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I have some confusion as to the use and history of the word octave in the context of music and tuning systems.

The (western) tuning system comes from the Pythagorean division of the octave, based on consonant frequencies, found by simple ratios. This already produced a division of an octave into 12 pitches. Later on, within the roman civilization and gregorian chants, this system was still applied and a notation system invented which contained 7 note names (today A to G), probably to refer to the common (ionian) major scale.

It seems to me that the term "octave" refers to these 8 notes within the 12 note system, and was used in retrospect after some music theory was already in the making (during definition and use of the ionian major scale -I know modes weren't defined up until much much later-).

All answers to questions related to the name "octave" just explain the history of modern tuning systems, but I haven't yet seen an explanation of why we use the word "octave" for a 12-tone system, which was never an 8-tone system.

When did people start using the word octave, and what does it really refer to?

To clarify: I am not asking why an octave is called an octave (meaning eight) if we have 7 notes. This is clear. My question is about the chronology of a 12 semi-tone tuning system, the use of an 8-note scale (major scale), and the introduction of the word 'octave' to refer to those 8 notes (7 notes + root). If I want to build an 8-note system, I can divide an octave into 7 intervals using a harmonic series. This division will give me 7 pitches + an octave, but this will sound nothing like the major eight-note scale we know today. As the modern major scale arises from the division of an octave into 12 intervals (12 notes + an octave), and selecting a sub-set of 8 out of it. However in this case we require a 12 semi-tone tuning system. If the term octave refers to the 8 notes in a major scale, then in is placed in retrospect, and after the development of enough music theory to define diatonic scales.

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    Related question: Etymology of word “Octave” – Aaron Oct 28 '20 at 0:39
  • @Aaron thank you. Yes that is probably the same question rephrased, which has yet not accepted answer. Most answers relate to "counting" of pitches in the diatonic scale. But the diatonic scale was defined much later than the creation of the 12-tone tuning, so as my question implies, the word octave must have been created in retrospect. Which promotes the other question: how was an octave called before the diatonic scale? – hirschme Oct 28 '20 at 0:46
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    "But the diatonic scale was defined much later than the creation of the 12-tone tuning": do you have any evidence to show that ancient Greek music had 12 tones per octave? Regardless, even if it did, that system was not in use in medieval Europe when the term "octave" arose. – phoog Oct 28 '20 at 2:25
  • @phoog great point, actually it seems like it isn't the case. From wikipedia: "twelve tone Pythagorean temperament was developed by medieval music theorists using the same method of tuning in perfect fifths, however there is no evidence that Pythagoras himself went beyond the tetrachord". You say that the term "octave" arose in medieval europe. That is already part of an answer to my question. Would you mind elaborating on that origin? What system was in use in medieval Europe when the term octave arose, and what did the term octave refer to? – hirschme Oct 28 '20 at 2:35
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    The paper given as a reference to support the quote does not in fact support the conclusion that medieval theorists were theorizing about 12 tones. See for example the table in section 3.2, which shows Pythagorean tuning of a diatonic scale from F to F (with a B-natural, not a B-flat). I've edited the Wikipedia article accordingly. – phoog Oct 28 '20 at 6:16
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Was the term “octave” coined after the development of early music theory?

No. As shown below, it was already in use by the 11th century to denote the musical interval (although the principal name for the interval at that time seems still to have been diapason).

What system was in use in medieval Europe when the term octave arose, and what did the term octave refer to?

I'm a little hazy on the state of music theory before Guido d'Arezzo, who worked in the early 11th century, but it's pretty clear that the letter names A through G were in use before he published his system of solmization with the six syllables ut, re, mi, fa, sol, and la.

In his treatise Micrologus, published around 1026, he mostly refers to the interval that we now call the octave as the diapason, but it's also clear from chapter V that the term "octave" had come to its musical meaning. Keep in mind that the day one week after some other day was also called the octave, which is simply Latin for "eighth," and, as has been noted elsewhere, the periods of a week and two weeks are to this day called eight and fifteen days in at least some modern Romance languages.

As an aside, I note that the only difference between numbering the intervals of the second, third, fourth, fifth, sixth, and seventh, on one hand, and the octave on the other, is that we use our native (Germanic) ordinals for the first set of intervals and a word derived from the Latin ordinal for the last. There's no theoretical reason not to call it an eighth; it's just custom, perhaps influenced by the special status given to that interval by octave equivalence.

Guido's system had eight pitch classes, the seven letters of the "white key" scale, A through G, plus B-flat. To distinguish, B-natural was written with a square form (𝇒), while B-flat was written with the normal round form. The two forms of B would never be used at the same time, though, so there were seven distinct pitches in the diatonic scale.

The system comprised roughly two and a half octaves, the lowest in upper case letters, the middle in lower-case letters, and the upper in double lower-case letters. Oh, and because he needed one note below the lowest octave, that note was designated with a Greek letter. There were, in total, 21 notes:

Γ A B C D E F G a b 𝇒 c d e f g aa bb 𝇒𝇒 cc dd

Guido, Micrologus, chapter V, on octave equivalence:

Diapason autem est, in qua diatessaron et diapente iunguntur; cum enim ab A in D sit diatessaron, et ab eadem D in a acutam sit diapente, ab A in alteram a diapason existit: cuius vis est eamdem litteram in utroque habere latere, ut a B in 𝇒, a C in c, a D in d, et reliq. Sicut enim utraque vox eadem littera notatur, ita per omnia eiusdem qualitatis, perfectissimaeque similitudinis utraque habetur et creditur. Nam sicut finitis septem diebus eosdem repetimus, ut semper primum et octavum diem eumdem esse dicamus, ita octavas semper voces easdem figuramus et dicimus, quia naturali eas concordia consonare sentimus, ut D et d.

Corrections to this translation are welcome. To maintain the distinction between the Greek and Latin interval names in the original, I've used Greek and English names here. Diapente is fifth, and diatessaron is fourth:

The diapason is, moreover, that in which the diatessaron and diapente are joined, for from A to D is a diatessaron, and likewise from D to acute a is a diapente, so from A to the other a there is a diapason: whose effect is having the same letter on either side, from B to 𝇒, from C to c, from D to d, and the rest. Thus as both sounds are notated by the same letter, so both by their property they are ascribed the most perfect similarity. For as, after seven days, the same [day] repeats; as always the first [primum] and eighth [octavum] days are called the same, so octaves [octavas] are written and called alike, for we perceive their natural harmony together, as D and d.

The question assumes that the seven-note scale arose by taking a subset of the twelve equal divisions of the octave, but this is not the case. Guido's introduction of two forms of B was the beginning of what later came to be known as chromatic alteration. With the advent of harmony, alterations of notes other than b and 𝇒 became necessary. There was E♭, then A♭. As the raised leading tone became important, F♯ was invented, then C♯. Eventually the sharps and the flats met somewhere around D♯/E♭ or G♯/A♭. Although there were some experiments with split-key keyboards to enable those pairs of notes to have different pitches, the winning strategy in the long run was to devise acceptable temperaments in which they could have the same pitch, leaving us with the 12-key keyboard we know today. In other words, the 12-tone system arose as an extension of the diatonic system, which came first.

There is another misconception in the question:

(during definition and use of the ionian major scale -I know modes weren't defined up until much much later-).

In fact, the modern Ionian mode was not identified as such until the sixteenth century, along with the Aeolian. The weird Locrian mode came later still (weird because its fifth degree is diminished). The other four modes are in fact older, being directly descended from their medieval authentic counterparts, developed in the 9th century. (These modes were named after the Greek modes, but they don't descend directly from them.)

In fact, the major and minor tonalities that were firmly established in the Baroque period developed from the four medieval modes during the Renaissance period. The modes were much earlier, not much later. Major and minor tonality developed as a result of the influence of harmonic considerations on various contrapuntal conventions, and the Aeolian and Ionian modes were subsequently invented to reflect this.

The basis of minor keys is the Dorian and Phrygian modes (primarily the Dorian), while the basis of the major keys is the Lydian and Mixolydian modes. Harmonic considerations led to the raising of the leading tone in Mixolydian and Dorian, the lowering of the fourth degree in Lydian, the raising of the second degree in Phrygian, and the lowering of the sixth degree in Dorian (and even today, the sixth and seventh degrees are so commonly raised or lowered in minor keys that we are taught three different forms of the minor scale, one of which even changes depending on its direction).

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  • Worthwhile adding some dates for Guido? Wikipedia gives "991/992 – after 1033". – Aaron Oct 28 '20 at 6:02
  • @Aaron yes, thanks. I had intended to do that but spent so much time on the translation that I forgot. – phoog Oct 28 '20 at 6:03
  • Glad I'm not the only one who gets lost in my posts. :-) – Aaron Oct 28 '20 at 6:05
  • @phoog thank you for the great answer! For me this indicates that the term octave was indeed added in retrospect, as there was another initial term (diapason), and after some advances in music theory (definition of notes, scales, which led to 7 note scales and thus to the term octave). Is that correct? In my question I indicate "early music theory" which is vague and can be confused with modern music theory, but Guido's system already sounds like a development in that area. – hirschme Oct 28 '20 at 14:18
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Absent a source indicating when the term "octave" can into common use, here are some references to terms implying numerical descriptions of intervals.

Ancient Greek theorists used the term "diapason" to refer to what we call the "octave." However, they also used the term "diapente", a clear numerical reference, for the fifth. "Diatessaron", the fourth, is also a numerically based term used by the Greeks.

Strings with lengths in the ratio 2 : 1 produced the interval of an octave known to the ancient Greeks as diapason, Those in the proportion 3 : 2 produced the interval of the fifth, known to the Greeks as diapente. Strings of equal tension with length in the proportion 4 : 3 produced the interval of a fourth known to the Greeks as diatessaron. The Greek word dia meant between, through or across. (SOURCE)

See also: https://en.wiktionary.org/wiki/diapason

The term "diocto", clearly referencing an 8-based pitch system, was in use by 562 C.E.

The consonance diapason, also called diocto, results from the ratio 2:1 ... and includes eight sounds, hence also the name it takes of diocto or diapason -- since the citharas of the ancients had eight strings this consonance, including as it does all sounds, is called diapason (literally, through all).1

"Octave" (also "fourth" and "fifth") is used in translations of the (pseudo-)Aristotlian Problemata (ca. 300 B.C.E. -- 600 C.E.), but it's unclear whether these are used by the translator as terms of convenience, or whether they reflect a numerical usage in the original.

Why is the octave the finest concord? Is it because the ratios are between terms that are wholes, while those of the others are not between wholes? ... for the fifth, which is hemiolic, is not of whole numbers ... the case is similar with the fourth....2


1Oliver Strunk, Source Readings in Music History: From Classical Antiquity through the Romantic Era (1950, W. W. Norton and Company), page 89. The passage is from Cassiodorus's Institutiones, believed to have been written between 543 and 562 C.E. (combining dates giving in Wikipedia and Strunk). In a footnote, Strunk explains that Cassiodorus's statement is, in fact, incorrect. I include it here, because of its connection to the notion of eight pitches.

2 Andrew Barker (ed.), Greek Musical Writings: Volume II -- Harmonic and Acoustic Theory (1989, Cambridge University Press), page 93.

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Lots of repeatable things (or cyclic) are named as a closed interval. There are 7 notes in the Western scale but early theorists liked to include the closing note (giving 7 intervals) A-B-C-D-E-F-G-A. Compare with some Spanish dialects calling a week "ocho días."

Another reason comes from the old Greek theory of tetrachords; these are 4-note patterns (like ABCD or CDEF or EFGA) which can be abutted to make a scale. Two 4-note objects seem to have been called 8-notes (even though the overlap means only 7 different notes.) Combining CDEF with GABC yields 8 notes but octave equivalence means the first and last notes have the same name.

The acoustical almost-equivalence of frequency doubling leads to the term "octave" meaning "frequency doubling."

It's like starting counting with 0 or 1; each may make sense in the context used.

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  • My issue doesn't have to do with 7 notes and counting methods. It has to do with the chronology of a 12 semi-tone system related to the term octave. In my question I assume that the 12 tone system preceeds the octave, as the later seems to refers to an already ordered sub-set of a 12 tone system and is related to scales, which indicate development of some level of music theory. The interesting part of your answer is that you point out that in Greek music theory we already had a reason for an 8 note system, possibly preceeding a 12-note system, proving me wrong in my assumptions – hirschme Oct 28 '20 at 3:47

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