© 1999 by Joe Monzo
(Originally posted to the Onelist Tuning List, appearing in Tuning Digest # 423, Message 10. Revised and updated for this webpage.)
For the record, I'm all for Johnny Reinhard's work on Ives's tuning. His Pythagorean version of 'Unanswered Question' had the most awesomely beautiful sound in the string section that I've ever heard in any performance of that piece. At the same time, I agree with Dan Stearns that Ives's conception of tuning was far more complex than simply having one personal favorite tuning.
I looked up some stuff on Ives at the library back when this thread first started (wasn't that a couple of months ago?), thinking I might find some evidence for Pythagorean. I copied one double-page from something called 'Scrapbook' (I think it was part of the 'Memos' but at this point I don't remember), without having read it, because it had a scale diagram that caught my eye.
I thought at first that it was some kind of overtone graph, but upon finally reading it the other day, found that it was an interesting scale Ives had come up with, based on two pianos in his church that were about 1/4-tone out of tune and some 'glasses' his father had made that played 1/8-tones.
Here's the quote - if someone could provide details for the citation, I'd appreciate it. (I think it's from 'Memos'.)
The editing and footnotes by me have been identified as such; the unidentified ones are by the editor of the book. I've done my best to make an ASCII rendering of the musical example. I'm not sure what the numbers I put in brackets mean - they appear in the margins of the book, and apparently refer to the cataloging of the original Ives manuscripts.
> -------Charles Ives, from 'Scrapbook', p 108-110 -------- > > > [51m] > > ... as a boy I had heard some quarter-tone experiments of > Father, and this division or other divisions of the tone were > not entirely unfamiliar to me. In the Sunday-School room of > the Central Presbyterian Chruch [*5], New York, there were, > for a while, two pianos which happened to be just about a > quarter tone apart, and I tried out a few chords then. > > > [m51v] > > In this connection, and also referring to Father's glasses > tuned in different intervals larger and less than quarter tones, > after hearing the two pianos out of tune in Central Church (but > as near as I could tell by listening and with tuning forks, [they] > were about a quarter tone apart) - a scale (to knock the octaves > and fifths out by wider intervals, stretching [the] whole and > half tones a little, but keeping the proportions of the scale) > - it was started or suggested by these two pianos, and glasses > between [the quarter tones]. But one piano was moved before I > could get it well grasped [*6] in my ears. It was mostly worked > out on paper, which I have in part (see back of _The Indians_ > score)[*7] - taking C as basis, 5 quarter tones up = whole > interval, and divided in [the] middle by [a] glass = 2&1/2 > [quarter] tones - that is:
[monz note: The diagram was presented horizontally by Ives; I present it vertically, and add mathematical notation from two different perspectives to specify the tuning.]
48-tET Ives larger scale Semitones
>
> Notes in
> old scale
>
> Eb 31 -+---- 8 Doh 2^(60/48) = 2^(5/4) 15.00
> |
> 30 -+
> |
> D 29 -+
> |---- 7 Te 2^(55/48) = (2^(5/4))^(11/12) 13.75
> 28 -+
> |
> C# 27 -+
> |
> 26 -+
> |
> 8 C 25 -+
> |
> 24 -+
> |---- 6 Lah 2^(45/48) = (2^(5/4))^(9/12) 11.25
> 7 B 23 -+
> |
> 22 -+
> |
> A# 21 -+
> |
> 20 -+
> |
> 6 A 19 -+
> |---- 5 Soh 2^(35/48) = (2^(5/4))^(7/12) 8.75
> 18 -+
> |
> G# 17 -+
> |
> 16 -+
> |
> 5 G 15 -+
> |
> 14 -+
> |---- 4 Fah 2^(25/48) = (2^(5/4))^(5/12) 6.25
> F# 13 -+
> |
> 12 -+
> |
> 4 F 11 -+---- 3 Me 2^(20/48) = (2^(5/4))^(4/12) 5.00
> |
> 10 -+
> |
> 3 E 9 -+
> |-- Re
> 8 -+
> |
> D# 7 -+
> |
> 6 -+---- 2 Ray 2^(10/48) = (2^(5/4))^(2/12) 2.50
> |
> 2 D 5 -+
> |
> 4 -+
> |-- De
> C# 3 -+
> |
> 2 -+
> |
> 1 C 1 -+---- 1 Doh 2^(0/48) = (2^(5/4))^(0/12) 0.00
>
>
(listen to a MIDI-file of this scale)
> > > [m52v] > > - (playing larger scale and then regular one alternately several > times [monz: MIDI-file of this] - and it is quite an interesting > sound difference and makes a kind of musical sense). > > New octaves, that is: > > /\ > / > / > / O > / --- > / > / --- > / > / --- > / cycle > / --- -O- > / > /\ / -O- --- \ --- > | | / \ > --|-|---/--------------------------------\-------------------- > |/ | \ > --|----------------------------------------\------------------ > /| \ > |- \-----------------------------------------\---------------- > | / | \ \ > |-|-|-|-----------------------------------#O---\-------------- > \___|/ #O \ \ > ----|-----------------------------------\--------\------------ > \__/ \ 4 > \ > \ > \ > -_..._---------------------------------------\---------------- > ' '\. \ > \------|---------------------------------------\-------------- > \ / . \ O \ > -----/-------#O-------------------------\--------\------------ > / \ \ 3 > ---/--------------------------------------\------------------- > \ > --------------------------------------------\----------------- > \ > --- --- \ > \ > -O- -O- \ > \ 2 >
[monz note: The notes and numbers on the right of this example apparently illustrate another similar scale that is stretched slighly more, so that its 'octave' is the next semitone higher. The scale under discussion has cyclic properties based on 'minor 10ths', the notes on the right seem to be a cycle based on 'major 10ths'. More on this below.]
> > = no octaves nor 5ths during each four octaves, or no > octaves nor 5ths for 48 half-tones, and [the] only interval > in common is [the] lower 4th. But [the] trouble is: - [the] > augmented 9th, taken as a scale length, may be confused with > [the] minor 3rd. I had some other division, where the scale > ended on a quarter-tone - can't find it.
[monz note: this 'other division, where the scale ended on a quarter-tone', evidently ended on some quarter-tone pitch which would have been the equivalent of the 'octave' in that scale. It seems to me most likely to have an 'octave' either between or on either side of the scales in the above musical example. The equivalence in the former scale of the 'augmented 9th' with the 'minor 3rd' - i.e., 'minor 10th' - apparently bothered Ives and that's why he came up with the second scale. Too bad we don't know its structure, but I think most likely we can assume that Ives diagrammed several scales having the same structure and varying only in the degree of 'stretch' involved. Some speculations on these scales appear below too.]
> In this larger scale [monz: that is, the one presented in the > diagram], there are but three intervals of even-ratio (so called): > (1) the 4th [of the old scale] = [the] 3rd [of the larger scale]; > or (2) from [the old] 4th to the top [of the larger scale] = > minor 7th [monz: of the old scale] = [augmented]* 6th [monz: > of the new scale]; and [3] the sum of (1) + (2) = from C to Eb top > = minor 10th [monz: of the old scale = 'octave' of the larger scale]. >
*[monz note: The editor is wrong here; '6th' refers not to the equivalence of the 12-tET or meantone 'augmented 6th' ='minor 7th', but rather indicates that this is the large scale's analog of the 'major 6th'.
It may help to present Ives's statement unedited and in tabular format:
> 3 intervals of even-ratio (so called):
>
> 12-tET diatonic 48-tET-subset
> 'old scale' 'larger scale'
>
> {1} the 4th = 3rd
>
> (2) from 4th to the top
> = minor 7th = 6th
>
> [3] the sum of (1) + (2)
> = from C to Eb top
> = minor 10th. = 8ve]
>
> The other intervals are uneven - some way out from a simple
> ratio [as] 2/1 - for instance 261/712 etc. This, at first,
> seemed very disturbing, - but when the ears have heard more and
> more (and year after year) of uneven ratios, one begins to feel
> that the use, recognition, and meaning (as musical expression)
> of intervals have just begun to be heard and understood. The
> even ratios have been pronounced the true basis of music,
> because man limits his ear, and not because nature does. The
> even ratios have one thing that got them and has kept them in
> the limelight of humanity - and one thing that has kept the
> progress to wider and more uneven ratios very slow - (it is said
> [that] for the power of man's ear to stand up against the
> comparatively uneven 3rds, [when used] to the very even octaves
> and 5ths, was a matter of centuries) - in other words, consonance
> has had a monopolistic tyranny, for this one principal reason:
> - it is *easy* for the ear and mind to use and know them - and
> the more uneven the ratio, the harder it is. The old fight of
> evolution - the one-syllable, soft-eared boys are still on too
> many boards, chairs, newspapers, and concert stages!
>
> --------------------
> Editor's notes:
>
> [*5] Ives was organist there from April 1900 through 1 June 1902.
>
> [*6] This word is hard to make out, but is quite possibly
> "grasped".
>
> [*7] There are three copies of the following diagram differing
> only in minor details: (1) on a rejected title page of "last
> Chorus" of _The Celestial Country_ (Q1718), back of which is
> p. 1 of the score-sketch of _The Indians_ (Q2838) - (2) in m51v
> - and (3) in m52v.
>
[monz note: I'd certainly like to know where to find the manuscripts of those diagrams 'differing only in minor details', to analyze precisely what the differences are. Johnny Reinhard offers another 1/8-tone scale on a _Universe Symphony_ sketch.]
------------------- end quote ------------------------------
Ives's 'larger scale' is based on an approximate 48-tET division of the 'octave'. Its 'whole-step' = 2^(10/48) = 2^(5/24) = 2.50 Semitones, and 'half-step' = 2^(5/48) = 1.25 Semitones; in other words, each 'half-step' is stretched, so that it is 1/8-tone = 2^(1/48) larger than the usual 12-tET 'half-step' or semitone.
The 12-tET 'minor 10th' = 2^(5/4) is used in Ives's 'larger scale' as the cyclic equivalent of an 'octave', so that each 'step' in his scale may be represented more clearly as (2^(5/4))^(x/12), where x = the equivalent 'step' in 12-tET.
Here's an interval matrix (in Semitones) which displays Ives's '3 intervals of even-ratio (so called)':
Doh 15.00 12.50 10.00 8.75 6.25 3.75 1.25 0.00
Te 13.75 11.25 8.75 7.50 5.00 2.50 0.00 13.75
Lah 11.25 8.75 6.25 5.00 2.50 0.00 12.50 11.25
Soh 8.75 6.25 3.75 2.50 0.00 12.50 10.00 8.75
Fah 6.25 3.75 1.25 0.00 12.50 10.00 7.50 6.25
Me 5.00 2.50 0.00 13.75 11.25 8.75 6.25 5.00
Ray 2.50 0.00 12.50 11.25 8.75 6.25 3.75 2.50
Doh Ray Me Fah Soh Lah Te Doh
What he meant by that was that there are only 3 intervals in this 48-tET-subset 'larger scale' which have an exact equivalent in the regular 12-tET scale. This is made plain by the matrix: the only intervals possible between any two pitches in the 'larger scale' which are exactly the same size as 12-tET intervals are those of 0.00 (unison in both scales), 5.00 (old 4th = larger 3rd), 10.00 (old minor 7th = larger 6th), and 15.00 (old minor 10th = larger 8ve) Semitones.
Listening to the MIDI file of this scale makes it clear to me that it sounds nothing like any 'usual' scale, but that it has an obvious tetrachordal structure which gives it musical qualities that are easily recognizable to someone familiar with 12-tET diatonic music, or diatonic music in general in any tuning which represents it well.
Given all the talk in this forum [the Tuning List] about how Ives considered sharps to be higher in pitch than flats, I find it very interesting that his notation here of 'notes in the old scale' uses sharps for all of the chromatic notes except for the flat which marks the highest note, the 'octave' of the 'larger scale'. I don't know if there's any significance to that, but it seems noteworthy. Also note that in the musical example, the augmented 9th [= stretched 'octave'] *is* notated as 'D#'.
I also thought it would be interesting to explore what that other scale illustrated in the musical examples might be like.
This scale apparently repeats at the 'major 10th', or 2^(32/24) = 2^(4/3) = 16.00 Semitones, which is the equivalent of the 'octave'. Each stretched 'semitone' in this larger scale would thus be (2^(4/3))^(x/12).
Upon analyzing its ratios, we find that it can be described as powers of 2, with exponents reduced, as a subset of 9-tET:
Semitones
9-tET Ives larger scale decimal fractional
Doh 2^(12/9) 2^(4/3) 16.00 16
Te 2^(11/9) (2^(4/3))^(11/12) ~14.67 14.&2/3
Lah 2^( 9/9) (2^(4/3))^( 9/12) 12.00 12
Soh 2^( 7/9) (2^(4/3))^( 7/12) ~ 9.33 9.&1/3
Fah 2^( 5/9) (2^(4/3))^( 5/12) ~ 6.67 6.&2/3
Me 2^( 4/9) (2^(4/3))^( 4/12) ~ 5.33 5.&1/3
Ray 2^( 2/9) (2^(4/3))^( 2/12) ~ 2.67 2.&2/3
Doh 2^( 0/9) (2^(4/3))^( 0/12) 0.00
(listen to a MIDI-file of this scale)
This scale, too, would have probably bothered Ives because of the equivalence of its stretched 'octave' with the 'major 10'. Thus he would have been more interested in that tantalizingly lost other scale which 'ended on a quarter-tone'.
I have a hunch as to what the tuning of that 'other scale' might have been, based on my premise that the above scale does represent the scale illustrated on the right in the musical example.
Most likely, the quarter-tone upon which it ended would have been the one which falls right between the 'minor 10th' and 'major 10th' upon which the two scales in the musical example are based. This would be the 'neutral 10th', and it would make sense that Ives would choose this particular interval as his 'octave' after experimenting with both of the other two and being dissatisfied at their match with 12-tET.
So this stretched 'octave' would be 2^(31/24) = 15.50 Semitones.
As 31 is prime, there is no simpler way to describe this tuning as a power of 2, than as a subset of 288-tET:
Semitones
288-tET Ives larger scale decimal fractional
Doh 2^(372/288) 2^(31/24) 15.50 15.&1/2
Te 2^(341/288) (2^(31/24))^(11/12) ~14.21 14.&5/24
Lah 2^(279/288) (2^(31/24))^( 9/12) ~11.63 11.&5/8
Soh 2^(217/288) (2^(31/24))^( 7/12) ~ 9.04 9.&1/24
Fah 2^(155/288) (2^(31/24))^( 5/12) ~ 6.46 6.&11/24
Me 2^(124/288) (2^(31/24))^( 4/12) ~ 5.17 5.&1/6
Ray 2^( 62/288) (2^(31/24))^( 2/12) ~ 2.58 2.&7/12
Doh 2^( 0/288) (2^(31/24))^( 0/12) 0.00 0
(listen to a MIDI-file of this scale)
Alternatively, considering the oft-quoted (here) statement that Ives considered sharps to be higher in pitch than flats, perhaps that 'Eb' instead of 'D#' at the top of the diagram indicates that he 'heard' (in his mind) the stretched 'octave' as being slightly lower than the 'minor 10th'. Then perhaps the 'other scale' was based on a stretched 'octave' of 2^(29/24) = 14.50 Semitones.
As in the above scale, 29 is also prime, so the smallest fractional power of 2 to describe this scale is again 288-tET:
Semitones
288-tET Ives larger scale decimal fractional
Doh 2^(348/288) 2^(29/24) 14.50 14.&1/2
Te 2^(319/288) (2^(29/24))^(11/12) ~13.29 13.&7/24
Lah 2^(261/288) (2^(29/24))^( 9/12) ~10.88 10.&7/8
Soh 2^(203/288) (2^(29/24))^( 7/12) ~ 8.46 8.&11/24
Fah 2^(145/288) (2^(29/24))^( 5/12) ~ 6.04 6.&1/24
Me 2^(116/288) (2^(29/24))^( 4/12) ~ 4.83 4.&5/6
Ray 2^( 58/288) (2^(29/24))^( 2/12) ~ 2.42 2.&5/12
Doh 2^( 0/288) (2^(29/24))^( 0/12) 0.00 0
That little discrepancy between the Eb and the D# in the diagram seems to me to indicate that this is probably the 'other' larger scale Ives had in mind.
(listen to a MIDI-file of this scale)
Because Ives uses 'D#' for the first stretched-'octave' in this scale in the musical example, I submit that perhaps the musical example is an earlier conception of the scale, when he had tried both the 48-tET and 9-tET versions, and that the diagram is a later version, after he thought of the idea of using a quarter-tone interval as the cyclic repetition. This would be the opposite of what is implied by the editor's placement of these two fragments in the published version.
Altho my interpretations of these latter three scales may have been the tuning Ives had in his mind, it is more likely that he constructed them from a practical viewpoint out of various combinations of 1/4- and 1/8-tones. Even if Johnny Reinhard is correct that Ives was implying Pythagorean tuning with some of his prose comments, I have never seen *specific* reference from Ives to any tuning outside of 12-tET except for these two scales (24-tET and 48-tET).
Of course, it's important to keep in mind that this fragment only once specifies 48-tET, and there's no way to be sure how exact Ives's tuning of that was on his father's 'glasses'. He never really was more specific, at least not in these fragments, than to say that they were 'tuned in different intervals larger and less than quarter tones'.
Ives certainly did not manipulate the mathematics of these tunings as carefully as I have, but there's no reason to dismiss the possibility that he could have drawn geometrical diagrams of his stretched-'octaves' in various subdivisions and tuned the 'glasses' accordingly. Thus, it is certainly possibly that he tuned them to the 1/3-tones I analyzed as being in the second scale in the musical example, and tho less probable, still possible that he tuned them something like the other scales.
Here's yet another idea for the tuning of Ives's stretched-'octave' 'larger scale'. Let's assume Johnny Reinhard is correct that Pythagorean was Ives's 'own personal tuning', i.e., that the Pythagorean pitches and intervals were more-or-less generally the ones Ives thought of when he composed his music.
It will help, in visualizing the mathematical procedure carried out here, to once again use Ives's diagram. We add the Pythagorean ratios of the notated pitches on the left, and for the 'quarter-' and '1/8-tones', divide either the 'limma' [= (2^8)/(3^5) = 256/243 = ~0.90 Semitone] or the 'apotome' [= (3^7)/(2^11) = 2187/2048 = ~1.14 Semitones] semitone, whichever is appropriate at that particular spot in the gamut:
Notes in Divisions of Pythagorean intervals Semitones
old scale
(2^6)/(3^3) Eb 31 -+---- 8 Doh (2^6)/(3^3) ~14.94
/ |
limma 30 -+
\ |
(3^2)/(2^2) D 29 -+
/ |---- 7 Te (3^2)/(2^2)/(((2^8)/(3^5))^(1/4)) ~13.81
limma 28 -+
\ |
(3^7)/(2^10) C# 27 -+
/ |
apotome 26 -+
\ |
n^0 8 C 25 -+
/ |
limma 24 -+
\ |---- 6 Lah ((3^5)/(2^7))*(((2^8)/(3^5))^(1/4)) ~11.32
(3^5)/(2^7) B 23 -+
/ |
limma 22 -+
\ |
(3^10)/(2^15) A# 21 -+
/ |
apotome 20 -+
\ |
(3^3)/(2^4) 6 A 19 -+
/ |---- 5 Soh ((3^3)/(2^4))/(((2^8)/(3^5))^(1/4)) ~ 8.83
limma 18 -+
\ |
(3^8)/(2^12) G# 17 -+
/ |
apotome 16 -+
\ |
(3^1)/(2^1) 5 G 15 -+
/ |
limma 14 -+
\ |---- 4 Fah ((3^6)/(2^9))*(((2^8)/(3^5))^(1/4)) ~ 6.34
(3^6)/(2^9) F# 13 -+
/ |
apotome 12 -+
\ |
(2^2)/(3^1) 4 F 11 -+---- 3 Me (2^2)/(3^1) ~ 4.98
/ |
limma 10 -+
\ |
(3^4)/(2^6) 3 E 9 -+
/ |-- Re
limma 8 -+
\ |
(3^9)/(2^14) D# 7 -+
/ |
apotome 6 -+---- 2 Ray ((3^2)/(2^3))*(((3^7)/(2^11))^(1/2)) ~ 2.61
\ |
(3^2)/(2^3) 2 D 5 -+
/ |
limma 4 -+
\ |-- De
(3^7)/(2^11) C# 3 -+
/ |
apotome 2 -+
\ |
n^0 1 C 1 -+---- 1 Doh n^0 0.00
(listen to a MIDI-file of this scale)
That should give performers of Ives considering alternative tunings plenty to try out. Perhaps a good research project would be to search thru his music looking for melodic patterns that come close to fitting these intervallic structures. He may have had these tunings in mind for some of that music, and just wrote it down in the usual (Pythagorean/meantone-based) notation because of lack of instruments so tuned.
updated:
1999.12.6-7
2002.09.13 - reformatted visually