals

In the section on intervals using equal temperament, maybe there could be a table with the name of the interval, its abbrevation, and a few examples of well-known songs that start with that interval? That would be helpful for people learning to recognize intervals. Skybrian 21:02, 19 February 2006 (UTC)

The names of the intervals are already in the table. So I am confused what you ask. The list of well known songs is a good idea, but is a project on its own, deserving a separate page. −Woodstone 21:28, 19 February 2006 (UTC)

Yes, the examples are what I'm most interested in. A separate page is fine, but I'm not sure what to call it or how it relates to the main article. There's already a List of Musical Intervals which is mostly unusual intervals, and separate pages for each equal temperament interval. - Skybrian 18:47, 20 February 2006 (UTC)

You may want to have a look at songs by interval. I have no idea about the copyright status of this site, so be careful in converting it to wikipedia. −Woodstone 19:24, 20 February 2006 (UTC)

[edit] Diatonic and chromatic intervals

Some time ago I deplored the almost universal shoddy treatment accorded the terms diatonic and chromatic as applied to intervals (see by searching above, and note my newly inserted comment about New Grove getting it wrong in yet another way). I have now been moved to add a new subsection to the article making two quite distinct usages clear, with examples (and new redirects for both terms). I have positioned it in such a way that it can be found readily enough, and so that it fits logically into the overall exposition. Note that chromatic interval and diatonic interval are used very often, generally on the web and in textbooks but also in Wikipedia, without there ever being clear disambiguations or explanations. As far as I can tell, Wikipedia will now be the first readily accessible resource to remedy this. Noetica 02:27, 4 June 2006 (UTC)

From your description above it sounds like original research. Hyacinth 21:20, 4 June 2006 (UTC)

No, it isn't original research. It's just clarity in disambiguation, and careful respect for traditional usage, through research in the literature. That traditional usage vaguely informs current usage; but current usage is all over the place with these terms chromatic and diatonic. I'll now provide references in the article, as you have requested with your annotation there. One will be the New Grove, for its sentence that I have quoted above: ("An interval is said to be chromatic if it is not part of a diatonic scale (e.g. F–F#, B–Eb)." The definitional substance of that sentence is literally true; only the examples are inadequate (the first because it gives a case that is too clear and therefore not discriminating enough; the second because it contradicts the definition itself, since B and Eb are to be found in the same diatonic scale). Noetica 21:41, 4 June 2006 (UTC)

The article gives the traditional definition of a diatonic interval as one that can be found in either a major or harmonic minor scale., How can that be? Surely, a traditional definition must exclude all chromatic alteration, given that diatonic usage predates the introduction of the harmonic minor scale. Can anyone tell me how or when this illogically expanded definition came about, and is it accepted by respected authorities on music theory? From Noetica's post, above, it would seem that Groves doesn't accept it, which is why they say that B - Eb doesn't belong to any diatonic scale. Thanks (Mark 23 November 2006)

Mark, you write concerning diatonic intervals: Surely, a traditional definition must exclude all chromatic alteration.... But what is meant by chromatic alteration, here? Chromatic alteration depends on context. Is B♭ chromatically altered? The question is unanswerable without knowledge of the key in which the note occurs. In A major or A minor B♭ is chromatically altered; in F major or F minor it is not. The definition presented in the article is not "illogically expanded"; it is just a fully understandable and established use of the term diatonic interval. (From what would it be "expanded", by the way? Have you a particular unexpanded definition in mind? And next ask: what makes that unexpanded definition logical?) At present, sources are all over the place with diatonic interval and chromatic interval. Some, including the New Grove, contradict themselves. The article cites Grove as a source for the traditional definition because it says this: An interval is said to be chromatic if it is not part of a diatonic scale. That's quite explicit, and only a little ambiguous. (It should strictly exclude the melodic minor, since if you take both ascending and descending forms of the melodic minor all sorts of intervals get included as diatonic: in G minor, the melodic has F# ascending and F natural descending, but this does not make F-F# diatonic, by the first of Grove's own examples.) Unfortunately Grove them contradicts itself with the examples: (e.g. F–F#, B–E♭), the second of which belongs in C harmonic minor. But that's just carelessness on the part of Grove. That august work is often careless; but this does not license us to be careless. I now see that I myself was careless months ago, and inadvertently wrote F where I meant C, etc., where all this is discussed above. I'll fix that next. – Noetica 21:59, 23 November 2006 (UTC)

Sorry Noetica, I should have been clearer. By chromatically altered, I don't just mean a note inflected with an accidental. I mean a note that has been actively altered from its diatonic origin, e.g., the note B in the scale of C harmonic minor. My point is that the original (traditional) definition of diatonic would be as is explained in the Wikipedia article diatonic and in that respect, I agree with Roivas's statement below. Unlike Roivas, however, I'm prepared to acknowledge a modern, looser, expanded definition that includes those chromatically altered intervals of the harmonic minor scale. I say 'illogically expanded' as it seems to have been done simply as an expedience with no acoustical justification.

Furthermore, Grove says B - Eb is a chromatic interval because it doesn't occur in a diatonic scale. You feel that that's an error as B - Eb does occur in the diatonic scale of C harmonic minor. But how do you know that Grove accepts that harmonic minor scales ARE diatonic? Perhaps, like Roivas, they don't. If you have access to Grove perhaps you (or anyone) could report their definition. I'd like to know how 'official' is this expanded definition among acknowledged authorities. Thanks for your prompt response to my original query. (Mark - 24 November 2006)

Mark, first: why don't you get a user-name here at Wikipedia? It makes signing your contributions, and having them documented and easy to discern, much more straightforward. (You can then sign and date simply by typing in ~~~~.) Second, please read carefully what I say in my contributions below. I address some things that you have just remarked on – especially concerning Grove's usage (or usages). – Noetica 01:11, 24 November 2006 (UTC)

This is the Diatonic Form: 2-2-1-2-2-2-1 (2 = whole step / 1 = semitone). Any other form is not Diatonic. The modern major, the modern natural minor, and the Ecclesiastical modes are the only Diatonic scales.

See the discussion of diminished seventh for a full explanation with references.

This is an excerpt:

I found an interesting example from Goetschius in his "The Material Used in Musical Composition". In explaining the difference between whole step, diatonic half-step, and chromatic half-step, he states that "The diatonic half-step is the difference in pitch between tones that lie five harmonic degrees apart (harmonic degrees are fifths upwards C-G-D-A-E-B...etc. He now gives an example...which I ain't typin' out). The chromatic half-step is seven harmonic degrees removed."

When thus defined, by direct derivation from the key, the natural (or major) scale is found to represent the following succession of whole and half-steps; whole steps between scale-steps 1-2, 2-3, 4-5, 5-6, 6-7; half-steps between scale steps 3-4, 7-8."

When you arrange this series of notes into a scale, you get the primordial 2-2-1-2-2-2-1 arrangement.

--Roivas 22:46, 23 November 2006 (UTC)

Roivas, what is the relevance of all that to the section of the article under discussion here (Diatonic and chromatic intervals)? What, if anything, do you dispute in that section? Both adjectives, diatonic and chromatic, are notoriously hard to get clear about; both are used in different senses in diverse sources, when they are applied to intervals. Since the article goes on to use diatonic interval in a particular way that is not universally accepted, it is important to clarify things early. The section in question does that. – Noetica 23:20, 23 November 2006 (UTC)

I think I now understand. There are two senses of diatonic scale:

A kind of scale defined in the article Diatonic scale.
A kind of scale implicitly understood in the article Minor scale:

A minor scale in musical theory is a diatonic scale whose third scale degree is an interval of a minor third above the tonic. While some definitions of minor scale encompass modes with the minor third, such as Dorian mode, most musicians use the term to refer to the natural minor, harmonic minor, and melodic minor scales described below. Also, compare major and minor.

By sense 2, melodic minors and harmonic minors are kinds of diatonic scales. This is the usage I have assumed above; and indeed it is a usage that is very common, despite what may be said above, or in the extended discussion at Talk:Diminished_seventh. It is very common, in well-established older usage, to hear talk of a pianist practising her chromatic and diatonic scales, with the intention to refer to chromatic scales on one hand and harmonic and melodic minor scales on the other. Now, this usage seems to be waning. If that's the case, then the section in question in this article may need revision, but perhaps only slight revision (in the exact form of words). The distinction it makes is still important. It counters, for example, such nonsense as you will find here. Of course, you might think that B-E♭, for example, is a chromatic interval. Do you? If we ought to think this, then we ought to think that Grove was right all along. But it is not yet established in which sense Grove uses diatonic scale, or whether it is consistent in its usage. A moral to draw from all of this: A term may have many uses, and it is a mistake merely to assume one of those uses, ignoring the others; it is also a mistake to assume that any one of those uses counts as the right one, simply because of its historical priority, its acceptance in contemporary practice, or its acceptance in one out of several contemporary bodies of practice. – Noetica 00:29, 24 November 2006 (UTC)

You said:

And next ask: what makes that unexpanded definition logical?

I simply gave one of many examples of what the diatonic scale is. As far as modern and traditional definitions, there's really only one meaning of the word. I don't know if our definitions disagree. I'm just showing that it's not all that ambiguous. The only problems are the misconceptions from online sources and a possible loose interpretation of the Oxford's Concise.

--Roivas 00:17, 24 November 2006 (UTC)

Ah, my contribution above (00:29) was posted without my seeing Roivas's latest contribution (00:17), so neither of these two contributions responds to the other. – Noetica 00:35, 24 November 2006 (UTC)

Incidentally, I referred not to the Grove's Concise but to the full online New Grove. Unfortunately I have no access to it right now. I would like to check its usage of diatonic scale, melodic minor, etc., throughout; not just at articles that might be headed with those terms. – Noetica 00:38, 24 November 2006 (UTC)

My objection is that "most musicians say" is not a valid reference. No published encyclopedia would settle for mere speculation and I don't see why any of us should. The harmonic minor scale is a chromatically altered scale and is never considered diatonic in any music theory book. People are using the term "Diatonic" out of context to make their assertions sound more "scientific." It is not a mistake to dismiss careless distortions of a term that has been well-defined for years. References can be found in the other discussion page I mentioned above.

The proper term for an interval derived from a diminished or augmented interval (like the diminished seventh) is "enharmonic interval" (see Walter Piston's "Harmony" pg. 8...my copy is third edition).

The definition that's being used here from Oxford Concise appears to be directly lifted from Piston's "Harmony" (page 3, after Ex. 1)) The word Diatonic isn't even mentioned on this page.

All it says: "Three scales are used as the basis of the music with which we are concerned"...and mentions major, minor (with it's two common alterations) and the chromatic.

So, I hope the whole basis of this perceived polarity isn't going to pivot on Oxford's Concise (edit: I had Grove's in there on accident).

If someone could let me know what Grove's says...I haven't seen this definition.

REALLY SORRY ABOUT THAT EDIT RE: THE GROVE'S/OXFORD MIXUP!!!

Hope this helps.--Roivas 03:31, 24 November 2006 (UTC)

Roivas, this dialogue is all good and healthy, I say. I certainly agree with you that "most musicians say" is not a valid reference, for many purposes. In fact, though, it does carry weight for some purposes. Usages in musical theory are changing all the time, and some that are becoming well-established are not yet reflected in standard works of reference. A highly relevant example: reference authorities for diatonic interval in the second sense given in the article (the distance spanned in a diatonic scale) are extraordinarily hard to track down. I personally have never used diatonic interval in that sense, but it is found from an excellent editor later in the article, and there are instances from other reputable sources. One example is to be found here.

Music syllabuses of universally recognised examining bodies give evidence in print that diatonic scales is a common way of referring to major scales, and harmonic and melodic minor scales. In the nature things it is hard to find material that is in print online, but this from the Royal College of Music is certainly relevant:

The number of required diatonic and tremolando scales in Grades 6 and higher has been reduced. For example, in the previous Syllabus Grade 8 candidates were required to prepare two- and three-octave scales in every major and minor key. In the 2004 Edition, this is reduced to six major scales and their relative minors.

This clearly is not intended to restrict diatonic scale so that the harmonic and melodic minor scales are excluded. In fact they, along with major scales, are certainly the only "diatonic scales" required in the guitar syllabus under discussion.

You mention the Oxford Concise (after you correct yourself). But it still isn't clear what you mean to refer to. Is it The Concise Oxford English Dictionary, or a music dictionary? Concerning it, you quote and gloss:

"Three scales are used as the basis of the music with which we are concerned"...and mentions major, minor (with it's [make that its] two common alterations) and the chromatic.

But this doesn't support the contention that the forms of the minor are not diatonic! If anything, it distances all forms of the minor from the chromatic: quite properly, but it doesn't say that the harmonic is a chromatically altered scale, to use your words.

Finally, what are we to make of diatonic harmony, in common use? It is surely harmony that uses only the notes of the major, the melodic minor, and the harmonic minor to construct its chords. It is called diatonic for just that reason. It is most definitely not harmony that uses only the resources of the major and the natural minor.

Anyway, my point has been that the situation is not all clear. I do not agree with you that the harmonic minor scale is a chromatically altered scale and is never considered diatonic in any music theory book. In older books especially, and in discussion of syllabuses such as the one I cite from an eminently respectable source, it is common to classify things this way. I also don't agree when you speak of the proper interpretation of anything. Usage, sadly, varies. A failure to document and acknowledge such variation can only perpetuate confusions and misunderstandings. If we want uniformity in an article (and in Wikipedia as a whole), we have to set it up explicitly, not simply assume it, or lay it down in a doctrinaire fashion. – Noetica 05:19, 24 November 2006 (UTC)

That section about the Oxford Concise Dictionary of Music was intended for Mark (as he referenced that in the Dim 7 article's discussion page. I thought you were mentioning the same thing. My brain was stuck in gear. Sorry.--Roivas 05:33, 24 November 2006 (UTC)

"The number of required diatonic and tremolando scales in Grades 6 and higher has been reduced. For example, in the previous Syllabus Grade 8 candidates were required to prepare two- and three-octave scales in every major and minor key. In the 2004 Edition, this is reduced to six major scales and their relative minors."

How is this supposed to provide a defense for the harmonic minor scale being diatonic? Is it implied by its very omission?--Roivas 05:40, 24 November 2006 (UTC)

"Three scales are used as the basis of the music with which we are concerned"...and mentions major, minor (with it's [make that its] [thank you very much for catching that] two common alterations) and the chromatic.

My point was that there was no mention of "Diatonic" in the text for a good reason, so any use of this to justify Mark's position that the word had been "expanded" would be invalid. That's all.

Please cite the "older book" that supports your position. Something from a book that explicitly states that a harmonic minor scale is diatonic would be greatly appreciated. Thanks.

I don't know what you mean by "Diatonic Harmony." In the Goetschius book, The Material Used in Musical Composition, pg. 134, it states:

If the last chord of the original key is one which belongs also to the prospective key (in other harmonic meaning, of course), the modulation will be gradual, and there need be no chromatic inflection. Consequently, such are called Diatonic modulations. Each chord-movement, before, during and after the change of key, pursues the track of a diatonic scale.

This is from an "older book" first published in 1889 by Schirmer. I mentioned other references in the Dim 7 discussion as well.--Roivas 05:45, 24 November 2006 (UTC)

What is the point of the webpage you linked us to: homepages.cae.wisc.edu? Please use an example from a book on music theory, acoustics, or tuning. I'm sure you can find quite a few at your local library. If you have access to a University, that's even better. This "online syllabus" thing is not adding anything to the discussion. Do the administrators at Wikipedia want contributions based on idle speculation and unreliable online sources?--Roivas 09:47, 24 November 2006 (UTC)

You say:

Usages in musical theory are changing all the time, and some that are becoming well-established are not yet reflected in standard works of reference.

I'm frustrated by the lack of academic integrity being shown on Wikipedia. I don't understand why these completely groundless assumptions are being clung to and defended. Please, for my own sanity, give me a source from a respected book on music theory that verifies what you are claiming to be true. If it isn't "reflect in a standard work of reference," then it's simply not valid (especially since this lack of precision accomplishes nothing).--Roivas 09:56, 24 November 2006 (UTC)

OK, Roivas. You've said quite a bit. I note that you've retracted and corrected a couple of things, and I'll mention them no more. I will, however, address all of the other points you make. I quoted the following, from the highly respected Royal College of Music:

You replied with this:

How is this supposed to provide a defense for the harmonic minor scale being diatonic? Is it implied by its very omission?

I am truly surprised if you can't see this. Obviously, in the guitar syllabus under discussion, there are major, melodic minor, and harmonic minor scales required. This we know as background, surely. That's the nature of such syllabuses in practical music. Clearly all of these are designated diatonic for short. Do you dispute that? If you do, what do you imagine the guitarists are to play – natural minor scales? Not at all likely! It could, with some effort, be checked – if you really do think that.

You also write:

My point was that there was no mention of "Diatonic" in the text for a good reason, so any use of this to justify Mark's position that the word had been "expanded" would be invalid. That's all.

Sorry. I don't follow that. Is it important? Try again if it is.

Please cite the "older book" that supports your position. Something from a book that explicitly states that a harmonic minor scale is diatonic would be greatly appreciated. Thanks.

I am away from my large collection of music reference books. But here's one, from the Concise Oxford Dictionary of Music (an older edition: 1964):

Scale: ...[First an account of the major scale, then the melodic minor and the harmonic minor scales, then:]... The major and minor scales are spoken of as DIATONIC SCALES, as distinct from a scale using nothing but semitones, which is the CHROMATIC SCALE. [Capitals in the original]

Happy?

I don't know what you mean by "Diatonic Harmony."

You don't? There are several books bearing the title Diatonic Harmony, or having it as part of their title. (To see some, go to Amazon and search for books using the phrase diatonic harmony. It means tonal harmony other than chromatic harmony, and therefore includes the use of all chords formed with notes from the major, harmonic minor, or melodic minor scales. It is called diatonic harmony for the good reason that many people call those three sorts of scales diatonic scales. Schönberg appears to be among those who speak of diatonic harmony in this sense.

Next you mention a book by Goetschius, but I can't see the relevance of it, still less how it might count against diatonic being understood as I say it often is.

Next you write:

What is the point of the webpage you linked us to: homepages.cae.wisc.edu?

I have made that perfectly clear above. It shows just one example of diatonic interval being used, at a respectable university (University of Wisconsin), in the rather new sense of number of notes spanned in a diatonic scale. The point I was quite explicitly making was this: it's hard to track down reference works that warrant that usage, but it is nevertheless out there in academic currency.

Please use an example from a book on music theory, acoustics, or tuning.

For this particular usage (which is not my own usage, note) I cannot. That was the whole point! For the usage with scales, see above. Is Oxford University Press good enough for you?

I'm sure you can find quite a few at your local library. If you have access to a University, that's even better.

Don't patronise. You don't know how unwarranted that is.

This "online syllabus" thing is not adding anything to the discussion. Do the administrators at Wikipedia want contributions based on idle speculation and unreliable online sources?

Elitist and antiquated nonsense. In any case, hardcopy can be found, for such syllabuses.

I'm frustrated by the lack of academic integrity being shown on Wikipedia. I don't understand why these completely groundless assumptions are being clung to and defended.

This is your response to the effort I have put into this? This, in response to my suggestion that this dialogue is healthy and worthwhile? Are you accusing me of a failure of academic integrity? I have met all of the challenges you offer, remote as I am at the moment from my resources. I invite you to reconsider what have written here.

Please, for my own sanity, give me a source from a respected book on music theory that verifies what you are claiming to be true.

Done (see above). Sane now?

And last:

If it isn't "reflect in a standard work of reference," then it's simply not valid (especially since this lack of precision accomplishes nothing).

An interesting, if unfounded, assertion. I argue against it above. In any case, the usages I have claimed to be prevalent clearly are in print from reputable sources. As for precision, I am strongly in favour of the most exacting precision. That's the whole point of my efforts at this article, and at several others. But a word of advice to you: a presumption of uniform usage, in the face of palpable evidence against such uniformity, is not to be confused with precision. It is, rather, a form of prejudice of the most unacademic kind. – Noetica 11:47, 24 November 2006 (UTC)

You haven't put much effort into this. It doesn't take much effort to Google a word.

The definition you posted, yet again, does not state that harmonic minor is diatonic. The octatonic scale isn't chromatic, so by your bizarre interpretation of Oxford Concise, it's diatonic.

You have not "met my challenge" by providing a reference that explicitly states that harmonic minor is diatonic.

See the Knud Jeppesen book (The Style of Palestrina and the Dissonance) for a clear example of what diatonic means.

Here is one of many basic descriptions of the word "Diatonic" which can be found in a sea of books on musicology:

"The Greeks had three genera - diatonic, chromatic, and enharmonic. A diatonic tetrachord contained two tones and a semitone, variously arranged, the Dorian tetrachord having the order shown above, as A G F E. In the chromatic tetrachord the second string (as G) was lowered until the two lower intervals in the tetrachord were equal. Thus A Gb F E represents the process of formaion better than the more commonly shown A F# F E. In the enharmonic tetrachord the second string was lowered still further until it was in unison with the third string; the third string was then tuned half way between the second and fourth strings...etc."

My valid reference: Tuning and Temperament, A Historical Survey, by J. Murray Barbour, ISBN 0-486-43406-0

Any occurance of "diatonic" in reference to harmonic minor or ascending minor scales need to be cleaned up from this and any other article. This is why:

Say where you got it: It is improper to copy a citation from an intermediate source without making clear that you saw only that intermediate source. For example, you might find some information on a web page which says it comes from a certain book. Unless you look at the book yourself to check that the information is there, your reference is really the web page, which is what you must cite. The credibility of your article rests on the credibility of the web page, as well as the book, and your article must make that clear.

Wikipedia:Verifiability The threshold for inclusion in Wikipedia is verifiability, not truth. By insisting that only facts, assertions, theories, ideas, claims, opinions, and arguments that have already been published by a reputable publisher may be published in Wikipedia, the no-original-research and verifiability policies reinforce one another.

"The obligation to provide a reliable source lies with the editors wishing to include the material, not on those seeking to challenge and/or remove it."

Reliable sources Wikipedia:Verifiability Any material that is challenged or likely to be challenged must be accompanied by a reliable source. Material that counts as "original research" within the meaning of this policy is material for which no reliable source can be found and which is therefore believed to be the original thought of the Wikipedian who added it. The only way to show your work is not original research is to produce a reliable published source who writes about the same claims or advances the same argument as you.

The excerpt you provided from Oxford Concise does not verify what you are claiming to be true. You are simply implying it. Once again, please provide a valid source.

--Roivas 19:08, 24 November 2006 (UTC)

OK, Roivas. Once more I'll go through what you have written and respond to everything (which is more than you have done for me, note):

You haven't put much effort into this. It doesn't take much effort to Google a word.

That is inaccurate and insulting. Look again at all that I have posted on this page, and at my work on the article. Reflect on the effort that goes on in the background. I have done a lot more than Google a word, as you rudely assert.

The definition you posted, yet again, does not state that harmonic minor is diatonic. The octatonic scale isn't chromatic, so by your bizarre interpretation of Oxford Concise, it's diatonic.

I see that a similar quote, but presumably from a later edition of the Oxford Concise Dictionary of Music, turns up at Talk:Diminished seventh. There too, having called for a standard reference work that would support a view that opposed your own, you rejected the citation when it was produced. Once more, here, you fail to understand the context in which the influential and respected authority (Percy Scholes) was writing. My interpretation of what he writes, far from being bizarre, is the only coherent interpretation available. Clearly Scholes is writing not about Greek theory, or theory appropriate to modal practice in European music: he is writing about the dominant later system in which the usual scales (excluding the chromatic) are major, melodic minor, and harmonic minor. These he clearly calls diatonic. How else are we to interpret what he writes? (The major and minor scales are spoken of as DIATONIC SCALES, as distinct from a scale using nothing but semitones, which is the CHROMATIC SCALE.) Scholes simply does not address the earlier scale that you call octatonic. Nor should he, for the modern context he writes in.

You have not "met my challenge" by providing a reference that explicitly states that harmonic minor is diatonic.

I clearly have. See immediately above. It is diatonic, in one of the accepted meanings of the term diatonic. Your refusal to accept that a term may have more than one sense does you no credit.

See the Knud Jeppesen book (The Style of Palestrina and the Dissonance) for a clear example of what diatonic means.

You refer here to that other discussion (Talk:Diminished_seventh). Please take care not to send us on such a chase. As it happens, I have that work in my collection, along with something else of Jeppesen's. I respect his authority, of course, and have cited him elsewhere in Wikipedia. But he is writing about an earlier period, one that Scholes is not writing about. Diatonic evolved new senses, just as the system of scales itself evolved.

Next you cite a work on Greek theory of music. Lovely! And undisputed. But again, evidence that a term has one meaning does not show that it has no other meanings.

Then you say something with which I almost agree:

Any occurance [sic] of "diatonic" in reference to harmonic minor or ascending minor scales need [sc. needs] to be cleaned up from this and any other article.

Well, all of this terminology needs to be sorted out. That was the whole point of my introducing a section clarifying the terms diatonic and chromatic, used of intervals. They need to be sorted out in many contexts, in fact – not just when they are used of intervals. Given that you have edited and discussed at Tritone, it surprises me that you passed over in silence Template:Diatonic_intervals, which is included there. In it, even the augmented unison and the diminished third are listed as diatonic intervals. Now, neither of us would find that acceptable. But I understand that the authors of the template had yet another meaning in mind for diatonic interval. Can you work out what that meaning is? There is not one single "right" interpretation. We just need to say how we are going to use our terms, with what sound precedents, and then work towards consistency. But consistency cannot be imposed on the spurious ground that there is a "right" way to use terms.

It is improper to copy a citation from an intermediate source without making clear that you saw only that intermediate source. For example, you might find some information on a web page which says it comes from a certain book. Unless you look at the book yourself to check that the information is there, your reference is really the web page, which is what you must cite. The credibility of your article rests on the credibility of the web page, as well as the book, and your article must make that clear.

OK, that's a worthy principle. Not always achievable, given one's immediate circumstances, but worthy nonetheless.

The threshold for inclusion in Wikipedia is verifiability, not truth. By insisting that only facts, assertions, theories, ideas, claims, opinions, and arguments that have already been published by a reputable publisher may be published in Wikipedia, the no-original-research and verifiability policies reinforce one another.

"The obligation to provide a reliable source lies with the editors wishing to include the material, not on those seeking to challenge and/or remove it."

So what? I cite verifiable sources (as does Mark, elsewhere), and they reveal that there is diversity in usage. You refuse to respect those sources, and refuse to acknowledge that diversity.

You next talk about Wikipedia's policy on original research, but I can't see the relevance here. Who's doing original research?

The excerpt you provided from Oxford Concise does not verify what you are claiming to be true. You are simply implying it. Once again, please provide a valid source.

Palpable nonsense. The only fault in the vicinity is your own. When your challenge to provide a respected source is fully and unambiguously met (here and also at Talk:Diminished_seventh), you cannot accept that you may have been mistaken.

There is little point going on with this. I suggest you stand back a little and take stock. You may find that you don't have a monopoly on truth, and on academic or Wikipedic rectitude. – Noetica 22:40, 24 November 2006 (UTC)

To correct another mistake of yours, "tonal" and "diatonic" are not interchangeable terms.

If you cannot come up with a published source, then accept that this erroneous definition of diatonic cannot be a part of this article. This is an encyclopedia, not your personal sounding board. Thanks.--Roivas 00:17, 25 November 2006 (UTC)

I have come up with a published source. So has Mark (elsewhere). The definition is not erroneous, it has been in use for some considerable time, and still is. You are being completely unreasonable. It is, on the evidence plainly visible here and at Talk:Diminished_seventh, ridiculous to assert that I am merely expressing an opinion of my own without published support. Somehow contrive to get the wisdom to see this. Until you wise up, you are not worth talking to, and are wasting your time and others' time, while doing damage to articles along the way. – Noetica 00:43, 25 November 2006 (UTC)

I don't believe that the definition means what you think it does. On top of this, the dictionary you cite does not say "the harmonic minor scale is diatonic." I would think you'd be able to find it in a music theory book if this was the case. It must be explicit and not implied.--Roivas 00:47, 25 November 2006 (UTC)

Let me raise the bar just slightly. The phrase "harmonic minor" has to occur SOMEWHERE on the page and be somewhat relevant to the matter at hand. Can we, at least, do that?--Roivas 01:27, 25 November 2006 (UTC)

Back to the Oxford Concise:

Scale: ...[First an account of the major scale, then the melodic minor and the harmonic minor scales, then:]... The major and minor scales are spoken of as DIATONIC SCALES, as distinct from a scale using nothing but semitones, which is the CHROMATIC SCALE. [Capitals in the original]

I noticed that the Lydian mode is not mentioned here. It is close to the major scale, but it's not the major scale. It seems to be completely omitted from this definition. Isn't the Lydian mode a Diatonic Scale?

How does the (modern) Locrian Mode fit into this definition? It's not the major, minor, or chromatic scale.

Locrian Mode: |"| | | : "| | | |

It has a minor third between 1 & 3, but a major third between 5 & 6. No leading tone between 7 & 8! It doesn't fit into this definition at all. Is the Locrian Mode Diatonic? Does the Oxford omit the Ecclesiastical Modes for a reason?

As you put it, the Oxford provides "the only coherent interpretation available." Come again?

Another source I've found:

Altered Chords in Minor

"The alterations in minor are reckoned from the harmonic form of the scale, and they tend chiefly to remove the unmelodious progressions between the 6th and 7th scale-steps. This interval embraces one and a half step, and therefore does not conform to the principle of scale-formation (par. 16)."

Here's par. 16:

"The diatonic half-step is the difference in pitch between tones that lie five harmonic degrees apart (harmonic degrees are fifths upwards C-G-D-A-E-B...etc. The chromatic half-step is seven harmonic degrees removed."

"When thus defined, by direct derivation from the key, the natural (or major) scale is found to represent the following succession of whole and half-steps; whole steps between scale-steps 1-2, 2-3, 4-5, 5-6, 6-7; half-steps between scale steps 3-4, 7-8."

Goetschius: The Material Used in Musical Composition, pg. 121 & pg. 5.

This is all very clearly stated, which is why Goetschius is a great introduction to basic music theory. I know you are concerned that I am hiding or omitting something by adhering to some antiquated Greek definition, but Goetschius is writing about the "dominant later system" that you have in mind.

Once again, here's the Diatonic Form: 2-2-1-2-2-2-1.

You said:

In older books especially...it is common to classify things this way.

Please give me, at least, a title of a book or an author and I will try to help you find an example. You said "common," right? That's really hard to believe.

I feel that my references are being dismissed as well, so...

--Roivas 07:43, 25 November 2006 (UTC)

Not really a big fan of using dictionaries to prove a point (they are usually not the best places to go for a firm understanding of music theory), but here's a definition I found in The Oxford Companion to Music:

Scale

3. Diatonic Scale: The sixth and seventh degrees of the minor scale are unstable and result in two forms, neither of them diatonic: the harmonic minor, with the characteristic interval of an augmented 2nd; and the melodic minor...

Pg. 1106, ISBN: 0198662122

This addresses the harmonic minor scale in a direct way and leaves no room for ambiguity and misinterpretation.

I don't know if the two people who have argued the most with me are still involved in this discussion, but in case I come up against this in the future, I want to make sure that my point has been clearly made.

--Roivas 05:22, 26 November 2006 (UTC)

Dr. Alan Crosier's definition has been removed from the article.--Roivas 20:50, 26 November 2006 (UTC)

Roivas has, here and at Talk:Diminished_seventh, rashly attributed to me statements and opinions that are not mine, and until that's sorted out I'll not discuss these matters with Roivas. Some of Roivas's recent comments here I deal with at Talk:Diminished_seventh, in a post made some minutes ago. – Noetica 20:53, 26 November 2006 (UTC)

Okay, other than a few things that I edited, I'm not sure what you are talking about.--Roivas 00:46, 28 November 2006 (UTC)

Roivas has just made an unwarranted personal attack. And as I wrote above, Roivas has, here and at Talk:Diminished_seventh, rashly attributed to me statements and opinions that are not mine, and until that's sorted out I'll not discuss these matters with Roivas. Here I will only add a couple of points concerning process:

1. Since June 2006, when I added a disambiguating section, I have not edited the present article except to correct or clarify things quite uncontroversially at a couple of points, and except to apply a {{{Neutrality}}} marker at the top. Roivas has now, in effect, removed that disambiguating section, along with its three references (one of which was the New Grove).

2. I have attempted to engage in fruitful discussion with Roivas. So has Mark. Neither of us has had much success. I see no way forward until we can all be more respectful. This would mean avoiding personal attacks, etc.

3. I look forward to being able to engage in rational dialogue concerning the different senses in which diatonic and chromatic are used, especially as they are applied to scales and intervals. Right here would seem to be a suitable place for such dialogue.

– Noetica 23:58, 27 November 2006 (UTC)

What's most amusing about this is that the most basic, "traditional" definition wasn't even in your "disambiguation." Kind of odd.--Roivas 16:55, 28 November 2006 (UTC)

You are right. We are all getting a little excited. I have no interest in "personally attacking" people and if you'd kindly refer me to the offending comments I'll gladly reword or remove them. I will even admit that I'm probably the cause of most of the bile. I will also try not to be sarcastic.

Please address this: The section I removed was pasted from a website quoting a certain Dr. Crosier. Am I correct in asserting this? --Roivas 00:12, 28 November 2006 (UTC)

Re: Moral Highground. Are you going to make edits where you've made assumptions on my part, or is this a one way street?--Roivas 01:09, 28 November 2006 (UTC)

The most important point is getting lost in these ridiculous arguments. What is the best way to introduce these ideas to a beginner or disinterested party? This is the only thing I'm trying to get across in the article. I think this "searching for truth" aspect of the argument is incredibly pretentious and out of place in an encyclopedia.--Roivas 19:11, 28 November 2006 (UTC)

Pretentious? moi? You're probably referring to a phrase I used over at the diminished seventh discussion page. Yes, hands up! It did come across that way as I noticed later to my dismay. I should have said that I'm just trying to get the true facts. And not just an accurate definition of 'diatonic' but the various usages of the term, whether careless or precise, correct or incorrect.

However it wasn't so much the phrase you attacked as the idea. That's completely confused me - I thought getting the most accurate information possible was exactly the point of any encyclopedia. To your credit you've shown me that Dr Alan Crozier isn't acknowledged as a source of such accurate information. So I'll keep at it, and mostly offline. (Mark - 28 November 2006)

Great. Now I'm attacking ideas.

Look, I'm just going to make my edits and cite my reasons. U Cool wit dat?--Roivas 23:33, 28 November 2006 (UTC)

Noetica said:

My interpretation of what he writes, far from being bizarre, is the only coherent interpretation available.

Just read this sentence a few times. How am I supposed to have a rational debate with someone who says things like this?--Roivas 23:52, 28 November 2006 (UTC)

Grove Music Online (wording is exactly the same in print as well...see page 295 in the actual text):

Diatonic (from Gk. dia tonos: ‘proceeding by whole tones’).

Based on or derivable from an octave of seven notes in a particular configuration, as opposed to Chromatic and other forms of Scale. A seven-note scale is said to be diatonic when its octave span is filled by five tones and two semitones, with the semitones maximally separated, for example the major scale (T–T–S–T–T–T–S) [which is the scale form I have been referring to all along]. The natural minor scale and the church modes (see Mode) are also diatonic.

An interval is said to be diatonic if it is available within a diatonic scale. The following intervals and their compounds are all diatonic: minor 2nd (S), major 2nd (T), minor 3rd (TS), major 3rd (TT), perfect 4th (TTS), perfect 5th (TTST), minor 6th (STTTS), major 6th (TTSTT), minor 7th (TSTTTS), major 7th (TTSTTT) and the octave itself. The Tritone, in theory diatonic according to this definition, has traditionally been regarded as the alteration of a perfect interval, and hence chromatic; it may be either a semitone more than a perfect 4th (augmented 4th: TTT) or a semitone less than a perfect 5th (diminished 5th: STTS).

Very clear definition.

--Roivas 08:13, 3 December 2006 (UTC)

[edit] Golden ratio

Is the interval produced by two notes with frequencies with the golden ratio musically significant?--Lkjhgfdsa 19:26, 7 June 2006 (UTC)

Interesting question. The golden ratio is (1+sqrt(5))/2=1.618034, which corresponds to 833.09 cents. This is the interval from c to about a third of a semitone between g# and a. Checking the extensive list at List of musical intervals this does not appear to have any musical significance. Viewed from another angle, musical relations are usually ratios between small numbers, whereas φ is not a rational number. −Woodstone 20:34, 7 June 2006 (UTC)

It does have a special property though: in a certain sense it is the most dissonant interval because its continued fraction expansion 1+1/(1+1/(1+1/... converges the most slowly. Isn't it weird how paintings and architecture that use the golden ratio look beautiful, but the golden ratio sounds ugly as a musical interval? —Keenan Pepper 21:58, 7 June 2006 (UTC)

[edit] NPOV

As a global encyclopedia, a NPOV for Wikipedia means a global point of view. The discription of intervals here concentrates on European or western scales. There are many totaly different scales in othere cultres. For example, in the classical courtly music of Buganda, played on instruments like amadinda and akadinda xylophones and on the enanga harp the scale used is a roughly equidistant pentatonic scale which means that the octave is divided into five equal intervals of about 240 cents (2.4 semitones). Moreover, some musical cultures have a higher tollerance for variation in their scales, so an emic and etic distinction seems appropriate here (similar to the distinction of phonetics and phonology). So the article needs to be structured into a general part definig general concepts applicable in all cultures (like "interval", "cent" etc.) and a section or sections describing specific intervals or interval systems occurring in certain musical cultures (like classical european music or classical buganda music). Nannus 22:21, 5 August 2006 (UTC)

I couldn't agree more. This article needs some major work to avoid systemic bias. It should start out with the universals: the terminology, the mathematics, the octave (the only interval with a good claim to universality), and then it should have equal sections for all the musical traditions of the world. A lot of the material should probably be broken off into its own article, Intervals of common practice music or Intervals of the diatonic scale or whatever. —Keenan Pepper 03:18, 6 August 2006 (UTC)

I agree also, though some of my comments early in 2005 (see above) might seem to show that I disagree. I do think there needs to be a clearly marked and well-managed "core" of theory, which can then be applied in treatments of several different traditions. As things stand, the coverage of Western common practice is pretty thorough; but it tends to blend too seamlessly and therefore confusingly with talk of Pythagorean commas, schismata, and the like. One problem might be that few have the expertise to undertake the non-Western coverage. Myself, I know the common-practice side of things well, but for the rest I could only assist with the task of integration and copyediting for consistency. So how about you, Nannus and Keenan Pepper? (By the way, KP: did you mean "systematic bias"? That's the more usual expression, and seems to refer to POV bias. By the way, I'm not sure that NPOV is precisely what is lacking here: would anything that falls short of full global coverage be aptly called a lapse from NPOV? Nannus: do you mean "the distinction of phonetics and phonemics", which is the usual way of glossing the etic–emic distinction?) Noetica 04:23, 6 August 2006 (UTC)

Good assessment. I have access to a good music theory library (Allen library at FSU) and enough knowledge to use it. I thought it was systematic too, but Wikipedia:WikiProject Countering systemic bias uses the other word. —Keenan Pepper 04:56, 6 August 2006 (UTC)

[edit] Minor reformatting

I reformatted the explanations of perfect intervals to match the list style of major/minor intervals which is easier to read. I didn't change any content. --64.175.42.169 20:14, 23 November 2006 (UTC)

[edit] Compound intervals

Should this article contain information on common compound intervals and their enharmonic equivalents?

maybe we could say that these intervals are part of the western chromatic scale, cuz i doubt we can find any experts on the 22-tone or parche scales. Blueaster 03:00, 28 November 2006 (UTC)

[edit] Topic Rewrite

We seriously need a RCM (royal conservatory of music) official to rewrite the faulty parts of the article..... that way, there won't be half as much arguing over the topic.... however, I don't think that they would respect a free online version of conservatory theory.... therefore.... Ex-Conservatory members? [[User:]] 17:39, 2 December 2006 (UTC)Talflick

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Category: NPOV disputes