Talk:Just intonation/Archive 1

Archive 1

Update external link to JIN

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

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

Old URL: www.dnai.com/~jinetwk

New URL: www.justintonation.net

Please update you link.

thank you.

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

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

On the contrary - this link has nothing to do with Just Intonation and appears to be spam for personal injury lawyers. Removing. — Preceding unsigned comment added by 70.196.192.193 (talk) 14:59, 2 August 2012 (UTC)

Continued Fractions and the Just Intonation Major Scale

I'm putting this section up front because it can be used to illuminate most discussions about harmonic ratios. It also clearly shows the short comings of the original article in extremely objective terms. I suggest that a re-write may be in order.

What are continued fractions and why are they illuminating? Continued Fractions is a well-known mathematical method that can find the simplest and most accurate possible fractional approximation to any arbitrary ratio. In some sense the continued fraction approximations are the best fractional approximations to any arbitrary ratio. Any fractional approximation to a ratio which is NOT a continued fraction approximation will either have larger integers in the numerator and/or denominator of the approximating fraction, or the difference of the fractional approximation from the approximated value will be greater (than a continued fraction approximation).

In other words, continued fractions can expose how harmonic an arbitrary ratio is by finding its best fractional approximations and examining how simple and accurate they are. For example, the first continued fraction approximation of a tempered perfect fifth that is closer than a semi-tone is 3/2 and differs from the tempered perfect fifth by only 1.95 cents.

Rather than give you the whole continued fraction thing I will give a simple method to use it.

0) First key the ratio to be approximated into a hand calculator, preferably one with a reciprocal function.

1) If the number is greater than one, subtract out the integer part and write the integer part down.

2) Take the reciprocal of residue that results from the subtraction.

3) repeat 1) & 2) a few times using the last result of 2) as the starting number for the next 1). This will give you a series of integers you have written down and perhaps a residue on the calculator.

If you exactly reverse the just described steps by adding the last integer to the current residue and taking the reciprocal, and backup through the list of integers in a similar fashion, you can reconstruction the original ratio you started with. Even more interesting, if you approximate reversing the original steps by zeroing the residue and starting with any intermediate integer and working backwards as just described, you will get a APPROXIMATION of the original arbitrary ratio. On the calculator you will get the answer in the form of a decimal fraction, but by doing integer calculation with paper and pencil you can get the answer as a "best" fractional approximation of the original ratio (there are a series of such approximations and which fraction in the series you get depends on which integer you decided to start with).

The more integers that you take, the more complex the fractional approximation and the closer it will be to the original number. If you use too many integers you will use up the precision of your calculator and the integers will be random and meaningless. If the original ratio was an exact fraction you will get just so many integers before all that is left as a residue is an integer with zero or almost zero value to the right of the decimal. (Often the result will not be exactly zero because of rounding errors due to the finite limit of precision of the calculator). Of course, the larger the numerator and denominator, the less harmonic significance a result has because it is more arbitrary than harmonic. Remember that the simpler the resulting fraction is, the more harmonic it "sounds".

In the fractional form it can be fun because you could start with a messy decimal fraction like 7/17 (e.g. 0.4117647...) which would give you the integers 2, 2, 3. Reconstructing would be 2 + 1/3, and then 2 + 3/7, and then 7/17. Or we could drop the last 3 and reconstruct just using 2, 2. This would give us 2 + 1/2 or 2/5. Notice that 2/5 is a good approximation to 7/17 with only an error of about 0.012...

Note that this process will work on any arbitrary ratio such as an irrational number that has no exact fractional representation because it would take an infinitely large integer numerator and denominator to "exactly" represent the irrational ratio.

Again, the fractional approximations you build in this way are in some sense the simplest and most accurate fractional approximations conceivable.

NOW COMES THE INTERESTING PART. If you take each note in a ET major scale, and find the first fraction of the corresponding continued fraction series (that is what you were finding with the hand calculator), that is closer than 1/2 the ratio between 2 adjacent half steps, THAT FRACTION WILL BE EXACTLY THE CORRESPONDING NOTE OF THE JI SCALE!!!!

What does this tell us? It tells us that several hundred years ago, musicians found the PUREST note (e.g. in some sense, the simplest harmonic ratio) corresponding to each note of the scale and put it together in what we now call the Just Intonation major scale. Now what is really going on some of the following comments make sense. The human ear could care less whether a not is one series or another, it only knows and recognizes harmonic relationships according to how simple the fractional ratio of a pair of tones is.

This shows that JI in the context of a continuing fraction analysis, is demonstratively the most harmonic scale possible based on 12 more or less equal intervals within an octave. This suggests that human ear has the capability of recognizing these harmonic relationships, if they are simple or "pure" enough. That makes these observations interesting. --HonestGent (talk) 03:25, 7 November 2009 (UTC)

JI/ horn deletion

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

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

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

Actually, I think the whole section (as it was) is rather silly (especially given that there are only two instruments, one of which is rather bizarre). There are thousands of instruments capable of just intonation not mentioned here, many quite conventional. The question is more whether or not they are ever asked to.

However, on the Natural Horn specifically, I think I'd have to argue with you. It produces a harmonic series quite easily, and quite naturally. To produce other notes, yes, there is a stopping technique that is used to flatten pitches. The seventh harmonic is actually used a fair bit, though you're right that the 11th is hardly ever used (Benjamin Britten's Serenade is a fun exception). However, in its standard technique, the harmonic tones which are used (1,2,3,4,5,6,(7),8,9,10,12,(14),15,16) are indeed just.

You might argue that if this is true, the natural horn can only be played just in one key then. This is also true. A quick study of natural-horn

Just tuning

I was going to merge the content below from Just tuning, but which "one possible scheme of implementing just intonation frequencies" does the table show? Hyacinth 10:29, 1 Apr 2005 (UTC)

It shows the normally used just intonation scale - I don't think that it has a special name. It can be constructed by 3 triads of 4:5:6 ratio that link to each other, e.g. F-A-C, C-E-G, G-B-D will make the scale of C. Yes, this should definitely be included. (3 April 2005)

Please Wikipedia:Sign your posts on talk pages. Thanks. Hyacinth 22:03, 3 Apr 2005 (UTC)

---

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

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

There are many semitones. A minor third is a chromatic semitone (25/24) smaller than a major third.

I can also give you a reductio ad absurdum for your reasoning. If 6/5 is an augmented second, then 5/4 * 6/5 = 3/2 is not a perfect fifth, but a doubly augmented fourth. Then if 4/3 is a perfect fourth, 4/3 * 3/2 = 2/1 is not an octave, but an augmented seventh. —Keenan Pepper 00:32, 14 April 2006 (UTC)

Why not split the difference, and call a semitone the twelfth root of 2, or 1.059463...? right between 16/15 at 1.066667 and 25/24 at 1.041667. Even better, one could call a minor third 300 cents, or the fourth root of 2, again smack between those silly over-simplified integer ratios. Of course I'm kidding; thanks, KP! Ivan, I'm not familiar with your Scholes reference; how completely does it treat the differences between just tunings and various temperaments? I'm guessing that's where the oversimplification may lie. Just plain Bill 01:49, 14 April 2006 (UTC)

Okay, I think I missed a key word in the Scholes Oxford Guide to Music text, approximately like this ...

If an inverval be chromatically increased a semitone, it becomes augmented.

I am looking at this as a mathematician and so the musical terminology throws me somewhat: dividing by 5 being called thirds and dividing by 3 being called fifths, etc. If I were to rewrite the Scholes text as shown below and use 25/24 as a definitiion for a 'chromatic semitone' as per Keenan Pepper's remarks, then I'd agree.

If an inverval be increased by a chromatic semitone, it becomes augmented.

The way this arose was me looking at the ratios with my mathematical background. Prime factorisation of integers is unique. The only primes less than 10 are 2,3,5, and 7. I gather that 7 is used for the 'blue' note in blues, and that most western music just uses or approximates ratios based on 2, 3 and 5. With 2 being used to determine octaves and with notes an octave apart being named similarly, that brings practical ratios down to just determining the power of 3 and the power of 5. I was making a 2 dimensional table of the intervals, and putting names to the numbers with the help of a borrowed book, but it appears the complex terminology for simple mathematics got the better of me.

I am happy to drop my remarks and delete all this in a few days time (including Keenan's and Bill's remarks), but I'll give you chance to read it and object before I do. Maybe some moderator will do that. Maybe you two are the moderators. Whatever. Anyway, thanks guys.

Ivan Urwin

You're mostly right about the primes. See Limit (music).

The labeling of intervals as "seconds", "thirds", etc. corresponds to the number of steps they map to in 7-per-octave equal temperament. 3/2 maps to four steps, so it's a "fifth", 5/4 maps to two steps, so it's a "third", 16/15 maps to one step, so it's a "second", and 25/24 maps to zero steps, so it's a kind of "unison" or "prime". Intervals separated by 25/24 have the same name, for example the 6/5 "minor third" and the 5/4 "major third".

Conversations on talk pages are usually never deleted, only archived when they become too long, so don't worry about that. —Keenan Pepper 05:04, 15 April 2006 (UTC)

I'm lost here (not a moderator, by the way, just another netizen) when you speak of "determining the power of 3 and the power of 5." The interval of a fifth is just the musical pitch space between the first and fifth notes of a scale. Because I happen to be used to vibrating strings, a perfect fifth being a 3/2 frequency ratio now seems as obvious to me as the fact that x^2+y^2=1 makes a unit circle, just a matter of familiarity. I'm equally happy to talk about it or to send it to oblivion as you suggest.

We are lucky to have folks around like Keenan Pepper who can quickly point out the discrepancy in types of semitone, for example. Just plain Bill 03:56, 15 April 2006 (UTC)

This is the sort of table I had. You can see that going up a major tone consists of going right couple of cells, so looking at the entry 10/9 (minor tone), I could quickly see that a mojor tone higher than that would be a third, and similarly a major tone higher than a semitone would be a minor third.

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

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

Ivan Urwin

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

---

content for merge:

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

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

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

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

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

There is a difference between Gb and F# which amounts to a ratio of ${3^{12}}/{2^{19}}$ = 1.0136433 as discovered by Pythagoras.

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

Meantone

Shouldn't the section on meantone be deleted? Meantone is a temperament. How does it belong here any more than a section on 12-TET? Caviare (talk) 04:14, 3 August 2015 (UTC)

I can see a certain logic from the context in which it is presented, since meantone compromises the perfect fifths in order to obtain just major thirds. The juxtaposition with Pythagorean tuning is instructive concerning the conflict created by combining just thirds and just fifths in any tuning system, though this aspect really needs to be explained in order to justify (no pun intended) the inclusion of meantone at all.—Jerome Kohl (talk) 17:50, 3 August 2015 (UTC)

Indian Scales

I will freely admit I am not an expert on Indian Classical music, but I study and play sitar and the Pythagorean Tuning is much closer to how my instrument is tuned than the Just Diatonic mentioned in the first sentence of the Indian Scales section. My instrument 'sounds correct' when Ga is ~404.3 cents versus 386.3 cents defined in the Just Diatonic tuning. A difference of 18 cents, while 404.3 is -3.5 cents relative to the Pythagorean tuning. Ni is also substantially 'off', 1104.6 cents on my sitar versus 1088.2 for Just Diatonic, a difference of 16.45 cents. And my Ni s 5 cents lower than the 1109.7 cents defined in Pythagorean. I am measuring the frequency of my sitar using the Peterson iStrobeSoft tuner. I tune it to have 0 beats relative to Sa for each note, and my sitar is in tune with my teacher's sitar, a master of the instrument. When I moved the Ga and Ni frets to match the Just Diatonic tuning, it sounds pretty awful so I don't think it is correct to say that the Indian scale is Just Diatonic.Chuckpwhite (talk) 04:01, 2 February 2016 (UTC)

The most commonly acknowledged system of shruti placement, keeping in mind that shrutis are not fixed pitches strictly speaking, posits the duality of ten possible pitch classes relative to a fixed tonic / Sa, and fifth / Pa. These may be considered as the Pythagorean twelve tone system together with the 5-limit alternatives of the ten degrees that are not tonic nor fifth. The 5-limit alternatives are a syntonic comma removed from their pythagorean relatives. Thus the twenty two shrutis may be represented by their relation to a theoretical twelve tone scale defined thusly; tonic / Sa / ratio 1:1 ; a 'semitone' / re / pythagorean ratio 256:243, and/or 5-limit ratio 16:15 ; 'whole tone' / Re / pythagorean ratio 9:8, and/or 5-limit ratio 10:9 ; 'semiditone' / Ga / pythagorean ratio 32:27, and/or 5-limit ratio 6:5 ; etcetera. This structure gives dual values for the intervals of re, Re, ga, Ga, Ma, ma, dha, Dha, ni, and Ni. These are useful to various contexts of interval movement in order to 'correct' intervals in terms of their origination and destination, and also in terms of placement and movement regarding the various deflections, undulations, glissandos and other grace ornamentation / gamak used in Indian classical music.

To add to these complications, one must also consider that variations of these intervals, such as the use of the septimal intervals in place of the Pythagorean, may be utilised in practice. This seems to be in keeping with the complex systems of 55 commas to the octave used by baroque western classical theory, including the various dieses amd commas, as well as the system of 66 shrutis to the octave. This last may indicate, not only an incorporatiom of higher n-limit interval ratios but also the possibility of superimposition of twenty two shruti systems. The mathematical legacies of Indian classical music are interesting to consider in conjunction with those of the Greek classical systems based upon tetrachord usage and mathematical string division. — Preceding unsigned comment added by Daniel Z. Franks (talk • contribs) 08:22, 29 February 2016 (UTC)

Staff notations

The current 'Staff Notations' section concentrates on the extended Helmholtz-Ellis notation, however in my opinion this section is incomplete since it does not mention several other important systems.

I've been researching notation systems for free Just Intonation over the last couple of years. I have developed my own system (Rational Comma Notation, or RCN), and I am also aware of at least two other systems which should be included: Sagittal (which can notate both just and tempered tunings) and Kite's color notation.

One reason to include these three, instead of just extended-HE notation, is because extended HE cannot notate the whole of free Just Intonation, but only up to a low prime limit, at the moment the prime 61. However, all of RCN, Sagittal and Kite's color notations can notate the whole of free-JI. This is due to defining 'prime commas' by algorithm by every higher prime number. Conversely, in extended-HE, there is no algorithm, no higher prime commas, and no general system for notation higher prime alterations.

Here are primary sources for each of the three notation systems: RCN: Sagittal: Kite's color notation:

Maybe someone else can provide secondary sources?

Thanks! Davidryan168 (talk) 13:34, 2 February 2017 (UTC)

I've researched into secondary sources for Sagittal, which is a system I've had a long term interest in as a music software developer as a potential way to show musical intervals to users. I think the citations I found make it sufficiently notable to justify a separate article on it, with several academic citations, and it's also included in the SMuF music font originally developed by Steinberg and now overseen by the W3D Music Notation Group. I propose making it as a separate article Sagittal (Music Notation System). The draft is in my user space here: User:Robertinventor/Sagittal. Corrections and citations welcome. I haven't been able to find much by way of secondary sources yet for the other two systems you mention, so they will depend on finding citations to support them. Robert Walker (talk) 14:30, 9 February 2017 (UTC)

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Just intonation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://web.archive.org/web/20120824020439/http://www.furious.com/perfect/experimentalstringinstruments.html to http://www.furious.com/perfect/experimentalstringinstruments.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 20:51, 29 April 2017 (UTC)

External links modified

Hello fellow Wikipedians,

I have just modified 2 external links on Just intonation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://web.archive.org/web/20110514002750/http://artofthestates.org/cgi-bin/genresearch.pl?genre=microtonal%2Fjust%20intonation to http://artofthestates.org/cgi-bin/genresearch.pl?genre=microtonal%2Fjust%20intonation
• Added archive https://web.archive.org/web/20100630223629/http://users.rcn.com/dante.interport/justguitar.html to http://users.rcn.com/dante.interport/justguitar.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 10:07, 3 December 2017 (UTC)

Mistake in audio example description?

This doesn't make sense:

"In the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz"

Humans can't hear that range! It should be kHz, right? — Preceding unsigned comment added by 71.230.123.65 (talk) 18:53, 16 January 2018 (UTC)

Hz is correct. Beat (acoustics) may be heard as pulsing, or volume variation, at those rates. Just plain Bill (talk) 20:03, 16 January 2018 (UTC)

Not Exactly....

This has been alluded to in a previous discussion of "cylce of fifths", but a little more is needed. It is not possible to have (for example) a perfect fifth at exactly 3:2 and an octave at exactly 2:1 AT THE SAME TIME, due to the mathematice theorem of Unique Prime Factorisation. it is kinda obviosu in this example - octaves are always going to be powers of two, so they can never be exactly divisible by three. I realize that this is not phrased very well, which is why I have left this here. — Preceding unsigned comment added by 121.212.139.63 (talk) 10:15, 11 March 2018 (UTC)

Intonate? Or tune?

The word "intonate" was used several in the article. I suspect this might be a Dutchism, and a better alternative might me "tune". Any objections?Redav (talk) 13:34, 14 March 2018 (UTC)

I’ve heard “intonating” used by a guitar tech to describe the specialized process of adjusting the bridge so each string’s stopped note at the twelfth fret was in unison with the second harmonic of the open string. In the context of this article, “justly tuned” seems like better diction. Just plain Bill (talk) 02:43, 15 March 2018 (UTC)

In my perception there is a difference between tuning and intonation. "Tuning" is a particular fixed system of tuning and the setting an instrument to such a system. Intonation is the adaptation of the pitch of a note while playing, depending of circumstances in the composition. −Woodstone (talk) 17:16, 15 March 2018 (UTC)

That's my sense of it as well. "Intonation" is a performance thing, while "tuning" has to do with a theoretical system or the adjustment of instruments. Just plain Bill (talk) 17:35, 15 March 2018 (UTC)

Not according to this article and, in my experience, "just intonation" is the correct expression for what is being described, whereas "tuning" is used to describe the treatment of individual intervals. Still, the verb "to intonate" is not appropriate in this context. According to the OED it had a now-obsolete sense meaning "to thunder forth; to utter with a loud voice like thunder", and retains a second sense "To recite in a singing voice", "to utter or pronounce with a particular tone", or (in phonetics, a rare usage) "to emit or pronounce with sonant vibration; to 'voice'".—Jerome Kohl (talk) 18:23, 15 March 2018 (UTC)

Thanks for bringing a general perspective to the thread. I admit being narrowly focused on "justly intonated" here. English idioms can be context-dependent, and even less logical than music theory. "Just intonation" is of course the correct term for the overarching topic. Just plain Bill (talk) 18:41, 15 March 2018 (UTC)

“Harmonics” as a unit of… something?

@Jerome Kohl: You say that the term harmonic, as used in the chart in § Diatonic scale, is explained in the lead. The word is used in the lead, but I don’t think it’s used in the same sense—i.e., what it means for, say, the tone of D (does it matter what octave?) to have 27 harmonics. The only other use of the word in relation to absolute numbers in this article is with fractions less than 1, when discussing the guqin. But whether or not the lead explains the word, it would still be helpful to have some mention of its meaning in the vicinity of the table that uses it. I’d add an explanation myself, but I have no idea what that line of the table is trying to say. —67.14.236.193 (talk) 01:36, 17 June 2018 (UTC)

If the 24th and 48th harmonics of some pedal tone (corresponding to 24x and 48x multiples of some base frequency) are mapped to the tonic and octave, then each note of the diatonic scale maps to a harmonic between them. Harmonics lower in the series are too widely spaced to give that kind of granularity. Explaining that in a few words to fit in that section will need some careful writing. Just plain Bill (talk) 04:06, 17 June 2018 (UTC)

Well, yes, exactly. However, at a much more basic level, is it really necessary to explain that "a harmonic" is one element of the "harmonic series" of any arbitrarily chosen pitch frequency? @67.14.236.193:, why do you think that there are only 27 harmonics in a harmonic series? I really don't understand your question.—Jerome Kohl (talk) 06:40, 17 June 2018 (UTC)

I have some idea of what that line of the table is trying to say, but I'm not sure what purpose it serves in this article. One of its implications amounts to "a natural horn player could play a justly tuned major scale whose tonic is the dominant of the horn's pedal tone raised by four octaves if only a horn's overtones were true harmonics." (Let's not burrow too far down into that just now.)

In other words, it is an abstract extension of the notion of small(ish) integer ratios making the intervals of just intonation. I happen to think it is an interesting, even amusing construction, and I thank 67.14.236.193 for calling it to my attention, but does it merit a line in the table?

(The interesting part is that 27:24 reduces to 9:8, 45:24 reduces to 15:8, 32:24 reduces to 4:3, and so forth.) Just plain Bill (talk) 13:00, 17 June 2018 (UTC)

@Jerome Kohl: I didn’t say anything about “only 27 in a series.” The table lists “27” under the “D” column of the “Harmonic” row. I was asking what that meant, or more precisely, why the article didn’t make clear what that meant. —67.14.236.193 (talk) 15:17, 17 June 2018 (UTC)

Perhaps this is a case of my being too familiar with the subject to understand why everybody doesn't see it ias clearly as I do. This can also make it difficult to understand your question, since "27" in a row labeled "Harmonic" obviously (to me) means harmonic number 27 in the harmonic series. How is this not clear, and what can be done to improve the description?—Jerome Kohl (talk) 22:45, 17 June 2018 (UTC)

Maybe something like this? If a reader tries to read this article without having an understanding of harmonic numbers and harmonic series (as I did), this should give them somewhere to start. But how much knowledge is safe to assume, and how much should be explained? Is this a solved problem on Wikipedia? We do have the “familiar with but not an expert” guideline, but I would think any musician would be familiar with tuning, but might be baffled by some parts of this article. —67.14.236.193 (talk) 00:49, 18 June 2018 (UTC)

So, what’s the word on jargon in this article and others like it? Should we do something? Is it fine as is? What knowledge is safe to assume? —67.14.236.193 (talk) 08:40, 28 June 2018 (UTC)

I agree with the comment that too much jargon is supposed to be understood. The naming conventions for intervals are based on a particular scale with uneven stepwidths. The names contain numbers like "octave", "fifth"and "third". The harmonics used for just tuning also are numbers, primarily 2, 3 and 5, but these connect to the scale names in a very awkward way. The fifth is related to the third harmonic and the third to the fifth. And why is a factor 2 named as eighth (octave)? Furthermore a (default) third is 4 semitones, a (perfect) fifth 7 semitones, and an (eighth) octave 12 semitones. Enough to confuse a novice.

I have tried in the past to eliminate the traditional interval naming from the first few sections, but was overridden. So who has the courage to make another attempt?

−Woodstone (talk) 10:38, 28 June 2018 (UTC)

Why not keep the naming, but explain it to avoid confusing readers unfamiliar with it? Unless of course the jargon is truly unnecessary, which I have no idea if it is. —67.14.236.193 (talk) 16:39, 30 June 2018 (UTC)

I’ve tagged the article with {{jargon}} in the hopes that, if no one in this duscussion takes the initiative, someone else knowledgeable notices. I’d handle it myself if I were. —67.14.236.193 (talk) 23:40, 30 June 2018 (UTC)

You might consider tagging Harmonic (mathematics) in the same way. However, I would not be surprised if you got an indignant response from some editor, to the effect that there is no everyday term that corresponds to Laplacians or eigenvalue, and explaining them at length in that article is pointless because there are links to those terms that serve the purpose. Whenever you are dealing with a technical subject, jargon terms are bound to present this difficulty. There are no simple-English replacements that accurately reflect their meaning, and making up a whole new terminological system to replace the established one is not acceptable on several grounds, not least because the "knowledgeable" reader will object to replacing perfectly serviceable words with ones that fail to evoke the right context for the discussion. I will continue to examine this text to see where this issue for the novice can be addressed without doing violence to technical precision, but it is a bit like trying to explain a motorcycle engine without using jargon words like "compression" or "cylinder".—Jerome Kohl (talk) 00:04, 1 July 2018 (UTC)

I’m not suggesting we replace the technical words or units with simpler or made-up ones. We’re not writing for idiots, but we’re not writing for experts, either. I’m just worried that a reader fmailiar with tuning an instrument, but not familiar with the science and math fields behind it, would be lost here. A complex article on motorcycle engines might explicitly direct readers to a more introductory article on engines, or an article on a given part or type, and otherwise assume familiarity with how engines work. For example, consider adding a brief background section with {{main}} tags pointing to the full treatment article(s) of the prerequisite notions; this approach is practical only when the prerequisite concepts are central to the exposition of the article's main topic and when such prerequisites are not too numerous. —67.14.236.193 (talk) 00:30, 1 July 2018 (UTC)

I do take your point (really!), but I am beginning to discover that the problem does not end with explaining the jargon terms to the layman. It seems that some of the "experts" who have contributed to this article either do not understand some of those terms, or are not very good at explaining things using them. Perhaps a Glossary section could be added, since there are at least a dozen, perhaps many more than a dozen such terms found in this article?—Jerome Kohl (talk) 00:53, 1 July 2018 (UTC)

Sorry for overstating my point, then! Hm… what’s WP’s position on glossaries? (Edit: they’re fine.)

Sounds like a solution to me! —67.14.236.193 (talk) 01:11, 1 July 2018 (UTC)

All right! I've just been looking at the guideline for glossaries, and it sounds like an embedded one should work here (more than five but fewer than twenty-five terms). Or would it? Perhaps we can begin here on the Talk page with a list of terms requiring explanation, in order to see how many we are actually talking about. "Harmonic" is obviously one. Bluelinked terms currently include "ratios" "fractions", "solidus", "hertz", "Pythagorean tuning", "perfect fifth", "cycle of fifths", "wolf fifth", "closure (mathematics)", "perfect fourth", "major third", "syntonic comma", "five-limit tuning", "guqin", "overtone", "diatonic scale", "chromatic scale", "semitone", "minor tone", "major tone", "major triad", "semiditone", "minor seventh", "enharmonic", "pitch", "irrational numbers", "diminished fifth", "Indian music", "shruti (music)", "modulation", "microtuner", "a cappella", "leading tone", "atonality", "prime number", "double octave", "square", "limit", "prepared guitar", "extended technique", "node (physics)", "multiphonic", "equal tempered", and "prime limits". That makes 44 terms, but not all of them are good candidates for a glossary (e.g., "Indian music", "guqin"). Some are defined within the article, but which ones are left dangling? Of these, which are essential to understanding the article?—Jerome Kohl (talk) 01:42, 1 July 2018 (UTC)

Restructuring the article

Would it be an improvement to restructure the article along the harmonics used. Only talk about second, third etc harmonic and avoid the (in this context) confusing words of the intervals "a third", "a fifth" and even "octave". For the explanations use only the supposedly known C, D ... names of the "white note" scale.

• 2nd harmonic: explain it is universally recognised as "the same" note.
• 3rd harmonic and Pythagorean tuning.
• 5th harmonic, point out the 4:5:6 relation in the simplest chords, building the diatonic scale.
• Pointing out the oneven steps in the scale and fill up to 5-limit chromatic scale.
• Only then connect for the first time to the conventional music theoretical interval names.
• 7th harmonic (add a section it's problematic)

What does everyone think? −Woodstone (talk) 07:45, 13 July 2018 (UTC)

Are you referring to the "Diatonic scale" section, or the entire article? If the latter, I would like to know how you mean to incorporate the material on Pythagorean tuning, Indian scales, and the like. As for "confusing words", how do you mean to describe the intervals represented by various pairs of harmonics if you do not use the conventional terms for intervals?—Jerome Kohl (talk) 17:52, 13 July 2018 (UTC)

I was aiming at the sections "history", "diatonic scale" and "twelve tone scale". The intervals are initially indicated by pairs of letters A to G, showing their sizes in a table (size in fractions and decimal, not yet cents). Then adding flats and sharps for the chromatic scale. After that, introduce the conventional names and comparison to cents for the rest of the article. Perhaps an intervening section about transpositions and resulting problems makes sense. a−Woodstone (talk) 13:59, 14 July 2018 (UTC)

That sounds reasonable to me, though I still wonder about delaying the interval names, especially since the overtone series runs into a problem already with the seventh overtone. Or maybe not, if we can assume readers are not fixated on 12-equal tuning. What are you planning on doing in the table with F♮?—Jerome Kohl (talk) 00:02, 15 July 2018 (UTC)

I propose to use the 4:5:6 ratio in the chords F:A:C, C:E:G, and G:B:D and set C arbitrarily to 1. That avoids having to use a series of very high harmonics to represent the scale. So it's not introduced as a selection of very high harmonics from one pedal tone, but a chain of harmonics from note to note (string, pipe etc). −Woodstone (talk) 08:34, 16 July 2018 (UTC)

Which means you are limiting your discussion to 5-limit JI. Well, as far as I am concerned, go ahead. If I see any problems, you will be the first to know!—Jerome Kohl (talk) 18:46, 16 July 2018 (UTC)

Actual Music

Maybe the moderators of this article would like to refer to www.ji5.nl for actual sound files demonstrating the alledged suitability of just itonation for organ music. Music by J.S. Bach, L. Boellmann, M. Reger and others. Kind regards, Erik Zuurbier, 22 October 2018 — Preceding unsigned comment added by 77.163.110.153 (talk) 14:51, 22 October 2018 (UTC)

New section "ratios"

The new section named "ratios" is definitely not an improvement on this article. Half of it is trivial primary school mathematics, the other half a rather esoteric view on just tuning. It will scare off almost all readers. I have a strong desire to scrap it. −Woodstone (talk) 05:16, 29 October 2018 (UTC)

Kind of agree. There might be a place for it somewhere, but not up top of this article. —Wahoofive (talk) 22:16, 29 October 2018 (UTC)

It might find a place at Harry Partch or a related article. Just plain Bill (talk) 00:48, 30 October 2018 (UTC)

Too technical

What parts of this article are too technical and possibly hard to understand? Anything specific that seems breezed over and not explained enough? Hyacinth (talk) 06:42, 25 October 2019 (UTC)

I'm not the one who tagged it, but I'd say the lede could have some esoteric detail removed, e.g.:

• The second paragraph with its references to 5-limit and 11-limit etc. is unnecessary for understanding the basic concept. The references in the third paragraph to "microtones" are similarly unhelpful for lay readers, who don't really need to know the difference between 10:9 and 9:8 whole steps to grasp the general idea. These details belong in the article, but they clutter up the lede with too much detail.
• All references to "cents" in the lede are unhelpful to readers who aren't already versed in the subject.
• The example in the first paragraph could be rewritten to reference the illustration.

—Wahoofive (talk) 00:17, 26 October 2019 (UTC)

Followed my own advice and fixed it. —Wahoofive (talk) 20:04, 28 October 2019 (UTC)

History needs fixing

I added a section on "terminology" which might help inexperienced readers. But I can't really fix the "history" section because I don't know much about it. The article Pythagorean tuning attributes it to Pythagoras, but a footnote on this page says it's Babylonian. The history section includes a bunch of unnecessary mathematical detail, but I'm loth to take it out without having some real history to include instead, especially history from non-Western cultures. Who can help on this? —Wahoofive (talk) 23:12, 7 November 2019 (UTC)

Formatting in boxes in tables

I recently made an edit "fixing" the formatting of the tables which was reverted by @Woodstone, however, if the boxes in the lower part of the tables are meant to span 1/2 of the boxes in the upper part then the formatting is completely broken for me in chrome, is this normal? IgnacioPickering (talk) 01:06, 18 July 2021 (UTC)

The table looks as intended in Edge. After the comment above I checked Chrome and indeed, it is wrong. I also checked the Brave browser and it presents the table correctly. In the past Chrome did this right, so it must be a new bug in Chrome. The intervals should span (about) half of one note and half of the next one. Before and after the intervals half a note box should be empty. Let's hope Chrome corrects this soon. I will do some tests to see if it can be fixed so that it looks good on all browsers. −Woodstone (talk) 05:00, 18 July 2021 (UTC)

I downloaded and tried Firefox and it displays as intended. Chrome is alone in this error. There is a solution, but not elegant. Insert a small row with only empty cells between the two parts. See below.

Note	Name	C		D		E		F		G		A		B		C
	Ratio from C	1:1		9:8		5:4		4:3		3:2		5:3		15:8		2:1
	Harmonic of Fundamental F	24		27		30		32		36		40		45		48
	Cents	0		204		386		498		702		884		1088		1200
Step	Name		T		t		s		T		t		T		s
	Ratio		9:8		10:9		16:15		9:8		10:9		9:8		16:15
	Cents		204		182		112		204		182		204		112

To make the bug very clear one can add in the line starting the empty row style="display: none" to make the error pop up again. Also heaving a row with colspan=18 repeats the error. What do you think: is this a reasonable temporary solution to support the most used browser? −Woodstone (talk) 06:54, 18 July 2021 (UTC)

Octaves

It is not true that: "Acoustic pianos are usually tuned with the octaves slightly widened". Octave are always pure in almost every tuning system ever devised. 2A02:A03F:69BA:5100:7C01:156C:7ACF:5C80 (talk) 11:28, 20 September 2022 (UTC)

Please Google "Railsback Curve", read what you find and come back here to apologize for what you said above. −Woodstone (talk) 15:09, 20 September 2022 (UTC)

@Woodstone: please be polite to your fellow contributors (even the anonymous ones). The-erinaceous-one (talk) 08:04, 25 September 2022 (UTC)

Scope? Accuracy?

There seems to be a lot here that isn't to do with Just Intonation itself. Why is Phythagorian tuning here? Why is there 20th Century stuff? This might be helpful *context* for Just Intonation, but this isn't explained and is frankly confusing.

I don't see any mention of Mersenne, Zarlino and so on who discussed these things quite a long time ago and should presumably be definitive sources...

A look in Groves' (at Just Intonation, and Temperament) might help quite a lot here. Sorry I don't have time to take this on myself. — Preceding unsigned comment added by 94.1.250.30 (talk) 14:39, 7 March 2020 (UTC) Why is Pythagorean intonation in here? It's arguably the very basis for the rest and *is* just intonation. The theorists mentioned there are standing on it, and came long after. I would say feel free to edit and add that, there are numerous authors of this, but I'm not sure you're the person for that job.J Civil 17:49, 9 November 2021 (UTC)

Hello. I've just made some improvements and one or two corrections here. Harry Partch was a modern champion of just tuning, worth a link and mention I would have thought. I'll be back. Thelisteninghand (talk) 14:39, 4 June 2022 (UTC)

Something I feel like is worth noting is that, although Pythagorean tuning is technically equivalent to 3-limit JI, it has different goals behind it, at least in almost every presentation of it I've ever encountered. (I have multiple academic sources on hand that actually contrast what they call "Pythagorean tuning" and "just intonation," not treating one as part of the other at all—one even treats them as kind of opposite.) My understanding is that the point of Pythagorean tuning is so that you have maximally in-tune fifths (wolf aside), despite what it does to the major thirds. This comes from a historical perspective that puts the fifth significantly higher than the major third in consonance, which constrasts with modern sensibilities. JI, as I understand it, is more about having your tuning closely approximate the harmonic series "overall," at least for the lower harmonics (with whatever value of "lower" the composer prefers). Those who are partisan to JI often argue for its superiority based on its stronger consonances compared to equal temperaments, and it would be kind of strange to prefer Pythagorean tuning on that basis, since its thirds are more out-of-tune than in 12-TET. Because of this, it doesn't surprise me that I haven't usually seen Pythagorean tuning referred to as a kind of JI outside of this article and the Xenharmonic wiki. It's not wrong to do so per se, at least for a certain formulation of JI, but it seems strange to me to simply say that and not give more historical context beyond noting that Pythagorean tuning is very old; it kind of gives the impression that JI is "timeless," which I feel is pouring old wine into new bottles a bit. (The Online Etymology Dictionary gives the earliest usage of "just" in a musical sense as 1850, which is also around the time that the modern interest in and controversy over JI began.) I'd really like it if the "History" section here made all of this more clear, and in some ways the whole presentation of JI's development and goals throughout the article. Mesocarp (talk) 18:37, 26 November 2022 (UTC)

Just to be sure, reading over this now I worry it comes off as a bit anti-JI. I'm not at all (I love Harry Partch's music for example) and don't see any reason why anyone should even stick to a single tuning system or anything; I just wish there was more context given for the ideas, which I think would benefit everyone. Mesocarp (talk) 20:04, 26 November 2022 (UTC)