Isoharmonic chord
IMPORTED REVISION FROM WIKISPACES
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Original Wikitext content:
=isoharmonic chords= In [[JustIntonation|just intonation]], Isoharmonic chords are build by successive jumps up the [[OverToneSeries|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all equal-hertz chords are isoharmonic chords, since the ratios between the notes need not be integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency. ===class i=== The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this: || harmonic || 1 || || 2 || || 3 || || 4 || || 5 || || 6 || || 7 || || 8 || || 9 || || 10 || || 11 || || 12 || || 13 || || 14 || || 15 || || 16 || || cents diff || || 1200 || || 702 || || 498 || || 386 || || 316 || || 267 || || 231 || || 204 || || 182 || || 165 || || 151 || || 139 || || 128 || || 119 || || 112 || || Some "scales" built this way: [[otones12-24]], [[otones20-40]]... ===class ii=== The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics): || harmonic || 1 || || 3 || || 5 || || 7 || || 9 || || 11 || || 13 || || 15 || || 17 || || 19 || || 21 || || 23 || || 25 || || 27 || || 29 || || 31 || || cents diff || || 1904 || || 884 || || 583 || || 435 || || 347 || || 289 || || 248 || || 217 || || 193 || || 173 || || 157 || || 144 || || 133 || || 124 || || 115 || || ===class iii=== Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii: || harmonic || 1 || || 4 || || 7 || || 10 || || 13 || || 16 || || 19 || || 22 || || 25 || || 28 || || 31 || || 34 || || 37 || || 40 || || 43 || || 46 || || cents diff || || 2400 || || 969 || || 617 || || 454 || || 359 || || 298 || || 254 || || 221 || || 196 || || 176 || || 160 || || 146 || || 135 || || 125 || || 117 || || || harmonic || 2 || || 5 || || 8 || || 11 || || 14 || || 17 || || 20 || || 23 || || 26 || || 29 || || 32 || || 35 || || 38 || || 41 || || 44 || || 47 || || cents diff || || 1586 || || 814 || || 551 || || 418 || || 336 || || 281 || || 242 || || 212 || || 189 || || 170 || || 155 || || 142 || || 132 || || 122 || || 114 || || ===class iv=== || harmonic || 1 || || 5 || || 9 || || 13 || || 17 || || 21 || || 25 || || 29 || || 33 || || 37 || || 41 || || 45 || || 49 || || 53 || || 57 || || 61 || || cents diff || || 2786 || || 1018 || || 637 || || 464 || || 366 || || 302 || || 257 || || 224 || || 198 || || 178 || || 161 || || 147 || || 136 || || 126 || || 117 || || || harmonic || 3 || || 7 || || 11 || || 15 || || 19 || || 23 || || 27 || || 31 || || 35 || || 39 || || 43 || || 47 || || 51 || || 55 || || 59 || || 63 || || cents diff || || 1467 || || 782 || || 537 || || 409 || || 331 || || 278 || || 239 || || 210 || || 187 || || 169 || || 154 || || 141 || || 131 || || 122 || || 114 || || ===class v=== || harmonic || 1 || || 6 || || 11 || || 16 || || 21 || || 26 || || 31 || || 36 || || 41 || || 46 || || 51 || || 56 || || 61 || || 66 || || 71 || || 76 || || cents diff || || 3102 || || 1049 || || 649 || || 471 || || 370 || || 306 || || 259 || || 225 || || 199 || || 179 || || 162 || || 148 || || 136 || || 126 || || 118 || || || harmonic || 2 || || 7 || || 12 || || 17 || || 22 || || 27 || || 32 || || 37 || || 42 || || 47 || || 52 || || 57 || || 62 || || 67 || || 72 || || 77 || || cents diff || || 2169 || || 933 || || 603 || || 446 || || 355 || || 294 || || 251 || || 219 || || 195 || || 175 || || 159 || || 146 || || 134 || || 125 || || 116 || || || harmonic || 3 || || 8 || || 13 || || 18 || || 23 || || 28 || || 33 || || 38 || || 43 || || 48 || || 53 || || 58 || || 63 || || 68 || || 73 || || 78 || || cents diff || || 1698 || || 841 || || 563 || || 424 || || 341 || || 284 || || 244 || || 214 || || 190 || || 172 || || 156 || || 143 || || 132 || || 123 || || 115 || || || harmonic || 4 || || 9 || || 14 || || 19 || || 24 || || 29 || || 34 || || 39 || || 44 || || 49 || || 54 || || 59 || || 64 || || 69 || || 74 || || 79 || || cents diff || || 1404 || || 765 || || 529 || || 404 || || 328 || || 275 || || 238 || || 209 || || 186 || || 168 || || 153 || || 141 || || 130 || || 121 || || 113 || ||
Original HTML content:
<html><head><title>isoharmonic chords</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="isoharmonic chords"></a><!-- ws:end:WikiTextHeadingRule:0 -->isoharmonic chords</h1>
<br />
In <a class="wiki_link" href="/JustIntonation">just intonation</a>, Isoharmonic chords are build by successive jumps up the <a class="wiki_link" href="/OverToneSeries">harmonic series</a> by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all equal-hertz chords are isoharmonic chords, since the ratios between the notes need not be integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="isoharmonic chords--class i"></a><!-- ws:end:WikiTextHeadingRule:2 -->class i</h3>
The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:<br />
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>1<br />
</td>
<td><br />
</td>
<td>2<br />
</td>
<td><br />
</td>
<td>3<br />
</td>
<td><br />
</td>
<td>4<br />
</td>
<td><br />
</td>
<td>5<br />
</td>
<td><br />
</td>
<td>6<br />
</td>
<td><br />
</td>
<td>7<br />
</td>
<td><br />
</td>
<td>8<br />
</td>
<td><br />
</td>
<td>9<br />
</td>
<td><br />
</td>
<td>10<br />
</td>
<td><br />
</td>
<td>11<br />
</td>
<td><br />
</td>
<td>12<br />
</td>
<td><br />
</td>
<td>13<br />
</td>
<td><br />
</td>
<td>14<br />
</td>
<td><br />
</td>
<td>15<br />
</td>
<td><br />
</td>
<td>16<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>1200<br />
</td>
<td><br />
</td>
<td>702<br />
</td>
<td><br />
</td>
<td>498<br />
</td>
<td><br />
</td>
<td>386<br />
</td>
<td><br />
</td>
<td>316<br />
</td>
<td><br />
</td>
<td>267<br />
</td>
<td><br />
</td>
<td>231<br />
</td>
<td><br />
</td>
<td>204<br />
</td>
<td><br />
</td>
<td>182<br />
</td>
<td><br />
</td>
<td>165<br />
</td>
<td><br />
</td>
<td>151<br />
</td>
<td><br />
</td>
<td>139<br />
</td>
<td><br />
</td>
<td>128<br />
</td>
<td><br />
</td>
<td>119<br />
</td>
<td><br />
</td>
<td>112<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
Some "scales" built this way: <a class="wiki_link" href="/otones12-24">otones12-24</a>, <a class="wiki_link" href="/otones20-40">otones20-40</a>...<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="isoharmonic chords--class ii"></a><!-- ws:end:WikiTextHeadingRule:4 -->class ii</h3>
The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the <a class="wiki_link" href="/BP">Bohlen-Pierce</a> tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):<br />
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>1<br />
</td>
<td><br />
</td>
<td>3<br />
</td>
<td><br />
</td>
<td>5<br />
</td>
<td><br />
</td>
<td>7<br />
</td>
<td><br />
</td>
<td>9<br />
</td>
<td><br />
</td>
<td>11<br />
</td>
<td><br />
</td>
<td>13<br />
</td>
<td><br />
</td>
<td>15<br />
</td>
<td><br />
</td>
<td>17<br />
</td>
<td><br />
</td>
<td>19<br />
</td>
<td><br />
</td>
<td>21<br />
</td>
<td><br />
</td>
<td>23<br />
</td>
<td><br />
</td>
<td>25<br />
</td>
<td><br />
</td>
<td>27<br />
</td>
<td><br />
</td>
<td>29<br />
</td>
<td><br />
</td>
<td>31<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>1904<br />
</td>
<td><br />
</td>
<td>884<br />
</td>
<td><br />
</td>
<td>583<br />
</td>
<td><br />
</td>
<td>435<br />
</td>
<td><br />
</td>
<td>347<br />
</td>
<td><br />
</td>
<td>289<br />
</td>
<td><br />
</td>
<td>248<br />
</td>
<td><br />
</td>
<td>217<br />
</td>
<td><br />
</td>
<td>193<br />
</td>
<td><br />
</td>
<td>173<br />
</td>
<td><br />
</td>
<td>157<br />
</td>
<td><br />
</td>
<td>144<br />
</td>
<td><br />
</td>
<td>133<br />
</td>
<td><br />
</td>
<td>124<br />
</td>
<td><br />
</td>
<td>115<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="isoharmonic chords--class iii"></a><!-- ws:end:WikiTextHeadingRule:6 -->class iii</h3>
Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:<br />
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>1<br />
</td>
<td><br />
</td>
<td>4<br />
</td>
<td><br />
</td>
<td>7<br />
</td>
<td><br />
</td>
<td>10<br />
</td>
<td><br />
</td>
<td>13<br />
</td>
<td><br />
</td>
<td>16<br />
</td>
<td><br />
</td>
<td>19<br />
</td>
<td><br />
</td>
<td>22<br />
</td>
<td><br />
</td>
<td>25<br />
</td>
<td><br />
</td>
<td>28<br />
</td>
<td><br />
</td>
<td>31<br />
</td>
<td><br />
</td>
<td>34<br />
</td>
<td><br />
</td>
<td>37<br />
</td>
<td><br />
</td>
<td>40<br />
</td>
<td><br />
</td>
<td>43<br />
</td>
<td><br />
</td>
<td>46<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>2400<br />
</td>
<td><br />
</td>
<td>969<br />
</td>
<td><br />
</td>
<td>617<br />
</td>
<td><br />
</td>
<td>454<br />
</td>
<td><br />
</td>
<td>359<br />
</td>
<td><br />
</td>
<td>298<br />
</td>
<td><br />
</td>
<td>254<br />
</td>
<td><br />
</td>
<td>221<br />
</td>
<td><br />
</td>
<td>196<br />
</td>
<td><br />
</td>
<td>176<br />
</td>
<td><br />
</td>
<td>160<br />
</td>
<td><br />
</td>
<td>146<br />
</td>
<td><br />
</td>
<td>135<br />
</td>
<td><br />
</td>
<td>125<br />
</td>
<td><br />
</td>
<td>117<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>2<br />
</td>
<td><br />
</td>
<td>5<br />
</td>
<td><br />
</td>
<td>8<br />
</td>
<td><br />
</td>
<td>11<br />
</td>
<td><br />
</td>
<td>14<br />
</td>
<td><br />
</td>
<td>17<br />
</td>
<td><br />
</td>
<td>20<br />
</td>
<td><br />
</td>
<td>23<br />
</td>
<td><br />
</td>
<td>26<br />
</td>
<td><br />
</td>
<td>29<br />
</td>
<td><br />
</td>
<td>32<br />
</td>
<td><br />
</td>
<td>35<br />
</td>
<td><br />
</td>
<td>38<br />
</td>
<td><br />
</td>
<td>41<br />
</td>
<td><br />
</td>
<td>44<br />
</td>
<td><br />
</td>
<td>47<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>1586<br />
</td>
<td><br />
</td>
<td>814<br />
</td>
<td><br />
</td>
<td>551<br />
</td>
<td><br />
</td>
<td>418<br />
</td>
<td><br />
</td>
<td>336<br />
</td>
<td><br />
</td>
<td>281<br />
</td>
<td><br />
</td>
<td>242<br />
</td>
<td><br />
</td>
<td>212<br />
</td>
<td><br />
</td>
<td>189<br />
</td>
<td><br />
</td>
<td>170<br />
</td>
<td><br />
</td>
<td>155<br />
</td>
<td><br />
</td>
<td>142<br />
</td>
<td><br />
</td>
<td>132<br />
</td>
<td><br />
</td>
<td>122<br />
</td>
<td><br />
</td>
<td>114<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="isoharmonic chords--class iv"></a><!-- ws:end:WikiTextHeadingRule:8 -->class iv</h3>
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>1<br />
</td>
<td><br />
</td>
<td>5<br />
</td>
<td><br />
</td>
<td>9<br />
</td>
<td><br />
</td>
<td>13<br />
</td>
<td><br />
</td>
<td>17<br />
</td>
<td><br />
</td>
<td>21<br />
</td>
<td><br />
</td>
<td>25<br />
</td>
<td><br />
</td>
<td>29<br />
</td>
<td><br />
</td>
<td>33<br />
</td>
<td><br />
</td>
<td>37<br />
</td>
<td><br />
</td>
<td>41<br />
</td>
<td><br />
</td>
<td>45<br />
</td>
<td><br />
</td>
<td>49<br />
</td>
<td><br />
</td>
<td>53<br />
</td>
<td><br />
</td>
<td>57<br />
</td>
<td><br />
</td>
<td>61<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>2786<br />
</td>
<td><br />
</td>
<td>1018<br />
</td>
<td><br />
</td>
<td>637<br />
</td>
<td><br />
</td>
<td>464<br />
</td>
<td><br />
</td>
<td>366<br />
</td>
<td><br />
</td>
<td>302<br />
</td>
<td><br />
</td>
<td>257<br />
</td>
<td><br />
</td>
<td>224<br />
</td>
<td><br />
</td>
<td>198<br />
</td>
<td><br />
</td>
<td>178<br />
</td>
<td><br />
</td>
<td>161<br />
</td>
<td><br />
</td>
<td>147<br />
</td>
<td><br />
</td>
<td>136<br />
</td>
<td><br />
</td>
<td>126<br />
</td>
<td><br />
</td>
<td>117<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>3<br />
</td>
<td><br />
</td>
<td>7<br />
</td>
<td><br />
</td>
<td>11<br />
</td>
<td><br />
</td>
<td>15<br />
</td>
<td><br />
</td>
<td>19<br />
</td>
<td><br />
</td>
<td>23<br />
</td>
<td><br />
</td>
<td>27<br />
</td>
<td><br />
</td>
<td>31<br />
</td>
<td><br />
</td>
<td>35<br />
</td>
<td><br />
</td>
<td>39<br />
</td>
<td><br />
</td>
<td>43<br />
</td>
<td><br />
</td>
<td>47<br />
</td>
<td><br />
</td>
<td>51<br />
</td>
<td><br />
</td>
<td>55<br />
</td>
<td><br />
</td>
<td>59<br />
</td>
<td><br />
</td>
<td>63<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>1467<br />
</td>
<td><br />
</td>
<td>782<br />
</td>
<td><br />
</td>
<td>537<br />
</td>
<td><br />
</td>
<td>409<br />
</td>
<td><br />
</td>
<td>331<br />
</td>
<td><br />
</td>
<td>278<br />
</td>
<td><br />
</td>
<td>239<br />
</td>
<td><br />
</td>
<td>210<br />
</td>
<td><br />
</td>
<td>187<br />
</td>
<td><br />
</td>
<td>169<br />
</td>
<td><br />
</td>
<td>154<br />
</td>
<td><br />
</td>
<td>141<br />
</td>
<td><br />
</td>
<td>131<br />
</td>
<td><br />
</td>
<td>122<br />
</td>
<td><br />
</td>
<td>114<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="isoharmonic chords--class v"></a><!-- ws:end:WikiTextHeadingRule:10 -->class v</h3>
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>1<br />
</td>
<td><br />
</td>
<td>6<br />
</td>
<td><br />
</td>
<td>11<br />
</td>
<td><br />
</td>
<td>16<br />
</td>
<td><br />
</td>
<td>21<br />
</td>
<td><br />
</td>
<td>26<br />
</td>
<td><br />
</td>
<td>31<br />
</td>
<td><br />
</td>
<td>36<br />
</td>
<td><br />
</td>
<td>41<br />
</td>
<td><br />
</td>
<td>46<br />
</td>
<td><br />
</td>
<td>51<br />
</td>
<td><br />
</td>
<td>56<br />
</td>
<td><br />
</td>
<td>61<br />
</td>
<td><br />
</td>
<td>66<br />
</td>
<td><br />
</td>
<td>71<br />
</td>
<td><br />
</td>
<td>76<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>3102<br />
</td>
<td><br />
</td>
<td>1049<br />
</td>
<td><br />
</td>
<td>649<br />
</td>
<td><br />
</td>
<td>471<br />
</td>
<td><br />
</td>
<td>370<br />
</td>
<td><br />
</td>
<td>306<br />
</td>
<td><br />
</td>
<td>259<br />
</td>
<td><br />
</td>
<td>225<br />
</td>
<td><br />
</td>
<td>199<br />
</td>
<td><br />
</td>
<td>179<br />
</td>
<td><br />
</td>
<td>162<br />
</td>
<td><br />
</td>
<td>148<br />
</td>
<td><br />
</td>
<td>136<br />
</td>
<td><br />
</td>
<td>126<br />
</td>
<td><br />
</td>
<td>118<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>2<br />
</td>
<td><br />
</td>
<td>7<br />
</td>
<td><br />
</td>
<td>12<br />
</td>
<td><br />
</td>
<td>17<br />
</td>
<td><br />
</td>
<td>22<br />
</td>
<td><br />
</td>
<td>27<br />
</td>
<td><br />
</td>
<td>32<br />
</td>
<td><br />
</td>
<td>37<br />
</td>
<td><br />
</td>
<td>42<br />
</td>
<td><br />
</td>
<td>47<br />
</td>
<td><br />
</td>
<td>52<br />
</td>
<td><br />
</td>
<td>57<br />
</td>
<td><br />
</td>
<td>62<br />
</td>
<td><br />
</td>
<td>67<br />
</td>
<td><br />
</td>
<td>72<br />
</td>
<td><br />
</td>
<td>77<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>2169<br />
</td>
<td><br />
</td>
<td>933<br />
</td>
<td><br />
</td>
<td>603<br />
</td>
<td><br />
</td>
<td>446<br />
</td>
<td><br />
</td>
<td>355<br />
</td>
<td><br />
</td>
<td>294<br />
</td>
<td><br />
</td>
<td>251<br />
</td>
<td><br />
</td>
<td>219<br />
</td>
<td><br />
</td>
<td>195<br />
</td>
<td><br />
</td>
<td>175<br />
</td>
<td><br />
</td>
<td>159<br />
</td>
<td><br />
</td>
<td>146<br />
</td>
<td><br />
</td>
<td>134<br />
</td>
<td><br />
</td>
<td>125<br />
</td>
<td><br />
</td>
<td>116<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>3<br />
</td>
<td><br />
</td>
<td>8<br />
</td>
<td><br />
</td>
<td>13<br />
</td>
<td><br />
</td>
<td>18<br />
</td>
<td><br />
</td>
<td>23<br />
</td>
<td><br />
</td>
<td>28<br />
</td>
<td><br />
</td>
<td>33<br />
</td>
<td><br />
</td>
<td>38<br />
</td>
<td><br />
</td>
<td>43<br />
</td>
<td><br />
</td>
<td>48<br />
</td>
<td><br />
</td>
<td>53<br />
</td>
<td><br />
</td>
<td>58<br />
</td>
<td><br />
</td>
<td>63<br />
</td>
<td><br />
</td>
<td>68<br />
</td>
<td><br />
</td>
<td>73<br />
</td>
<td><br />
</td>
<td>78<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>1698<br />
</td>
<td><br />
</td>
<td>841<br />
</td>
<td><br />
</td>
<td>563<br />
</td>
<td><br />
</td>
<td>424<br />
</td>
<td><br />
</td>
<td>341<br />
</td>
<td><br />
</td>
<td>284<br />
</td>
<td><br />
</td>
<td>244<br />
</td>
<td><br />
</td>
<td>214<br />
</td>
<td><br />
</td>
<td>190<br />
</td>
<td><br />
</td>
<td>172<br />
</td>
<td><br />
</td>
<td>156<br />
</td>
<td><br />
</td>
<td>143<br />
</td>
<td><br />
</td>
<td>132<br />
</td>
<td><br />
</td>
<td>123<br />
</td>
<td><br />
</td>
<td>115<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td>harmonic<br />
</td>
<td>4<br />
</td>
<td><br />
</td>
<td>9<br />
</td>
<td><br />
</td>
<td>14<br />
</td>
<td><br />
</td>
<td>19<br />
</td>
<td><br />
</td>
<td>24<br />
</td>
<td><br />
</td>
<td>29<br />
</td>
<td><br />
</td>
<td>34<br />
</td>
<td><br />
</td>
<td>39<br />
</td>
<td><br />
</td>
<td>44<br />
</td>
<td><br />
</td>
<td>49<br />
</td>
<td><br />
</td>
<td>54<br />
</td>
<td><br />
</td>
<td>59<br />
</td>
<td><br />
</td>
<td>64<br />
</td>
<td><br />
</td>
<td>69<br />
</td>
<td><br />
</td>
<td>74<br />
</td>
<td><br />
</td>
<td>79<br />
</td>
</tr>
<tr>
<td>cents diff<br />
</td>
<td><br />
</td>
<td>1404<br />
</td>
<td><br />
</td>
<td>765<br />
</td>
<td><br />
</td>
<td>529<br />
</td>
<td><br />
</td>
<td>404<br />
</td>
<td><br />
</td>
<td>328<br />
</td>
<td><br />
</td>
<td>275<br />
</td>
<td><br />
</td>
<td>238<br />
</td>
<td><br />
</td>
<td>209<br />
</td>
<td><br />
</td>
<td>186<br />
</td>
<td><br />
</td>
<td>168<br />
</td>
<td><br />
</td>
<td>153<br />
</td>
<td><br />
</td>
<td>141<br />
</td>
<td><br />
</td>
<td>130<br />
</td>
<td><br />
</td>
<td>121<br />
</td>
<td><br />
</td>
<td>113<br />
</td>
<td><br />
</td>
</tr>
</table>
</body></html>